diff --git a/-tFLT4oBgHgl3EQfvS-t/content/tmp_files/2301.12159v1.pdf.txt b/-tFLT4oBgHgl3EQfvS-t/content/tmp_files/2301.12159v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..59060c38acbc175aecfeafd2e7dd740846cfbada --- /dev/null +++ b/-tFLT4oBgHgl3EQfvS-t/content/tmp_files/2301.12159v1.pdf.txt @@ -0,0 +1,1306 @@ +arXiv:2301.12159v1 [cs.CV] 28 Jan 2023 +ClusterFuG: Clustering Fully connected Graphs by Multicut +Ahmed Abbas 1 Paul Swoboda 1 2 +Abstract +We propose a graph clustering formulation based +on multicut (a.k.a. weighted correlation cluster- +ing) on the complete graph. +Our formulation +does not need specification of the graph topology +as in the original sparse formulation of multicut, +making our approach simpler and potentially bet- +ter performing. In contrast to unweighted corre- +lation clustering we allow for a more expressive +weighted cost structure. In dense multicut, the +clustering objective is given in a factorized form +as inner products of node feature vectors. This al- +lows for an efficient formulation and inference in +contrast to multicut/weighted correlation cluster- +ing, which has at least quadratic representation +and computation complexity when working on +the complete graph. We show how to rewrite clas- +sical greedy algorithms for multicut in our dense +setting and how to modify them for greater ef- +ficiency and solution quality. In particular, our +algorithms scale to graphs with tens of thousands +of nodes. Empirical evidence on instance seg- +mentation on Cityscapes and clustering of Ima- +geNet datasets shows the merits of our approach. +1. Introduction +Graph-based clustering approaches, primarily among them +multicut (Chopra & Rao, 1993), are theoretically appeal- +ing: They do not need specification of the number of clus- +ters, but infer them as part of the optimization process. +They allow for a flexible clustering objective with attrac- +tive and repulsive costs between pairs of nodes. +They +are also theoretically well-understood as optimization prob- +lems with intensively studied polyhedral descriptions. Effi- +cient solvers that scale well and give high quality solutions +have also been developed. +As a drawback, graph-based clustering approaches need +specification of the underlying graph topology. In prac- +1MPI for Informatics, Saarland Informatics Campus, Germany +2University of Mannheim, Germany. Correspondence to: Ahmed +Abbas . +Preprint. +tice, this means an additional engineering effort as well as +the possibility to not get it right, which would decrease the +downstream task performance. Naively circumventing this +challenge by using the complete graph is not scalable – the +number of edges grows quadratically. One approach to re- +solve this conundrum is graph structure learning, which in- +fers the graph topology as part of the inference process, but +adds considerable additional complexity. +We propose a method to solve graph clustering efficiently +on complete graphs. Our formulation will use the well- +known edge-based multicut formulation and only restrict +the way edge costs can be computed: they need to be based +on inner products of node features. This has two advan- +tages: First, it reduces storage requirements. Instead of +storing a full adjacency matrix of edge costs as in multicut, +which grows quadratically with the number of nodes, we +only need to store a linear number of node features and can +compute edge costs on demand. Second, operations needed +in multicut algorithms can be made scalable. Instead of +operating on the complete graph we can sparsify it adap- +tively during the solving process. This allows to simulate +the workings of multicut algorithms on complete graphs by +working on a small subset of it. The key technical ingre- +dient to obtain these sparse subgraphs will be fast nearest +neighbor search, for which efficient and scalable implemen- +tations exist (Johnson et al., 2019). In effect, this allows us +to solve large dense multicut instances in moderate time, +which is not possible with existing solvers. In detail, our +contribution is as follows: +Formulation: We propose multicut on complete graphs +with factorized edge costs as an efficiently repre- +sentable graph clustering formalism. +Algorithm: We propose scalable algorithms for solving +the dense multicut problems, one mimicking exactly +the original greedy additive edge constraction (GAEC) +algorithm (Keuper et al., 2015), the other a more effi- +cient variant in the spirit of the balanced edge contrac- +tion heuristic (Kardoost & Keuper, 2018)1. +Empirical: We show efficacy in terms of memory and run- +time of our solvers and show the merit of using them +1Our +code +is +available +at +https://github.com/aabbas90/cluster-fug + +Clustering Fully connected Graphs by Multicut +for image segmentation on Cityscapes and clustering +of ImageNet classification dataset. +2. Related work +Multicut and correlation clustering: +The original mul- +ticut problem is formulated as an extension of the min- +cut problem to multiple terminals with non-negative edge +costs (Hu, 1963). +In machine learning the multicut +problem is defined differently and is equivalent (up to +variable involution) to the correlation clustering prob- +lem (Demaine et al., 2006), i.e. arbitrary edges costs and +no terminals. For the purpose of this work we will use the +latter definition of multicut. The polyhedral geometry of +the multicut problem has been studied in (Deza et al., 1992; +Chopra & Rao, 1993; Oosten et al., 2001). +Although the multicut problem is NP-Hard (Bansal et al., +2004; Demaine et al., 2006), greedy algorithms perform +well in practice for computer vision and machine learn- +ing tasks (Keuper et al., 2015; Levinkov et al., 2017; +Bailoni et al., 2022). +More involved algorithms in- +clude message passing in the dual domain for multi- +cut, studied in (Swoboda & Andres, 2017; Lange et al., +2018; Abbas & Swoboda, 2022). +These algorithms give +lower bounds and improved primal solutions. +Another +line of efficient primal heuristics is based on move- +making (Beier et al., 2014; 2015). All these graphs, while +efficient, scale with the number of edges, making them +unsuitable for very large dense graphs. +Algorithms for +correlation clustering on complete graphs were proposed +in (Pan et al., 2015; Veldt, 2022). However, they only al- +low unweighted edges. In this paper we consider efficient +algorithms on full graphs and with weighted edges. +K-Means: +The K-means problem (Lloyd, 1982) is sim- +ilar to our approach in that it works directly on feature +representations and its objective is based on L2-distances +between features. Similarly to our algorithm, large num- +ber of points are handled by efficiently computing kNN- +graphs (Qaddoura et al., 2020), thereby reducing run time. +In contrast to multicut, the number of clusters must be +given a-priori, while in multicut it is derived as part of the +optimization process. +Other clustering approaches: +There are a number of +other paradigms for clustering. A prominent approach is +spectral clustering, in which a weighted graph is given and +a clustering is computed with the help of the eigenvec- +tors of the graph Laplacian (Von Luxburg, 2007; Jia et al., +2014). The work (Dhillon et al., 2007) shows connections +between weighted k-means and multiple spectral clustering +approaches. As for K-means and unlike multicut, spectral +clustering requires the number of clusters to be specified. +i +j +fi fj +(0, 0) +Figure 1: Example illustration of dense multicut prob- +lem (3) on 5 nodes. Each node i is associated with a vec- +tor fi ∈ R2 and all possible edges between distinct nodes +are considered (i.e. the complete graph). The edge cost be- +tween a pair of nodes i, j is measured by ⟨fi, fj⟩ and attrac- +tive/repulsive edges are colored green/red. Edge thickness +represents absolute edge cost. Also shown is the optimal +partitioning to 2 clusters with cut edges denoted by dashed +lines. +3. Method +A decomposition (or clustering) of a weighted graph G = +(V, E, c) with vertices V , edges E and edge costs c ∈ RE +can be obtained by solving the following multicut problem +min +y∈MG +� +ij∈E +cijyij. +(1) +We say that an edge ij with cij > 0 is attractive. Its end- +points prefer to be in the same cluster. In the opposite case +cij < 0 we call the edge repulsive. The set MG enumer- +ates all possible partitions of G defined as +MG = + + +δ(V1, . . . , Vn) : +n ∈ N +Vi ∩ Vj = ∅ +∀i ̸= j +V1 ˙∪ . . . ˙∪Vn = V + + + . +(2) +where δ(·, . . . , ·) ⊆ E is the set of edges straddling distinct +components. +The goal of our work is to consider the scenario when the +graph G is complete i.e. E = {ij : i ∈ V, j ∈ V \ {i}}. +For large graphs storage and processing of edge costs c be- +comes prohibitive. To address this issue we instead require +as input a feature vector fi ∈ Rd for each node i in V . The +edge costs between a pair of nodes i and j can then be mea- +sured on-demand through some function s(fi, fj) → R. In +this case the multicut problem becomes +min +y∈MG +� +i∈V +� +j∈V \i +s(fi, fj)yij, +(3) + +Clustering Fully connected Graphs by Multicut +which we term as dense multicut problem. An illustration +of our formulation is given in Figure 1. In the following +we first revisit an algorithm to approximately solve (1) and +show its extensions for dense multicut problem (3). +3.1. Greedy Additive Edge Contraction +The +greedy +additive +edge +contraction +(GAEC) +scheme +(Keuper et al., +2015) +computes +approximate +solution of the multicut problem (1) as given in Algo- +rithm 1. It initializes each node as a separate cluster and +iteratively contracts a pair of nodes i, j with the largest +non-negative cost cij (if it exists). Let m be the node i and +j are contracted to. The edge costs of edges incident to m +are +cml = cil + cjl, l ∈ Ni ∪ Nj \ {i, j}, +(4) +where costs of non-existing edges are assumed to be 0 +and Ni corresponds to neighbours of i in graph G. For +complete graphs directly applying this algorithm by oper- +ating on edge costs is computationally expensive. More- +over, since each node is connected to all other nodes (Ni = +V \ {i}), cost updates (4) during edge contraction take +O(|V |) instructions. +Algorithm 1: GAEC (Keuper et al., 2015) +Data: Weighted graph G = (V, E, c) +Result: Clusters V +1 while maxuv∈E cuv ≥ 0 do +2 +m := ij = arg maxuv∈E cuv +// Aggregate edge costs +3 +cml = cil + cjl, l ∈ Ni ∪ Nj \ {i, j} +// Update edges +4 +E = (E∪{ml|l ∈ Ni∪Nj})\{il}l∈Ni ∪{jl}l∈Nj +// Update nodes +5 +V = (V ∪ m) \ {i, j} +Contraction on complete graphs: +We show how to per- +form a more efficient (and equivalent) contraction by oper- +ating on the node features f by our formulation (3) for the +particular case of s(·, ·) defined as +s(fi, fj) = ⟨fi, fj⟩. +(5) +From now on, unless stated otherwise, our edge costs will +be given by (5). +Lemma 3.1 (Contraction with node features). Assume +edge costs are measured by (5) and nodes i and j are con- +tracted to m. Then features of node m given by +fm = fi + fj +(6) +produce contracted edge costs according to (4). +Proof. By applying (5) for l ∈ V and comparing with (4) +we get +s(fm, fl) = ⟨fm, fl⟩ = ⟨fi, fl⟩ + ⟨fj, fl⟩ += s(fi, fl) + s(fj, fl) . +Next we will build on the previous result to devise heuris- +tics for solving dense multicut problem (3) efficiently. +GAEC for complete graphs: +We devise an algorithm +which exactly imitates GAEC (Keuper et al., 2015) but +is applicable to our formulation on complete graphs (3). +Specifically to make GAEC efficient with node features +and a complete graph, we sparsify the original graph G by +working on its directed k-nearest neighbours (NN) graph +(V, A). The NN graph stores candidate edges for contrac- +tion. +The arc set A is populated by nearest neighbour +search w.r.t. feature similarity (5) and is updated on each +edge contraction. We denote by N + +i +the set of outgoing +neighbours of i in the NN graph i.e. {l|(l, i) ∈ A} and simi- +larly by N − +i the incoming neighbours. Moreover we define +N + +ij as N + +i ∪ N + +j . The complete strategy to obtain a fea- +sible solution of dense multicut problem is described in Al- +gorithm 2. It imitates Algorithm 1 by iteratively searching +and contracting the most attractive edge, but it restricts its +search only to the NN graph thereby reducing computation. +After contraction, the NN graph is updated (lines 5-8) by +only recomputing nearest neighbors of nodes which were +affected by the contraction in the NN graph. +Algorithm 2: Dense GAEC +Data: Node features fi, ∀i ∈ V ; Number of nearest +neighbours k +Result: Clusters V +// Find nearest neighbours of each node +1 A = {(i, j)|i ∈ V, j ∈ arg top-ki′̸=i⟨fi, fi′⟩} +2 while max(u,v)∈A⟨fu, fv⟩ ≥ 0 do +3 +m := (i, j) = arg top-k(u,v)∈A⟨fu, fv⟩ +// Aggregate node features +4 +fm = fi + fj +// Update nodes +5 +V = (V ∪ m) \ {i, j} +// Nodes with i, j as NN +6 +H = {(q, arg maxl∈V \q⟨fm, fl⟩)|q ∈ N − +ij } +// NN of merged node +7 +H = H ∪ {(m, r)|r = arg top-kl∈V \m⟨fm, fl⟩} +// Update arcs +8 +A = (A ∪ H) \ {(q, i)}q∈N − +i ∪ {(q, j)}q∈N − +j +Proposition 3.2 (Dense Greedy Contraction). Algorithm 2 +always merges a pair of nodes i and j with the largest edge + +Clustering Fully connected Graphs by Multicut +cost i.e. +(i, j) ∈ arg max +(u,v)∈A +⟨fu, fv⟩ =⇒ ⟨fi, fj⟩ ≥ max +u,v̸=u⟨fu, fv⟩. +(7) +Proof. The statement is trivially satisfied before any merge +operation is performed since A is constructed by nearest +neighbour search over all nodes in line 1 of the algorithm. +We now show that after each merge operation (i.e. after +line 8 of the algorithm) the statement (7) still holds. We +define Q = m ∪ {q|q ∈ N − +ij } to be the set of nodes using i +or j as their nearest neighbours. Two cases can arise: +Case 1: {i, j}∩Q ̸= ∅: +Due to nearest neighbour search +for all nodes in Q at lines 6 and 7, the statement holds. +Case 2: {i, j} ∩ Q = ∅: +In this case if i is the con- +tracted node m from the last edge contraction operation +then (i, j) ∈ A due to line 6. If i ̸= m then it remains +connected to its nearest neighbours either due to the initial +NN search at line 1 or the NN update at lines 6 and 7. +The above result guarantees that the most attractive edge +will always be present in the nearest neighbour graph +thus foregoing the need to search in the complete graph. +This proves that the Algorithm 2 performs locally optimal +merges as proposed in (Keuper et al., 2015) and is also scal- +able to large complete graphs. As a downside the algorithm +requires costly nearest neighbour search after every edge +contraction. Since computing nearest neighbours and con- +tracting edges is not commutative, in the worst case one +has to recompute the nearest neighbours on the contracted +graph from scratch. +Incremental nearest neighbours: +For faster nearest +neighbour updates after edge contraction we show how +to reuse more of the previously computed nearest neigh- +bors through the following two approaches. First, for all +nodes whose nearest neighbours are merging nodes (i.e. +line 6 of Alg. 2), we check if merged node m is already +a nearest neighbour without requiring exhaustive search. +Specifically assume a contracting node i was a k-nearest +neighbour of some other node q ∈ V \ i. +Then the +merged node m is a k-nearest neighbour of q if ⟨fq, fm⟩ ≥ +minl∈N + +q ⟨fq, fl⟩. This check can be cheaply performed for +all such nodes thereby reducing computation. Second, we +devise a criterion which can allow to efficiently populate +nearest neighbours of the contracted node m. +Proposition 3.3 (Incremental nearest neighbours). Let the +k-nearest neighbours N + +i , N + +j +of nodes i and j be given. +Assume that nodes i, j are merged to form a new node m. +Then edge costs between nodes v ∈ V \ N + +ij and m are +i +j +N + +i +N + +j +N − +ij +Figure 2: Illustration of nearest neighbour graph and an +edge ij being contracted. The set N + +ij = N + +i +∪ N + +j +is +searched first to find nearest neighbours of the merged node +efficiently (Prop. 3.3). The nodes in set N − +ij need to update +their nearest neighbours since their current nearest neigh- +bour nodes i and j are getting contracted. Only the arcs +to/from i and j are shown. +bounded from above by +bij := min +p∈N + +i +⟨fi, fp⟩ + min +q∈N + +j +⟨fj, fq⟩ +Proof. Since neighbours of i are computed by nearest +neighbours search we have for all nodes p′ /∈ N + +i +⟨fi, fp′⟩ ≤ min +p∈N + +i +⟨fi, fp⟩, +and similarly for node j. +Then by definition of v and +Lemma 3.1 we obtain +⟨fm, fv⟩ = ⟨fi, fv⟩ + ⟨fj, fv⟩ +≤ min +p∈N + +i +⟨fi, fp⟩ + min +q∈N + +j +⟨fj, fq⟩ . +The above proposition gives an upper bound of feature sim- +ilarity (i.e. edge cost) of merged node m with all nodes not +in N + +ij . Thus if a node in N + +ij exceeds this upper bound it +is more similar to m than all nodes not in N + +ij . This allows +to possibly skip recomputing the nearest neighbors of m in +Alg. 2 (line 7). +Lemma 3.4. If +|{p ∈ N + +ij : ⟨fm, fp⟩ ≥ bij}| ≥ k +(8) +then +k-nearest +neighbour +of +node +m +given +by +arg top-kv∈V \{i,j,m}⟨fm, fv⟩ +can +be +chosen +as +arg top-kp∈N + +ij ⟨fm, fp⟩. +Proof. Since the elements of N + +ij already satisfy the bound +bij from Prop. 3.3 and there are at least k many such el- +ements, the k-nearest neighbours of node m can be taken +from N + +ij . + +Clustering Fully connected Graphs by Multicut +Both of these approaches for efficiently updating the NN +graph after contraction are used in Alg. 3. This algorithm +can be used instead of lines 6 and 7 in Alg. 2 for improved +performance. See Figure 2 for an illustration on nearest +neighbour graph and edge contraction update. +Algorithm 3: Incremental NN update +Data: Contracting nodes i, j; Contracted node m; NN +graph (V, A); Node features fi, ∀i ∈ V ; Num. +of neighbours k; +Result: Nearest neighbour arcs H to add in A +// NNs of m by Prop. 3.3 +1 H = {(m, l)|l ∈ N + +ij , ⟨fm, fl⟩ ≥ bij} +// Keep at most k NN +2 H = arg top-k(m,l)∈H⟨fm, fl⟩ +3 if H = ∅ then +4 +H = {(m, r)|r = arg top-kl∈V \m⟨fm, fl⟩} +5 for q ∈ N − +ij \ {i, j} do +// Check if m a NN of q +6 +if ⟨fq, fm⟩ ≥ minl∈N + +q ⟨fq, fl⟩ then +7 +H = H ∪ (q, m) +8 +else +9 +H = H ∪ {(q, arg maxl∈V \q⟨fq, fl⟩)} +3.2. Lazy Edge Contraction +We further forego the need for nearest neighbours recom- +putation after edge contraction by lifting the restriction of +performing only greedy moves. This allows to maximally +utilize the NN graph: the algorithm performs contractions, +including non-greedy ones, until no contraction candidates +are present in the NN graph. Specifically we do not per- +form the exhaustive search in lines 4 and 9 and only return +the nearest neighbours which are easily computable. The +NN graph is repopulated as lazily as possible i.e. when no +contraction candidates are left. In addition to being more +efficient this strategy is reminiscent of the balanced edge +contraction approach of (Kardoost & Keuper, 2018). The +authors normalized the edge costs with cluster size of two +end-points. These normalized edge costs were used to find +the edge to contract. This strategy encouraged consecutive +contractions to occur at different regions of the graph. As +our lazy approach does not always make the nearest neigh- +bours of the contracted node available thus contractions can +only be done to nodes other than the contracted node. This +also produces contractions in different regions. +Lastly we also utilize efficient methods for approximate +nearest neighbour search (Malkov & Yashunin, 2018) for +populating the possibly large initial NN graph. For later +nearest neighbour searches we still use exact methods as +the search space is reduced due to previous contractions. +3.3. Varying Affinity Strength +Our basic edge costs computed by ⟨fi, fj⟩ for two features +fi and fj have one fundamental limitation: Clusters will by +default occupy whole quadrants. In other words, whenever +two features have angle lower than 90◦ they are attractive +and will prefer to be in the same cluster, see Figure 3. In +order to let our formulation favor larger or smaller clusters, +we modify our original similarity function s(·, ·) by adding +an additional term indicated by α-variables: +f i = [fi; αi], +(9) +s(f i, fj) = ⟨fi, fj⟩ ± αi · αj , +(10) +where we choose positive sign for favoring larger clusters +and negative for smaller clusters. In our experiments we +will set αi = α with α > 0 a constant. +We note that the contraction mechanism carries over di- +rectly to our extended setting. +Lemma 3.5. Aggregating features of the contracted node +m by fm = f i + f j is equivalent to setting edge costs as +per (4) on complete graph. +Proof. Similar to the proof of Lemma 3.1 as follows +s(f m, fl) = ⟨fm, fl⟩ ± αm · αl += ⟨fi + fj, fl⟩ ± (αi + αj) · αl += ⟨fi, fl⟩ ± αi · αl + ⟨fj, fl⟩ ± αj · αl += s(fi, f l) + s(f j, fl) . +Large clusters: +If we want to allow for larger clusters +(corresponding to choosing + in (10)), we work directly on +the extended feature set fi = [fi; αi] and use it in the NN +graph. +Small clusters: +If we want to allow for smaller clusters +(corresponding to choosing − in (10)), we must modify our +algorithms slightly. In order to construct NN graphs we +will use two sets of features: First, the query nodes will +have their features defined by ˆfi = [fi, −αi] and second, +the pre-existing nodes j ∈ V in the graph will keep the +same features fj from (10). In order to search for nearest +neighbors of node i in the graph V the modified similarity +function (10) can be implemented by an inner product as +s(f i, f j) = ⟨ ˆfi, f j⟩ . +(11) +4. Experiments +We study the benefits of the multicut on complete +graphs (3) and compare possible algorithms on the + +Clustering Fully connected Graphs by Multicut +fd +fe +ff +fg +fh +fa +fb +fc +⟨fc, fb⟩ > 0 +⟨fc, fd⟩ > 0 +Figure 3: Illustration of edge costs between 8 nodes where +feature vectors of each node i is in two-dimensional space +i.e. fi ∈ R2. If we want each node to be a separate cluster +then the edge costs measured by (5) are not suitable. This is +because there will always be atleast two vectors with pos- +itive costs preferring to be in the same cluster. Using an +large enough positive value of α through (10) this issue can +be resolved. +tasks of ImageNet (Deng et al., 2009) clustering and +Cityscapes (Cordts et al., 2016) panoptic segmentation. +The algorithms are +GAEC: The greedy additive edge contraction algorithm +from (Keuper et al., 2015)(Alg. 1) is run on the com- +plete graph where all edge costs are precomputed and +then passed to the algorithm. +RAMA: We also compare with the recent GPU-based mul- +ticut solver of (Abbas & Swoboda, 2022). Similar to +GAEC we run it on the complete graph. The solver +uses dual optimization for better solution quality and +also gives lower bounds to the multicut objective (1). +As a drawback this solver cannot handle large in- +stances due to high memory requirement of complete +graphs. For running the solver we use an NVIDIA +A40 GPU with 48GB of memory. +DGAEC: Our Algorithm 2 which operates on node features +and performs contractions according to Lemma 3.1. +The nearest neighbour graph is updated by exhaustive +search after edge contraction. The number of nearest +neighbours k is set to 1. +DGAECInc: Our Algorithm 2 which additionally makes +use of Algorithm 3 for incrementally populating near- +est neighbours after edge contraction. The value of k +is set to 5. +DLAEC: A variant of our DGAEC where non-greedy moves +are also allowed as described in Sec. 3.2. The value of +k is set to 5. +DAppLAEC: Another variant of our DLAEC where initial +nearest neighbours are computed by approximate near- +est neighbour search method +(Malkov & Yashunin, +2018) through the implementation (Johnson et al., +2019). +For all multicut algorithms on all datasets we set the value +of affinity strength αi in (11) to 0.4, preferring small clus- +ters. All CPU algorithms are run on an AMD 7502P CPU +with a maximum of 8 threads to allow for faster NN search. +4.1. ImageNet clustering +We evaluate clustering of the ImageNet (Deng et al., 2009) +validation set containing 50k images. +Each image in +the dataset acts as a node for our dense multicut for- +mulation. +The features of each image are computed +by a ResNet50 (He et al., 2016) backbone trained by +MoCov3 (Chen et al., 2021) in unsupervised fashion by a +constrastive loss on the training split of ImageNet. The fea- +tures have a dimension of 2048 and are normalized to have +unit L2 norm. We create two problem instances containing +5k and 50k images by considering 100 and all 1000 classes +respectively. +Clustering quality: +Before comparing our algorithmic +contributions we first test the efficacy of our dense +multicut formulation by comparing its clustering re- +sult with k-means (Lloyd, 1982) using the implemen- +tation from (Pedregosa et al., 2011) and initialization +of (Arthur & Vassilvitskii, 2007). Since k-means requires +the number of clusters to be known beforehand we set it to +the number of classes in the problem instance. For an ad- +ditional comparison we also run k-means on the number of +clusters given by our dense multicut algorithm. The quality +of clustering results are evaluated using normalized mutual +information (NMI) and adjusted mutual information (AMI) +metrics (Vinh et al., 2010). The results are given in Table 1. +We observe that although our formulation does not require +the number of clusters to be specified, the results are on par +with k-means. Additionally the value of affinity strength +α does not need to be changed for different problem in- +stances. As compared to k-means our algorithms are much +faster especially on the larger instance. The RAMA solver +of (Abbas & Swoboda, 2022) performs better than all other +approaches on the smaller instance but runs out of mem- +ory for the larger one. Lastly, our formulation creates more +clusters than the number of classes. This is mainly due to +presence of outliers in the feature space as the feature ex- +tractor is trained without any groundtruth information. +Algorithms comparison: +We compare different algo- +rithms for solving dense multicut problem (3) for imageNet +clustering in Table 2. Firstly, we see that on the smaller + +Clustering Fully connected Graphs by Multicut +Table 1: Comparison of clustering obtained by different +methods on ImageNet validation set. t [s]: compute time +in seconds, NMI: normalized mutual information, AMI: ad- +justed mutual information, # clusters: number of clusters, †: +out of GPU memory. For k-means the number of clusters +was specified as input. +Method +t [s] ↓ +NMI ↑ +AMI ↑ +# clusters +ImageNet-100 (|V | = 5k) +k-means +16 +0.42 +0.27 +100 +k-means +32 +0.53 +0.26 +333 +RAMA +0.9 +0.57 +0.29 +639 +DGAECInc +42 +0.43 +0.22 +343 +DAppLAEC +3.2 +0.47 +0.26 +333 +ImageNet-1000 (|V | = 50k) +k-means +701 +0.54 +0.2 +1000 +k-means +1801 +0.61 +0.19 +2440 +RAMA +† +† +† +† +DGAECInc +2964 +0.49 +0.19 +2488 +DAppLAEC +65 +0.56 +0.26 +2440 +instance the GPU based solver RAMA (Abbas & Swoboda, +2022) gives the best performance. Secondly using incre- +mental nearest neighbour search through Alg. 3 gives bet- +ter run time than exhaustive search. Lastly our non-greedy +algorithms give the best run time among all CPU-based al- +gorithms although with slightly worse objectives. +On the smaller instance, RAMA outperforms other algo- +rithms in terms of the objective value (3) and also gives bet- +ter clustering quality as compared to k-means. As a draw- +back RAMA cannot handle large dense multicut instances. +This shows multicut on complete graphs can be a suitable +alternative to k-means. We speculate that algorithmic im- +provements on top of our proposed algorithms will further +improve clustering quality for large graphs. +4.2. Panoptic segmentation +We +evaluate our method +on +the +task +of +panoptic +segmentation (Kirillov et al., 2019) on the Cityscapes +dataset (Cordts et al., 2016). The panoptic segmentation +task consists of assigning a class label to each pixel and +partitioning different instances of classes with object cat- +egories (e.g. +car, person etc.). +We focus on the task +of partitioning for which the multicut formulation (1) +has been used by (Kirillov et al., 2017; Abbas & Swoboda, +2021). +The latter work used a carefully crafted graph +structure. +Our dense multicut (3) formulation foregoes +the need for finding a suitable graph structure. We use +the pretrained Axial-ResNet50 (Wang et al., 2021) network +Table 2: Comparison of algorithms for solving dense mul- +ticut problem on two splits of Imagenet validation set. t [s]: +compute time in seconds, Obj: objective value of cluster- +ing (3), †: out of GPU memory, ⋆: no result within a 3 hour +time limit. +ImageNet-100 +ImageNet-1000 +Method +t [s] ↓ +Obj ↓ +t [s] ↓ +Obj ↓ +GAEC +24 +-6.84e5 +2605 +-9.353e7 +RAMA +0.8 +-6.95e5 +† +† +DGAEC +132 +-6.84e5 +⋆ +⋆ +DGAECInc +42 +-6.84e5 +2934 +-9.353e7 +DLAEC +5 +-6.83e5 +341 +-9.332e7 +DAppLAEC +3.2 +-6.83e5 +65 +-9.332e7 +from (Yu et al., 2022), made available by (Weber et al., +2021) to compute the node features. Specifically, the net- +work computes L2-normalized instance discriminative fea- +tures in its intermediate stages which we use for our study +without any training. +For our evaluation we first compute semantic class predic- +tions and then create a dense multicut instance for each se- +mantic category with objects (i.e., car, person etc.). Such +classes are also known as thing classes. The goal of the +multicut problem is then to partition all nodes belonging +to same semantic class to different objects. This strategy +creates a total of 1631 dense multicut problem instances +of varying sizes from 500 images of the Cityscapes valida- +tion set. The largest problem instance contains around 43k +nodes. +Clustering quality: +As a first point of comparison we +check whether formulating a multicut problem on the com- +plete graph by (3) is beneficial as compared to a hand- +crafted sparse graph structure. We take the sparse graph +structure from (Abbas & Swoboda, 2021) as a baseline. +Their graph also includes long-range edges for dealing with +occlusions leading to about 10 ·|V | edges in total. We com- +pute the edge costs in this sparse graph in the same way as +for our dense multicut formulation. For solving this multi- +cut problem (1) we use Alg. 1. +In Table 3 we compare the quality of clustering through +the panoptic quality metric (Kirillov et al., 2019). We ob- +serve that our dense multicut formulation performs better +than multicut on the sparse handcrafted graph. This im- +provement is significant for classes which can have many +instances of the same class within an image (i.e. person, +car) thus making the partitioning problem difficult. For +classes with large objects (e.g. truck) having more edges +does not help since the sparse graph can already capture + +Clustering Fully connected Graphs by Multicut +Table +3: +Comparison +of +panoptic +segmentation +on Cityscapes dataset. +Multicut on sparse graph +of (Abbas & Swoboda, 2021) is computed by Alg. 1. +For dense multicut we use the DAppLAEC algorithm. +PQth: Average panoptic quality of all thing classes. +Panoptic quality (%) ↑ +Category +Sparse multicut +Dense multicut +Person +40.0 +46.9 +Rider +53.0 +54.4 +Car +50.7 +60.5 +Truck +52.7 +52.3 +Bus +72.1 +71.1 +Train +65.6 +62.9 +Motorcycle +47.0 +46.8 +Bicycle +45.7 +46.9 +PQth +53.3 +55.2 +most inter-pixel relations. On average our dense multicut +formulation gives better results than sparse multicut while +alleviating the need for designing a graph structure. +Algorithms comparison: +We compare dense multicut al- +gorithms for the panoptic segmentation task in terms of ob- +jective value and run time. We were not able to run RAMA +since the GPU could not store large graphs. The compar- +ison of performance to the remaining algorithms averaged +over all problem instances is given in Table 4. Moreover, +in Figure 4 we compare performance of the algorithms on +all large problem instances. +In terms of run time, we see that our most naive algorithm +DGAEC is slower than GAEC which directly operates on +edge costs. Our other algorithms surpass GAEC reaching up +to an order of magnitude run time improvement with lazy +edge contractions and approximate initial nearest neigh- +bours search. In terms of objective value we see slight im- +provement by our lazy contraction algorithms as compared +to the greedy ones. +Sensitivity of affinity strength: +In Table 5 we study the +effect of changing the value of α from (10). We observe +even better panoptic quality using a value of 0.3 as com- +pared to our default of 0.4. As the edge costs lie in [−1, 1] +due to L2-normalized node features, values of α close to +0 or 1 gives more performance degradation. Last, we see +further improvement if the value of α is set differently for +each class. We refer to the Appendix for further results. +−1.4 −1.2 +−1 +−0.8 −0.6 −0.4 −0.2 +0 +← Objective value (×108) +10−1 +100 +101 +102 +103 +104 +← time [s] +GAEC +DGAEC +DGAECInc +DLAEC +DAppLAEC +Figure 4: Comparison of algorithms on large dense multi- +cut instances (|V | ≥ 5000) from Cityscapes validation set. +Overlaid bars mark the 0.25, 0.5 and 0.75-quantile. +Table 4: Comparison of algorithms for solving dense mul- +ticut problem on Cityscapes validation set. (t [s]): average +compute times in seconds, (Obj): average objective value +of clustering (3). The average is calculated over all problem +instances. +Method +t [s] ↓ +Obj (×106) ↓ +GAEC +10.0 +-6.338 +DGAEC +84.1 +-6.338 +DGAECInc +3.2 +-6.338 +DLAEC +2.1 +-6.340 +DAppLAEC +1.5 +-6.341 +Table 5: Results of panoptic segmentation via dense multi- +cut with different values of attraction/repulsion strength α +in (10). PQth: Avg. panoptic quality over all thing classes. +α +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +PQth +54.5 +55.8 +55.2 +55.0 +54.1 +52.0 +49.3 + +Clustering Fully connected Graphs by Multicut +5. Conclusion +We have demonstrated that optimizing multicut on large +complete graphs is possible when using factorized edge +costs through inner products of features. We speculate that +further algorithmic improvements are possible e.g. by per- +forming dual optimization directly on the node features. +As a potential theoretical advantage our approach sidesteps +the need for learning graph structure. This offers a possibil- +ity to embed it as a differentiable layer in neural networks, +using e.g. the work (Vlastelica et al., 2019). +References +Abbas, A. and Swoboda, P. Combinatorial optimization for +panoptic segmentation: A fully differentiable approach. +Advances in Neural Information Processing Systems, 34: +15635–15649, 2021. +Abbas, A. and Swoboda, P. +RAMA: A Rapid Multicut +Algorithm on GPU. In Proceedings of the IEEE/CVF +Conference on Computer Vision and Pattern Recognition +(CVPR), pp. 8193–8202, June 2022. +Arthur, D. and Vassilvitskii, S. K-means++: The advan- +tages of careful seeding. +In Proceedings of the Eigh- +teenth Annual ACM-SIAM Symposium on Discrete Al- +gorithms, SODA ’07, pp. 1027–1035, USA, 2007. So- +ciety for Industrial and Applied Mathematics. +ISBN +9780898716245. +Bailoni, A., Pape, C., H¨utsch, N., Wolf, S., Beier, T., +Kreshuk, A., and Hamprecht, F. A. GASP, a General- +ized Framework for Agglomerative Clustering of Signed +Graphs and Its Application to Instance Segmentation. In +Proceedings of the IEEE/CVF Conference on Computer +Vision and Pattern Recognition, pp. 11645–11655, 2022. +Bansal, N., Blum, A., and Chawla, S. Correlation cluster- +ing. Machine learning, 56(1-3):89–113, 2004. +Beier, T., Kroeger, T., Kappes, J. H., Kothe, U., and Ham- +precht, F. A. Cut, glue & cut: A fast, approximate solver +for multicut partitioning. In Proceedings of the IEEE +Conference on Computer Vision and Pattern Recogni- +tion, pp. 73–80, 2014. +Beier, T., Hamprecht, F. A., and Kappes, J. H. +Fusion +moves for correlation clustering. +In Proceedings of +the IEEE Conference on Computer Vision and Pattern +Recognition, pp. 3507–3516, 2015. +Chen, X., Xie, S., and He, K. An empirical study of train- +ing self-supervised vision transformers. In Proceedings +of the IEEE/CVF International Conference on Computer +Vision, pp. 9640–9649, 2021. +Chopra, S. and Rao, M. R. The partition problem. Mathe- +matical Programming, 59(1-3):87–115, 1993. +Cordts, M., Omran, M., Ramos, S., Rehfeld, T., Enzweiler, +M., Benenson, R., Franke, U., Roth, S., and Schiele, B. +The cityscapes dataset for semantic urban scene under- +standing. In Proceedings of the IEEE conference on com- +puter vision and pattern recognition, 2016. +Demaine, E. D., Emanuel, D., Fiat, A., and Immorlica, N. +Correlation clustering in general weighted graphs. Theo- +retical Computer Science, 361(2-3):172–187, 2006. +Deng, J., Dong, W., Socher, R., Li, L.-J., Li, K., and +Fei-Fei, L. Imagenet: A large-scale hierarchical image +database. In 2009 IEEE conference on computer vision +and pattern recognition, pp. 248–255. Ieee, 2009. +Deza, M., Gr¨otschel, M., and Laurent, M. +Clique-web +facets for multicut polytopes. +Mathematics of Opera- +tions Research, 17(4):981–1000, 1992. +Dhillon, I. S., Guan, Y., and Kulis, B. Weighted graph cuts +without eigenvectors a multilevel approach. IEEE Trans- +actions on Pattern Analysis and Machine Intelligence, 29 +(11):1944–1957, 2007. doi: 10.1109/TPAMI.2007.1115. +He, K., Zhang, X., Ren, S., and Sun, J. Deep residual learn- +ing for image recognition. In Proceedings of the IEEE +conference on computer vision and pattern recognition, +pp. 770–778, 2016. +Hu, T. C. Multi-commodity network flows. Operations +research, 11(3):344–360, 1963. +Jia, H., Ding, S., Xu, X., and Nie, R. The latest research +progress on spectral clustering. Neural Comput. Appl., +24(7–8):1477–1486, jun 2014. ISSN 0941-0643. doi: +10.1007/s00521-013-1439-2. +Johnson, J., Douze, M., and J´egou, H. Billion-scale similar- +ity search with GPUs. IEEE Transactions on Big Data, +7(3):535–547, 2019. +Kardoost, A. and Keuper, M. Solving minimum cost lifted +multicut problems by node agglomeration. In Asian Con- +ference on Computer Vision, pp. 74–89. Springer, 2018. +Keuper, M., Levinkov, E., Bonneel, N., Lavou´e, G., Brox, +T., and Andres, B. +Efficient decomposition of image +and mesh graphs by lifted multicuts. In Proceedings of +the IEEE International Conference on Computer Vision, +2015. +Kirillov, A., Levinkov, E., Andres, B., Savchynskyy, B., +and Rother, C. +Instancecut: from edges to instances +with multicut. In Proceedings of the IEEE Conference +on Computer Vision and Pattern Recognition, 2017. + +Clustering Fully connected Graphs by Multicut +Kirillov, A., He, K., Girshick, R., Rother, C., and Doll´ar, P. +Panoptic segmentation. In Proceedings of the IEEE/CVF +Conference on Computer Vision and Pattern Recogni- +tion, pp. 9404–9413, 2019. +Lange, J.-H., Karrenbauer, A., and Andres, B. +Partial +optimality and fast lower bounds for weighted correla- +tion clustering. In International Conference on Machine +Learning, 2018. +Levinkov, E., Kirillov, A., and Andres, B. A comparative +study of local search algorithms for correlation cluster- +ing. In GCPR, 2017. +Lloyd, S. Least squares quantization in pcm. IEEE Transac- +tions on Information Theory, 28(2):129–137, 1982. doi: +10.1109/TIT.1982.1056489. +Malkov, Y. A. and Yashunin, D. A. Efficient and robust +approximate nearest neighbor search using hierarchical +navigable small world graphs. IEEE transactions on pat- +tern analysis and machine intelligence, 42(4):824–836, +2018. +Oosten, M., Rutten, J. H., and Spieksma, F. C. The clique +partitioning problem: facets and patching facets. Net- +works: An International Journal, 38(4):209–226, 2001. +Pan, X., Papailiopoulos, D., Oymak, S., Recht, B., Ram- +chandran, K., and Jordan, M. I. Parallel correlation clus- +tering on big graphs. In Cortes, C., Lawrence, N., Lee, +D., Sugiyama, M., and Garnett, R. (eds.), Advances in +Neural Information Processing Systems, volume 28. Cur- +ran Associates, Inc., 2015. +Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., +Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., +Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cour- +napeau, D., Brucher, M., Perrot, M., and Duchesnay, E. +Scikit-learn: Machine learning in Python. +Journal of +Machine Learning Research, 12:2825–2830, 2011. +Qaddoura, R., Faris, H., and Aljarah, I. An efficient cluster- +ing algorithm based on the k-nearest neighbors with an +indexing ratio. International Journal of Machine Learn- +ing and Cybernetics, 11(3):675–714, 2020. +Swoboda, P. and Andres, B. A message passing algorithm +for the minimum cost multicut problem. In Proceedings +of the IEEE Conference on Computer Vision and Pattern +Recognition, 2017. +Veldt, N. Correlation clustering via strong triadic closure +labeling: Fast approximation algorithms and practical +lower bounds. In International Conference on Machine +Learning, pp. 22060–22083. PMLR, 2022. +Vinh, N. X., Epps, J., and Bailey, J. +Information +theoretic +measures +for +clusterings +comparison: +Variants, +properties, +normalization +and +correc- +tion +for +chance. +Journal +of +Machine +Learn- +ing +Research, +11(95):2837–2854, +2010. +URL +http://jmlr.org/papers/v11/vinh10a.html. +Vlastelica, M., Paulus, A., Musil, V., Martius, G., and +Rol´ınek, M. Differentiation of blackbox combinatorial +solvers. arXiv preprint arXiv:1912.02175, 2019. +Von Luxburg, U. A tutorial on spectral clustering. Statistics +and computing, 17(4):395–416, 2007. +Wang, H., Zhu, Y., Adam, H., Yuille, A., and Chen, L.-C. +Max-deeplab: End-to-end panoptic segmentation with +mask transformers. +In Proceedings of the IEEE/CVF +conference on computer vision and pattern recognition, +pp. 5463–5474, 2021. +Weber, M., Wang, H., Qiao, S., Xie, J., Collins, M. D., Zhu, +Y., Yuan, L., Kim, D., Yu, Q., Cremers, D., Leal-Taixe, +L., Yuille, A. L., Schroff, F., Adam, H., and Chen, L.- +C. DeepLab2: A TensorFlow Library for Deep Labeling. +arXiv: 2106.09748, 2021. +Yu, Q., Wang, H., Qiao, S., Collins, M., Zhu, Y., Adam, H., +Yuille, A., and Chen, L.-C. k-means mask transformer. +In European Conference on Computer Vision, pp. 288– +307. Springer, 2022. + +Clustering Fully connected Graphs by Multicut +Appendix +A. Influence of affinity strength +On the Cityscapes dataset we compare panoptic quality on different object classes by varying the value of affinity strength +α in (11). The results are given in Table 6. We observe that for classes contain many small objects large value of α is +suitable whereas for classes with large objects small value of α is preferable. Although our default value of 0.4 already +makes dense multicut outperform the baseline, further improvement is still possible e.g. by tuning α. +Table 6: Comparison of panoptic segmentation on Cityscapes dataset for different values of affinity strength α (11). All +results are computed using the DAppLAEC algorithm. Largest values in each row are highlighted with bold. +Panoptic quality on varying values of α +Category +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Person +31.5 +38.1 +43.2 +46.9 +49.8 +52.6 +54.3 +55.0 +52.4 +Rider +51.1 +53.0 +53.9 +54.5 +55.5 +55.4 +53.9 +51.0 +45.5 +Car +45.6 +52.9 +57.8 +60.5 +63.3 +64.8 +64.1 +62.2 +57.8 +Truck +54.1 +53.7 +52.7 +52.3 +49.0 +47.8 +45.4 +41.5 +34.7 +Bus +75.1 +74.2 +73.5 +71.2 +69.3 +63.6 +58.5 +54.5 +47.3 +Train +75.0 +74.9 +71.5 +62.9 +56.3 +51.7 +45.1 +40.4 +32.3 +Motorcycle +45.5 +46.1 +48.0 +46.8 +48.7 +49.1 +47.8 +45.2 +39.8 +Bicycle +38.1 +43.2 +45.6 +46.9 +47.8 +48.0 +46.9 +44.6 +40.4 +Average (PQth) +52.0 +54.5 +55.8 +55.2 +55.0 +54.1 +52.0 +49.3 +43.8 + diff --git a/-tFLT4oBgHgl3EQfvS-t/content/tmp_files/load_file.txt b/-tFLT4oBgHgl3EQfvS-t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2ec47397f2119fce0ea369c9d8325b61a9b87f7b --- /dev/null +++ b/-tFLT4oBgHgl3EQfvS-t/content/tmp_files/load_file.txt @@ -0,0 +1,924 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfvS-t/content/2301.12159v1.pdf,len=923 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfvS-t/content/2301.12159v1.pdf'} +page_content='12159v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfvS-t/content/2301.12159v1.pdf'} +page_content='CV] 28 Jan 2023 ClusterFuG: Clustering Fully connected Graphs by Multicut Ahmed Abbas 1 Paul Swoboda 1 2 Abstract We propose a graph clustering formulation based on multicut (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfvS-t/content/2301.12159v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfvS-t/content/2301.12159v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfvS-t/content/2301.12159v1.pdf'} +page_content=' weighted correlation cluster- ing) on the complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfvS-t/content/2301.12159v1.pdf'} +page_content=' Our formulation does not need specification of the graph topology as in the original sparse formulation of multicut, making our approach simpler and potentially bet- ter performing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfvS-t/content/2301.12159v1.pdf'} +page_content=' In contrast to unweighted corre- lation clustering we allow for a more expressive weighted cost structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfvS-t/content/2301.12159v1.pdf'} +page_content=' In dense multicut, the clustering objective is given in a factorized form as inner products of node feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfvS-t/content/2301.12159v1.pdf'} +page_content=' This al- lows for an efficient formulation and inference in contrast to multicut/weighted correlation cluster- ing, which has at least quadratic representation and computation complexity when working on the complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfvS-t/content/2301.12159v1.pdf'} +page_content=' We show how to rewrite clas- sical greedy algorithms for multicut in our dense setting and how to modify them for greater ef- ficiency and solution quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfvS-t/content/2301.12159v1.pdf'} +page_content=' In particular, our algorithms scale to graphs with tens of thousands of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfvS-t/content/2301.12159v1.pdf'} +page_content=' Empirical evidence on instance seg- mentation on Cityscapes and clustering of Ima- geNet datasets shows the merits of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfvS-t/content/2301.12159v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfvS-t/content/2301.12159v1.pdf'} +page_content=' Introduction Graph-based clustering approaches, primarily among them multicut (Chopra & Rao, 1993), are theoretically appeal- ing: They do not need specification of the number of clus- ters, but infer them as part of the optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfvS-t/content/2301.12159v1.pdf'} +page_content=' They allow for a flexible clustering objective with attrac- tive and repulsive costs between pairs of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfvS-t/content/2301.12159v1.pdf'} +page_content=' They are also theoretically well-understood as optimization prob- lems with intensively studied polyhedral descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfvS-t/content/2301.12159v1.pdf'} +page_content=' Effi- cient solvers that scale well and give high quality solutions have also been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfvS-t/content/2301.12159v1.pdf'} +page_content=' As a drawback, graph-based clustering approaches need specification of the underlying graph topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfvS-t/content/2301.12159v1.pdf'} +page_content=' In prac- 1MPI for Informatics, Saarland Informatics Campus, Germany 2University of Mannheim, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFLT4oBgHgl3EQfvS-t/content/2301.12159v1.pdf'} +page_content=' Correspondence to: Ahmed Abbas 300 GeV, +where the POWHEG+Pythia events lead to a slightly harder +distribution than the events generated from the fixed order +NLO cross section. +160 +165 +170 +175 +180 +185 +190 +m [GeV] +0 +100 +200 +300 +400 +− log +� +L(m) +Lmax +� +mtrue = 173.2 GeV, 9232 analysed events +LO prediction: +ˆm2µ0 +0.5µ0 = 173.88 ± 1.22+2.13 +−2.57GeV +NLO prediction: +ˆm2µ0 +0.5µ0 = 177.93 ± 1.24+0.22 ++0.38GeV +FIG. 8. Analysis of 9232 POWHEG+Pythia events with fixed-order +likelihoods calculated at LO and NLO accuracy. +ˆmt ±∆stat +∆ +2µ0 +sys +∆ +µ0/2 +sys +[GeV] +likelihood +LO prediction +NLO prediction +L 173.88±1.22+2.13 +−2.57 177.93±1.24+0.22 ++0.38 +TABLE II. Extracted values for the estimator of the top-quark mass +from unweighted POWHEG+Pythia events following the NLO predic- +tion matched to a parton shower. +The result of the likelihood analysis using LO and NLO +cross section predictions is shown in Fig. 8 and summarized +in Tab. II. We do not study the extended likelihood, since the +extended likelihood leads to much larger systematic uncer- +tainties. Again the statistical uncertainties are very similar for +the LO and NLO analysis, while the systematic uncertainty is +significantly reduced when using NLO predictions. In both +cases we observe a shift of about 4 GeV compared to the re- +sults based on the event sample generated from the fixed-order +NLO predictions. The large shift shows the high sensitivity of +the MEM with respect to tiny changes in the distributions. In +a mass determination from events registered at the LHC this +shift must be taken into account via a calibration procedure. +It is remarkable that the shift is, taking the uncertainties into +account, independent from the perturbative order of the em- +ployed likelihood calculation. This is similar to what has been +observed in Refs. [24, 25]. The LO likelihood analysis repro- +duces the true mass value used in the POWHEG+Pythia analy- +sis. However, this ist most likely accidental and due to the fact +that the LO fixed-order results undershoots the true mass value +by about 4 GeV which is compensated by the aforementioned +shift. +IV. +CONCLUSION +In this work the MEM at NLO is applied to top-quark +pair production at the LHC. To investigate the potential of +the matrix element method to measure the top-quark mass, +the MEM at NLO is applied to pseudo-data: unweighted + +7 +events generated from the fixed-order NLO cross section +as well as events obtained using POWHEG+Pythia incorpo- +rating the parton shower effects. Using pseudo-data based +on POWHEG+Pythia allows to study the effect of the parton +shower and gives a more realistic simulation. Including the +NLO corrections in the likelihood calculation leads to a signif- +icant reduction of the theoretical uncertainties of the extracted +top-quark mass, while the statistical uncertainties remain al- +most unchanged compared to the LO analysis. We stress that +the uncertainties due to scale variation cannot be reduced by a +calibration. The reduction of the uncertainties associated with +the scale variation when going from LO to NLO thus presents +an important improvement and a strong argument in favour of +the the MEM at NLO accuracy. +Another important observation is the fact that the extended +likelihood yields a significant improvement in terms of the sta- +tistical uncertainties. However, in practical applications this +gain in precision is completely outweighed by the theoretical +uncertainties of the number of expected events. This can be +understood from the fact that, much as the NLO corrections +(see Fig. 3), the scale variations do not dramatically change +the shape of the kinematic distributions but mostly their nor- +malization (see Fig. 4) thereby making the extended likeli- +hood analyses more sensitive to their effect. Additionally, em- +ploying the extended likelihood requires precise knowledge of +the integrated luminosity of the LHC. The dependence on this +parameter introduces an additional source of systematic un- +certainty. This has to be taken into account for future experi- +mental applications of the MEM with realistic event numbers +for abundantly produced top-quark pairs at the LHC which +will most likely be dominated by systematic uncertainties. As +has already been stated before ([23–26]), for parameter infer- +ence with the MEM it is mandatory to perform the likelihood +calculation at least at NLO accuracy in order to properly fix +the renormalization scheme of the extracted parameter. +The application of the MEM at NLO to top-quark pair +events at the LHC can offer an alternative approach to deter- +mine the top-quark mass with high accuracy. As has been +demonstrated in this work, already for a few ten thousand +events the precision of the analysis becomes dominated by +systematic uncertainties. As the LHC produces millions of +top-quark pairs, the analysis could be performed with a rather +small fraction of cherry-picked events allowing to minimize +the overall systematic uncertainty. The results obtained in this +article suggest that top-quark mass determination with an un- +certainty below 1 GeV could be feasible. Of course, for a re- +alistic application of the MEM to experimental data, transfer +functions accounting for decays, additional radiation and de- +tector effects have to be considered. In addition, as the analy- +sis based on the events including parton shower effects shows, +a further calibration is required. +ACKNOWLEDGMENTS +This work was supported in part by the Bundesministerium +für Bildung and Forschung under contract 05H18KHCA1. +0 +10 +20 +30 +40 +50 +60 +dσNLO +dE1 +[pb GeV−1] +result +reference +0 +200 +400 +600 +800 +1000 +E1 [GeV] +−2σ +−σ +σ +2σ +pull +0 +10 +20 +30 +40 +50 +dσNLO +dη1 +[pb] +result +reference +−10.0 +−7.5 +−5.0 +−2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +η1 +−2σ +−σ +σ +2σ +pull +FIG. 9. Validation of the implementation: Comparison of differential +distributions of the top quark obtained in this work with results from +madgraph5 aMC@NLO. +Appendix A: Additional results on distributions used for the +validation +In this appendix we show further cross checks used for the +validation of the implementation. Fig. 9 shows comparisons +of NLO predictions for differential distributions calculated +in this work with distributions obtained from madgraph5 +aMC@NLO [37] which is based on the dipole subtraction +method [40, 41]. The pull distributions in the bottom plots +of Fig. 9 and Fig. 10 illustrate the agreement between both +implementations within statistical uncertainties. This compar- +isons serve as a further validation for the choice of the slicing +parameter. Fig. 11 shows the NLO corrections (upper part) +together with the k-factors (lower part) for the Mt¯t and the φ1- +distribution. Similar to what is shown in Fig. 3 again a flat k- +factor is observed. As a check of the event generation and the +unweighting procedure Fig. 12 shows distributions calculated +from the generated unweighted events compared with a calcu- +lation using madgraph5 aMC@NLO [37]. Similar to Fig. 7 we +show in Fig. 13 for further distributions the comparison of dis- +tributions obtained at fixed-order NLO accuracy with results +using POWHEG+Pythia. + +8 +0 +5 +10 +15 +dσNLO +dφ1 +[pb] +result +reference +−4 +−3 +−2 +−1 +0 +1 +2 +3 +4 +φ1 +−2σ +−σ +σ +2σ +pull +0 +10 +20 +30 +40 +dσNLO +dk⊥ +1 +[pb GeV−1] +result +reference +0 +100 +200 +300 +400 +500 +k⊥ +1 [GeV] +−2σ +−σ +σ +2σ +pull +FIG. 10. Same as Fig. 9 but for the φ1- and the k⊥ +1 -distribution. +0 +10 +20 +30 +40 +dσ +dMt¯t [pb GeV−1] +dσNLO +dσLO +300 +400 +500 +600 +700 +800 +900 +1000 +Mt¯t [GeV] +1.0 +1.5 +2.0 +dσNLO +dσLO +0 +5 +10 +15 +20 +dσ +dφ1 [pb] +dσNLO +dσLO +−4 +−3 +−2 +−1 +0 +1 +2 +3 +4 +φ1 +1.0 +1.5 +2.0 +dσNLO +dσLO +FIG. 11. Same as Fig. 3 but for the Mt¯t and the φ1-distribution. + +9 +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +1 +σNLO +dσNLO +dMt¯t +[pb GeV−1] +results +reference +300 +400 +500 +600 +700 +800 +900 +1000 +Mt¯t[GeV] +−2σ +−σ +σ +2σ +pull +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +1 +σNLO +dσNLO +dφ1 +[pb] +results +reference +−4 +−3 +−2 +−1 +0 +1 +2 +3 +4 +φ1 +−2σ +−σ +σ +2σ +pull +FIG. 12. Same as Fig. 5 but for the Mt¯t- and the φ1-distribution. +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +1 +σNLO +dσNLO +dE1 +[GeV−1] +POWHEG events +fixed-order NLO +0 +200 +400 +600 +800 +1000 +E1 [GeV] +0 +1 +2 +POWHEG +fixed-order NLO +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +1 +σNLO +dσNLO +dMt¯t +[GeV−1] +POWHEG events +fixed-order NLO +300 +400 +500 +600 +700 +800 +900 +1000 +Mt¯t [GeV] +0 +1 +2 +POWHEG +fixed-order NLO +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +1 +σNLO +dσNLO +dφ1 +POWHEG events +fixed-order NLO +−4 +−3 +−2 +−1 +0 +1 +2 +3 +4 +φ1 +0 +1 +2 +POWHEG +fixed-order NLO +FIG. 13. Same as Fig. 7 but for the energy, Mt¯t- and φ1-distribution. + +10 +[1] P. Nason, S. Dawson, and R. K. Ellis, Nucl. Phys. B 303, 607 +(1988). +[2] P. Nason, S. Dawson, and R. K. Ellis, Nucl. Phys. B 327, 49 +(1989), [Erratum: Nucl.Phys.B 335, 260–260 (1990)]. +[3] W. Beenakker, H. Kuijf, W. L. van Neerven, and J. Smith, Phys. +Rev. D 40, 54 (1989). +[4] W. Beenakker, W. L. van Neerven, R. Meng, G. A. Schuler, and +J. Smith, Nucl. Phys. B 351, 507 (1991). +[5] W. Bernreuther, A. Brandenburg, Z. G. Si, and P. Uwer, Nucl. +Phys. B 690, 81 (2004), arXiv:hep-ph/0403035. +[6] K. +Melnikov +and +M. +Schulze, +JHEP +08, +049 +(2009), +arXiv:0907.3090 [hep-ph]. +[7] M. Czakon, P. Fiedler, +and A. Mitov, Phys. Rev. Lett. 110, +252004 (2013), arXiv:1303.6254 [hep-ph]. +[8] M. Czakon, D. Heymes, and A. Mitov, Phys. Rev. Lett. 116, +082003 (2016), arXiv:1511.00549 [hep-ph]. +[9] M. Czakon, P. Fiedler, D. Heymes, and A. Mitov, JHEP 05, +034 (2016), arXiv:1601.05375 [hep-ph]. +[10] M. Beneke, M. Czakon, P. Falgari, A. Mitov, and C. Schwinn, +Phys. Lett. B690, 483 (2010), arXiv:0911.5166 [hep-ph]. +[11] M. Czakon, A. Mitov, +and G. F. Sterman, Phys. Rev. D80, +074017 (2009), arXiv:0907.1790 [hep-ph]. +[12] M. Beneke, P. Falgari, S. Klein, and C. Schwinn, Nucl. Phys. +B855, 695 (2012), arXiv:1109.1536 [hep-ph]. +[13] M. Cacciari, M. Czakon, M. Mangano, A. Mitov, and P. Nason, +Phys. Lett. B710, 612 (2012), arXiv:1111.5869 [hep-ph]. +[14] N. +Kidonakis, +Phys. +Part. +Nucl. +45, +714 +(2014), +arXiv:1210.7813 [hep-ph]. +[15] A. Ferroglia, B. D. Pecjak, and L. L. Yang, Phys. Rev. D86, +034010 (2012), arXiv:1205.3662 [hep-ph]. +[16] A. Ferroglia, S. Marzani, B. D. Pecjak, and L. L. Yang, JHEP +01, 028 (2014), arXiv:1310.3836 [hep-ph]. +[17] M. Czakon, A. Ferroglia, D. Heymes, A. Mitov, B. D. Pecjak, +D. J. Scott, X. Wang, and L. L. Yang, JHEP 05, 149 (2018), +arXiv:1803.07623 [hep-ph]. +[18] W. Beenakker, A. Denner, W. Hollik, R. Mertig, T. Sack, and +D. Wackeroth, Nucl. Phys. B411, 343 (1994). +[19] W. Bernreuther, M. Fuecker, +and Z.-G. Si, Phys. Rev. D74, +113005 (2006), arXiv:hep-ph/0610334 [hep-ph]. +[20] J. H. Kühn, A. Scharf, +and P. Uwer, Eur. Phys. J. C51, 37 +(2007), arXiv:hep-ph/0610335 [hep-ph]. +[21] S. Moretti, M. R. Nolten, and D. A. Ross, Phys. Lett. B639, +513 (2006), [Erratum: Phys. Lett.B660,607(2008)], arXiv:hep- +ph/0603083 [hep-ph]. +[22] D. Pagani, I. Tsinikos, and M. Zaro, Eur. Phys. J. C76, 479 +(2016), arXiv:1606.01915 [hep-ph]. +[23] T. +Martini +and +P. +Uwer, +JHEP +09, +083 +(2015), +arXiv:1506.08798 [hep-ph]. +[24] T. Martini and P. Uwer, (2017), arXiv:1712.04527 [hep-ph]. +[25] M. Kraus, T. Martini, and P. Uwer, Phys. Rev. D 100, 076010 +(2019), arXiv:1901.08008 [hep-ph]. +[26] M. Kraus, T. Martini, S. Peitzsch, +and P. Uwer, +(2019), +arXiv:1908.09100 [hep-ph]. +[27] P. Nason, JHEP 11, 040 (2004), arXiv:hep-ph/0409146. +[28] T. Sjostrand, S. Mrenna, +and P. Z. Skands, JHEP 05, 026 +(2006), arXiv:hep-ph/0603175. +[29] S. Frixione, P. Nason, and C. Oleari, JHEP 11, 070 (2007), +arXiv:0709.2092 [hep-ph]. +[30] S. Frixione, P. Nason, and G. Ridolfi, JHEP 09, 126 (2007), +arXiv:0707.3088 [hep-ph]. +[31] S. Alioli, P. Nason, C. Oleari, and E. Re, JHEP 06, 043 (2010), +arXiv:1002.2581 [hep-ph]. +[32] W. T. Giele, E. W. N. Glover, and D. A. Kosower, Nucl. Phys. +B403, 633 (1993), arXiv:hep-ph/9302225 [hep-ph]. +[33] S. Badger, R. Sattler, and V. Yundin, Phys. Rev. D 83, 074020 +(2011), arXiv:1101.5947 [hep-ph]. +[34] T. Kinoshita, J. Math. Phys. 3, 650 (1962). +[35] T. D. Lee and M. Nauenberg, Phys. Rev. 133, B1549 (1964). +[36] M. Aliev, H. Lacker, U. Langenfeld, S. Moch, P. Uwer, et al., +Comput.Phys.Commun. 182, 1034 (2011), arXiv:1007.1327 +[hep-ph]. +[37] J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, +O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli, and M. Zaro, +JHEP 07, 079 (2014), arXiv:1405.0301 [hep-ph]. +[38] J. von Neumann, in Monte Carlo Method, National Bureau of +Standards Applied Mathematics Series, Vol. 12, edited by A. S. +Householder, G. E. Forsythe, and H. H. Germond (US Gov- +ernment Printing Office, Washington, DC, 1951) Chap. 13, pp. +36–38. +[39] A. Butter, T. Heimel, T. Martini, S. Peitzsch, +and T. Plehn, +(2022), arXiv:2210.00019 [hep-ph]. +[40] S. Catani and M. Seymour, Nucl.Phys. B485, 291 (1997), +arXiv:hep-ph/9605323 [hep-ph]. +[41] S. Catani, S. Dittmaier, M. H. Seymour, +and Z. Trocsanyi, +Nucl.Phys. B627, 189 (2002), arXiv:hep-ph/0201036 [hep-ph]. + diff --git a/09E1T4oBgHgl3EQflAR-/content/tmp_files/load_file.txt b/09E1T4oBgHgl3EQflAR-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3b7f2c917d2874788a6d021fd79f3fe33039eaca --- /dev/null +++ b/09E1T4oBgHgl3EQflAR-/content/tmp_files/load_file.txt @@ -0,0 +1,764 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf,len=763 +page_content='Determination of the top-quark mass from top-quark pair events with the matrix element method at next-to-leading order: Potential and prospects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Till Martini∗ Fraunhofer Zentrum SIRIOS, Fraunhofer Institute for High-Speed Dynamics EMI, Berlin, Germany Turan Nuraliyev† and Peter Uwer‡ Humboldt-Universität zu Berlin, Institut für Physik, Newtonstraße 15, 12489 Berlin, Germany More than 25 years ago the matrix element method has been used in a pioneering work by D�0 to determine the top-quark mass from a handful of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Since then the method has been matured into a powerful analysis tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' While the first applications were restricted to leading-order accuracy, in the meantime also the extension to next- to-leading order (NLO) accuracy has been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In this article we explore the potential of the matrix element method at NLO to determine the top-quark mass using events with pair-produced top quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' We simulate a toy experiment by generating unweighted events with a fixed input mass and apply the matrix element method to construct an estimator for the top-quark mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Two different setups are investigated: unweighted events obtained from the fixed-order cross section at NLO accuracy as well as events obtained using POWHEG matched to a parton shower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The latter lead to a more realistic simulation and allow to study the impact of higher-order corrections as well as the robustness of the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' We find that the matrix element method in NLO accuracy leads to a significant reduction of the theoretical uncertainties compared to leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In view of the high luminosity phase of the LHC, this observation is especially relevant in analyses which are no longer dominated by statistical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' INTRODUCTION Regarding experimental as well as theoretical progress, hadronic top-quark pair production has evolved into one of the flagship processes at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' This development is pro- pelled by the expectation of the top quark to play a prominent role in extensions of the Standard Model due to it being by far the heaviest of the elementary particles with a life time sig- nificantly shorter than the time scale of hadronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The high production rate of top-quark pairs at the LHC as well as onward advances in experimental data taking enable for ever-decreasing statistical and systematic uncertainties in the recorded data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In order to make optimal use of this fact in ex- perimental analyses, the employed theoretical predictions are required to keep up in terms of uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The next-to-leading order QCD corrections for top-quark pair production have been calculated for the spin independent case more then 30 years ago [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Later, also the spin depen- dent cross sections were evaluated at NLO accuracy in QCD [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In a series of ground breaking articles also the next- to-next-to-leading order QCD corrections were calculated [7– 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Furthermore, beyond fixed order also the resummation of soft-gluon corrections has been studied in great detail ([10– 17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In addition to QCD corrections also weak and QED corrections have been calculated [18–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In summary, many detailed theoretical predictions for top-quark pair production are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' However, these might not be readily applicable in the experimental analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' It is thus important to put more ∗ Work on this article was conducted while employed at Humboldt- Universität zu Berlin, Institut für Physik, Berlin, Germany;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Till.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='Martini@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='hu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='de † Turan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='Nuraliyev@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='hu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='de ‡ Peter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='Uwer@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='hu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='de effort in improving the interface between experiment and the- ory to make optimal use of the increasing precision reached in both fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Multivariate analysis methods like the matrix element method (MEM), turn out to be particularly useful in making optimal use of the theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The MEM requires the calculation of event weights in terms of differential cross sections and is thus often formulated at lower-order accuracy only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' At leading order (LO), the MEM has been established as a powerful analysis tool for both signal searches as well as parameter inference by virtue of its optimal utilization of the information content of the available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Typically, the im- pact of higher-order QCD corrections on theoretical predic- tions can be significant while often simultaneously decreasing the theoretical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In the quest for accuracy and pre- cision to match experimental achievements, the MEM at next- to-leading order (NLO) represents a promising remedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' But when taking higher-order corrections into account, the calcu- lation of event weights constitutes a non-trivial task due to the intricate combination of virtual and real contributions to ob- tain meaningful finite results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The problem of extending the MEM beyond the Born approximation has been solved in the past by introducing modified jet algorithms on the one hand or sensible event definitions on the other hand ([23–25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' At the same time, the application of the MEM at NLO has been demonstrated for top-quark mass extraction from simulated single top-quark events ([23–25]) as well as anomalous cou- pling parameter determination from simulated Higgs boson events in association with a single top quark ([26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Addi- tionally, the effects of a parton shower applied to simulated single top-quark data has been investigated with the MEM at NLO ([25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In this work, we present the application of the MEM at NLO to top-quark pair production at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In contrast to the electroweak production mechanism of single top quarks studied before, top-quark pair production is QCD- induced at LO already with the two production channels of arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='03280v1 [hep-ph] 9 Jan 2023 2 quark-antiquark annihilation and gluon-gluon fusion consti- tuting the dominant source of top quarks at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Given the aforementioned prominent role of top-quark pair produc- tion in both experimental as well as theoretical advances at the LHC, it represents an ideal example to study higher-order effects within the MEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Furthermore, in view of the ongo- ing progress in top-quark mass measurements, the MEM at NLO accuracy could be an interesting alternative to existing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In section II the NLO QCD calculation of the differential cross section for top-quark pair production with the phase space slicing method and the subsequent generation of unweighted events are briefly re- viewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Section III focuses on the application of the MEM to the generated events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' To study parton shower effects, events generated with POWHEG+Pythia [27–31] are also analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The conclusions are presented in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' TOP-QUARK PAIR PRODUCTION AT THE LHC A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Implementing the NLO prediction with the phase space slicing method The MEM at NLO as presented in [23–25] requires the cross-section calculation at NLO to be carried out using the phase space slicing method [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The respective calculation is available in the literature [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Thus, in this section we only give a brief review of the important aspects of the calcula- tion and present the validation for the choice of the slicing pa- rameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In the phase space slicing method, the cross-section prediction at NLO accuracy dσNLO is formed of two contri- butions: First, the so-called hard part dσHard is just the ma- trix element for the real corrections evaluated for phase space points where all partons are resolved, that is the additional parton is neither collinear to the incoming partons nor soft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Second, a Born-like part is comprised of the Born contribution dσLO, the virtual corrections dσvirtual (taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [33]) as well as the so-called soft and collinear parts dσsoft/coll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' stemming from approximated real corrections integrated over phase space regions in which the additional parton is unre- solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The separation of the phase space for the real cor- rections into resolved and unresolved regions is mediated by the so-called slicing parameter xmin which acts as a scale to separate the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In the unresolved regions, well-known fac- torization properties of QCD real corrections can be employed allowing to analytically integrate over the additional radiation in the singular limits in an approximate way thereby reducing the respective phase space to Born-like kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The diver- gences of these integrations can be regularized within dimen- sional regularization leading to poles in the dimensional shift away from four space-time dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The outcome can be combined with the virtual contributions to cancel the respec- tive poles from the loop integration and yield finite results ac- cording to the Kinoshita-Lee-Nauenberg theorem ([34, 35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Since the real corrections are approximated in the unresolved (singular) regions, the result is only accurate up to deviations 10−6 10−5 10−4 10−3 xmin 703 704 705 706 707 708 709 σNLO[pb] reference σNLO FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Phase space slicing parameter (in-)dependence of the total cross section predicted at NLO accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The red line shows the reference value taken from HATHOR [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 0 10 20 30 40 dσNLO dk⊥ 1 [pb GeV−1] xmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0002 xmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0001 xmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='00005 0 100 200 300 400 500 k⊥ 1 [GeV] −2σ −σ σ 2σ pull xmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0002 vs xmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0001 xmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0001 vs xmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='00005 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Phase space slicing parameter (in-)dependence of the top- quark transverse momentum predicted at NLO accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' proportional to the slicing parameter xmin: dσNLO = dσHard + dσLO + dσvirtual + dσsoft/coll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' + O(xmin) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' (1) Additionally, the separation of the real phase space in terms of the slicing parameter introduces logarithmic dependencies of the hard and soft/collinear contributions on xmin which cancel in the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' However, when numerically integrating over the finite hard contribution, these logarithms can lead to numeri- cal instabilities if xmin is chosen too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Hence, the value of xmin has to be chosen as a compromise between numeri- cal stability and the demand that the deviation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' (1) is negligible compared to the statistical uncertainties of the total cross section as well as distributions calculated at NLO accu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 1 shows NLO predictions for the total cross section of top-quark pair production for different values of the slicing parameter xmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The total cross section as the sum of Born, virtual and real contributions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 1 is indeed finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' How- ever, it shows a systematic deviation from the reference value taken from HATHOR [36] for values xmin ⪆ 2 × 10−3 while for values xmin ⪅ 5 × 10−6 numerical instabilities dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Ac- cordingly, a value of xmin = 10−4 is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' As an example of a differential distribution the top-quark transverse momen- 3 0 10 20 30 40 dσ dk⊥ 1 [pb GeV−1] dσNLO dσLO 0 100 200 300 400 500 k⊥ 1 [GeV] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 dσNLO dσLO 0 10 20 30 40 50 dσ dη1 [pb] dσNLO dσLO −4 −2 0 2 4 η1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 dσNLO dσLO FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Differential distributions together with the respective k- factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' tum calculated at NLO accuracy is shown for three different choices of xmin in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In the lower plot we show for dif- ferent choices of xmin the differences in units of the statistical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' We conclude that all three choices lead to coher- ent predictions justifying the choice xmin = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In addition to the top-quark transverse momentum this has been checked also for the top-quark energy distribution and the rapidity dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Furthermore, the distributions calculated here have been cross checked with results from madgraph5 aMC@NLO [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The comparison is shown in appendix A, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 9 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The impact of the NLO corrections on kinematic dis- tributions is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 3 where NLO and LO predic- tions for kinematic distributions are compared and their ratios (the k-factor) are shown at the bottom of the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Results for further distributions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 11 in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' As can be seen from the rather constant k-factors, the NLO corrections only mildly affect the shapes of the kinematic dis- tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' However, the NLO corrections lead to a significant increase of the cross sections by a factor of roughly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 4 the impact of variations of the factorization scale µF and renormalization scale µR by a factor of 2 as a means to es- timate the effect of un-calculated higher orders are illustrated for the shapes of two representative kinematic distributions of the top quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' For moderate energy scales, one observes a significant reduction of the impact of the scale variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 0 200 400 600 800 1000 k⊥ 1 [GeV] 10−6 10−5 10−4 10−3 10−2 1 σ dσ dk⊥ 1 [GeV−1] LO NLO −4 −3 −2 −1 0 1 2 3 4 y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='30 1 σ dσ dy LO NLO FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Effect of scale variations on the shapes of kinematic distri- butions of the top quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Unweighted event generation From the calculation of the cross section at NLO accu- racy outlined in the previous section, event weights can be calculated which can be used to generate unweighted events which are distributed according to the NLO cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' As described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [25], a sensible event definition is manda- tory for obtaining meaningful event weights at NLO accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In particular, the event definition must not fix the invariant masses or the overall transverse momentum of the final-state objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' For top-quark pair production, we define events ⃗x in terms of the transverse momentum k⊥ 1 , azimuthal angle φ1 and pseudo rapidity η1 of the top quark as well as the pseudo rapidity of the antitop quark η2: ⃗x = (k⊥ 1 ,φ1,η1,η2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' (2) The two-particle Born phase space as well as the three-particle phase space for the real radiation can be parameterized in terms of these variables dR2 = k⊥ 1 3 coshη1 coshη2 8π2 E1 E2 shad dk⊥ 1 dφ1 dη1 dη2 , (3) dR3 = k⊥ 1 2 k⊥ 2 k⊥ 3 2 coshη1 coshη2 coshη3 128π5 E1 E2 E3 shad × dk⊥ 1 dφ1 dη1 dη2 dk⊥ 3 dφ3 dη3 , (4) 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='06 1 σNLO dσNLO dk⊥ 1 [pb GeV−1] results reference 0 100 200 300 400 500 k⊥ 1 [GeV] −2σ −σ σ 2σ pull 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='07 1 σNLO dσNLO dη1 [pb] results reference −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 η1 −2σ −σ σ 2σ pull FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Validation of the event generation: Comparison of differen- tial distributions of the top quark obtained from unweighted events with results from madgraph5 aMC@NLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' where Ei denotes the energy of particle i and shad is the hadronic center-of-mass energy squared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The additional ra- diation occurring in the real corrections is parametrized by the transverse momentum k⊥ 3 , the azimuthal angle φ3 and the pseudo rapidity η3 of the radiated parton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' These parametriza- tions allow together with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' (1) to calculate the event weight at NLO accuracy for each event ⃗x using d4σNLO dk⊥ 1 dφ1 dη1 dη2 = d4σLO dk⊥ 1 dφ1 dη1 dη2 + � d7σHard dk⊥ 1 dφ1 dη1 dη2 dk⊥ 3 dφ3 dη3 dk⊥ 3 dφ3 dη3 + d4σvirtual dk⊥ 1 dφ1 dη1 dη2 + d4σsoft/collinear dk⊥ 1 dφ1 dη1 dη2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' (5) The weights calculated in this way can also be used to gener- ate unweighted events with, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=', the von-Neumann acception- rejection method ([38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 5 shows the distribution of the unweighted events compared to kinematic distributions ob- tained with the madgraph5 aMC@NLO code [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The events obtained from the event weights defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' (5) are within the uncertainties in perfect agreement with the predictions ob- tained using madgraph5 aMC@NLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In appendix A, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 12 we show in addition the calculation of the Mt¯t- and the φ1- distribution with the same perfect agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The compari- son of the generated unweighted events with the results from madgraph5 aMC@NLO also serves as a further validation for the choice of the slicing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' APPLICATION: DETERMINATION OF THE TOP-QUARK MASS USING THE MEM AT NLO The event weights defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' (5) can be used in the MEM to calculate the likelihood at NLO accuracy for a given sample of N events {⃗xi}, i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=',N: L�{⃗xi} | mt � = 1 (σNLO(mt))N N � i=1 d4σNLO(mt) dk⊥ 1 dφ1 dη1 dη2 ������⃗x=⃗xi (6) where the dependence of the total and differential cross sections on the value of the top-quark mass is high- lighted—exemplarily for generic model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Here, the so-called transfer functions, parametrizing the probability of measuring a certain signal in the detector given a particular partonic configuration, are set to delta functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The trans- fer functions account for particle decays, additional radiation as well as detector effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Thus, this choice for the transfer functions corresponds to the assumption of a perfect detector which allows a perfect unfolding from the detector signals to partonic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' While for variables related to angles, set- ting the transfer function to delta function may give a reason- able approximation, this is not necessarily true in case of vari- ables sensitive to energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In future applications non-trivial transfer functions should thus be incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' This may be done using invertible neural networks trained to a full simula- tion as discussed in great detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' This is however beyond the scope of this work which focuses on exploring the potential of the method for top-quark mass measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Maximizing the likelihood with respect to the parameter mt yields an estimator for the top-quark mass ˆmt: L�{⃗xi} | ˆmt � = max mt �L�{⃗xi} | mt �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' (7) Because the event weights in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' (6) are normalized to yield probabilities, the MEM is only sensitive to the shapes of kine- matic distributions but not to the total number of events in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' To also benefit from the information of the total event number the so-called extended likelihood can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The extended likelihood is obtained from the likelihood in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' (7) by multiplying with the Poisson probability for observing N events when the expected number of events is given by the total cross section times the integrated luminosity Lint of the collider: Lext �{⃗xi} | mt � = (σNLO(mt) Lint)N N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' e−σNLO(mt) Lint L�{⃗xi} | mt �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' (8) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 6 we show the likelihood obtained analysing 9900 unweighted top-quark pair events distributed according to the NLO prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Likelihood (upper plot) as well as the ex- tended likelihood (lower plot) have been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The green curves correspond to likelihoods calculated at NLO accuracy 5 160 165 170 175 180 185 190 m [GeV] −50 −25 0 25 50 75 100 125 150 − log � L(m) Lmax � mtrue = 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='2 GeV, 9900 analysed events LO prediction: ˆm2µ0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5µ0 = 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='77 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='18+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='21 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='66GeV NLO prediction: ˆm2µ0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5µ0 = 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='65 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='20+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='30 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='17GeV 140 150 160 170 180 190 m [GeV] −100 0 100 200 300 400 500 600 700 800 − log � Lext(m) Lext,max � mtrue = 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='2 GeV, 9900 analysed events LO prediction: ˆm2µ0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5µ0 = 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='34−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='97 +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='86GeV NLO prediction: ˆm2µ0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5µ0 = 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='36−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='11 +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='69GeV FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Analysis of unweighted events following the fixed-order NLO prediction with (extended) likelihoods calculated at LO and NLO accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' ˆmt ±∆stat ∆ 2µ0 sys ∆ µ0/2 sys [GeV] likelihood LO prediction NLO prediction L 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='77±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='18+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='21 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='66 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='65±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='20+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='30 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='17 Lext 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='22±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='34−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='97 +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='86 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='68±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='36−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='11 +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='69 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Extracted values for the estimator of the top-quark mass from 9900 unweighted events following the fixed-order NLO predic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' using different choices for the factorization and renormaliza- tion scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The orange curves are obtained using only LO predictions again for different scale settings in the likelihood calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The analysed events are generated for an input value of the top-quark mass of mtrue = 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='2 GeV and the scale choice µF = µR = µ0 = mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The extracted values for the estima- tor of the top-quark mass together with statistical and system- atic uncertainties are summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The estimators ˆmt are determined from the minima of the parabolas fitted to the negative logarithms of the likelihood functions while the sta- tistical uncertainties ∆stat are estimated from their widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The systematic uncertainties ∆2µ0 sys , ∆µ0/2 sys are estimated by varying the scale by a factor 2 around µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' As can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 6 and Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' I, both the NLO and the LO analyses have similar statistical uncertainties of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='2 GeV and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='35 GeV de- pending on whether the likelihood or the extended likelihood is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' As expected, the statistical uncertainties are to good approximation independent from the perturbative order of the theoretical predictions of the cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Taking the statistical uncertainties into account, the extracted estimators from the NLO analyses are in perfect agreement with the input value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' For the likelihood as well as for the extended likelihood the NLO differential cross section matches the probability dis- tribution underlying the event sample thus leading to an unbi- ased estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Obviously, taking into account the information on the total number of events via the extended likelihood leads to a reduction of the statistical uncertainties as additional in- formation contained in the event sample is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Since the cross section shows a much stronger residual scale depen- dence than the normalized distributions, the extended likeli- hood leads however to a significantly larger systematic un- certainty due to uncalculated higher order corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In ad- dition, the uncertainty of the luminosity measurement which is not taken into account in the extended likelihood analysis leads to an additional uncertainty outweighing the gain in the reduced statistical uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The estimators from the LO analyses on the other hand show a bias of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='9×∆stat and 38×∆stat depending on whether the likelihood or the extended likelihood is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' It should be emphasized that the occurrence of a bias per se does not rule out the application of the MEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' It is well known, that the MEM typically leads to a bias if the probability distribu- tion used in the evaluation of the likelihood does not match the distribution underlying the event sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' However, via a calibration procedure it is possible to compensate the bias and obtain an unbiased determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Since the calibration can introduce additional uncertainties the preferred situation is that the probability distribution used in the likelihood de- termination matches the probability distribution of the event sample as best as possible thus reducing the need of addi- tional calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' As shown in section II, the NLO correc- tions dominantly alter the normalization of the kinematic dis- tributions rather than their shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Accordingly, the analysis employing extended likelihoods which is sensitive to the total cross section shows thus a much stronger separation between the results obtained from the NLO and LO predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Significant improvement from taking NLO corrections into account can be seen in their impact on the theoretical uncer- tainties: In the NLO analyses the theoretical uncertainties due to uncalculated higher order corrections are roughly halved with respect to the LO analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In order to further study the robustness of the approach and having a more realistic simulation, unweighted events ob- tained from a parton shower simulation matched to the NLO calculation can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The parton shower resums certain logarithmic corrections to all orders on top of the fixed-order NLO parton level calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Since these additional correc- tions present in the event sample are not accounted for in the fixed-order-only likelihood calculation based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' (5), there is a mismatch between the underlying probability distribution of the generated events and the basis of the likelihood calcula- tion (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' (5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' As seen before in case of the LO analysis, this 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='06 1 σNLO dσNLO dk⊥ 1 [GeV−1] POWHEG events fixed-order NLO 0 100 200 300 400 500 k⊥ 1 [GeV] 0 1 2 POWHEG fixed-order NLO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='08 1 σNLO dσNLO dη1 POWHEG events fixed-order NLO −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 η1 0 1 2 POWHEG fixed-order NLO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='06 1 σNLO dσNLO dy POWHEG events fixed-order NLO −4 −2 0 2 4 y 0 1 2 POWHEG fixed-order NLO FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Impact of the parton shower on the kinematic distributions of the top quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' mismatch can cause a systematic bias in the extracted estima- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 7 shows the distributions obtained using POWHEG+Pythia [27–31] to generate about the same number of events as in the case of the fixed-order analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The parton shower only mildly affects the kinematic distribu- tions relevant for the event definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Further distributions supporting this observation are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 13 in the ap- pendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Apart from minor differences in the k⊥ 1 distribution at low k⊥ 1 , a small difference is visible for k⊥ 1 > 300 GeV, where the POWHEG+Pythia events lead to a slightly harder distribution than the events generated from the fixed order NLO cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 160 165 170 175 180 185 190 m [GeV] 0 100 200 300 400 − log � L(m) Lmax � mtrue = 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='2 GeV, 9232 analysed events LO prediction: ˆm2µ0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5µ0 = 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='88 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='22+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='13 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='57GeV NLO prediction: ˆm2µ0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5µ0 = 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='93 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='24+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='22 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='38GeV FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Analysis of 9232 POWHEG+Pythia events with fixed-order likelihoods calculated at LO and NLO accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' ˆmt ±∆stat ∆ 2µ0 sys ∆ µ0/2 sys [GeV] likelihood LO prediction NLO prediction L 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='88±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='22+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='13 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='57 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='93±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='24+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='22 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='38 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Extracted values for the estimator of the top-quark mass from unweighted POWHEG+Pythia events following the NLO predic- tion matched to a parton shower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The result of the likelihood analysis using LO and NLO cross section predictions is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 8 and summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' We do not study the extended likelihood, since the extended likelihood leads to much larger systematic uncer- tainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Again the statistical uncertainties are very similar for the LO and NLO analysis, while the systematic uncertainty is significantly reduced when using NLO predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In both cases we observe a shift of about 4 GeV compared to the re- sults based on the event sample generated from the fixed-order NLO predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The large shift shows the high sensitivity of the MEM with respect to tiny changes in the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In a mass determination from events registered at the LHC this shift must be taken into account via a calibration procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' It is remarkable that the shift is, taking the uncertainties into account, independent from the perturbative order of the em- ployed likelihood calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' This is similar to what has been observed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The LO likelihood analysis repro- duces the true mass value used in the POWHEG+Pythia analy- sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' However, this ist most likely accidental and due to the fact that the LO fixed-order results undershoots the true mass value by about 4 GeV which is compensated by the aforementioned shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' CONCLUSION In this work the MEM at NLO is applied to top-quark pair production at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' To investigate the potential of the matrix element method to measure the top-quark mass, the MEM at NLO is applied to pseudo-data: unweighted 7 events generated from the fixed-order NLO cross section as well as events obtained using POWHEG+Pythia incorpo- rating the parton shower effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Using pseudo-data based on POWHEG+Pythia allows to study the effect of the parton shower and gives a more realistic simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Including the NLO corrections in the likelihood calculation leads to a signif- icant reduction of the theoretical uncertainties of the extracted top-quark mass, while the statistical uncertainties remain al- most unchanged compared to the LO analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' We stress that the uncertainties due to scale variation cannot be reduced by a calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The reduction of the uncertainties associated with the scale variation when going from LO to NLO thus presents an important improvement and a strong argument in favour of the the MEM at NLO accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Another important observation is the fact that the extended likelihood yields a significant improvement in terms of the sta- tistical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' However, in practical applications this gain in precision is completely outweighed by the theoretical uncertainties of the number of expected events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' This can be understood from the fact that, much as the NLO corrections (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 3), the scale variations do not dramatically change the shape of the kinematic distributions but mostly their nor- malization (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 4) thereby making the extended likeli- hood analyses more sensitive to their effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Additionally, em- ploying the extended likelihood requires precise knowledge of the integrated luminosity of the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The dependence on this parameter introduces an additional source of systematic un- certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' This has to be taken into account for future experi- mental applications of the MEM with realistic event numbers for abundantly produced top-quark pairs at the LHC which will most likely be dominated by systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' As has already been stated before ([23–26]), for parameter infer- ence with the MEM it is mandatory to perform the likelihood calculation at least at NLO accuracy in order to properly fix the renormalization scheme of the extracted parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The application of the MEM at NLO to top-quark pair events at the LHC can offer an alternative approach to deter- mine the top-quark mass with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' As has been demonstrated in this work, already for a few ten thousand events the precision of the analysis becomes dominated by systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' As the LHC produces millions of top-quark pairs, the analysis could be performed with a rather small fraction of cherry-picked events allowing to minimize the overall systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The results obtained in this article suggest that top-quark mass determination with an un- certainty below 1 GeV could be feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Of course, for a re- alistic application of the MEM to experimental data, transfer functions accounting for decays, additional radiation and de- tector effects have to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' In addition, as the analy- sis based on the events including parton shower effects shows, a further calibration is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported in part by the Bundesministerium für Bildung and Forschung under contract 05H18KHCA1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 0 10 20 30 40 50 60 dσNLO dE1 [pb GeV−1] result reference 0 200 400 600 800 1000 E1 [GeV] −2σ −σ σ 2σ pull 0 10 20 30 40 50 dσNLO dη1 [pb] result reference −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 η1 −2σ −σ σ 2σ pull FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Validation of the implementation: Comparison of differential distributions of the top quark obtained in this work with results from madgraph5 aMC@NLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Appendix A: Additional results on distributions used for the validation In this appendix we show further cross checks used for the validation of the implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 9 shows comparisons of NLO predictions for differential distributions calculated in this work with distributions obtained from madgraph5 aMC@NLO [37] which is based on the dipole subtraction method [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' The pull distributions in the bottom plots of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 9 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 10 illustrate the agreement between both implementations within statistical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' This compar- isons serve as a further validation for the choice of the slicing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 11 shows the NLO corrections (upper part) together with the k-factors (lower part) for the Mt¯t and the φ1- distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Similar to what is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 3 again a flat k- factor is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' As a check of the event generation and the unweighting procedure Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 12 shows distributions calculated from the generated unweighted events compared with a calcu- lation using madgraph5 aMC@NLO [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 7 we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 13 for further distributions the comparison of dis- tributions obtained at fixed-order NLO accuracy with results using POWHEG+Pythia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 8 0 5 10 15 dσNLO dφ1 [pb] result reference −4 −3 −2 −1 0 1 2 3 4 φ1 −2σ −σ σ 2σ pull 0 10 20 30 40 dσNLO dk⊥ 1 [pb GeV−1] result reference 0 100 200 300 400 500 k⊥ 1 [GeV] −2σ −σ σ 2σ pull FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 9 but for the φ1- and the k⊥ 1 -distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 0 10 20 30 40 dσ dMt¯t [pb GeV−1] dσNLO dσLO 300 400 500 600 700 800 900 1000 Mt¯t [GeV] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 dσNLO dσLO 0 5 10 15 20 dσ dφ1 [pb] dσNLO dσLO −4 −3 −2 −1 0 1 2 3 4 φ1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0 dσNLO dσLO FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 3 but for the Mt¯t and the φ1-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='06 1 σNLO dσNLO dMt¯t [pb GeV−1] results reference 300 400 500 600 700 800 900 1000 Mt¯t[GeV] −2σ −σ σ 2σ pull 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='025 1 σNLO dσNLO dφ1 [pb] results reference −4 −3 −2 −1 0 1 2 3 4 φ1 −2σ −σ σ 2σ pull FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 5 but for the Mt¯t- and the φ1-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='10 1 σNLO dσNLO dE1 [GeV−1] POWHEG events fixed-order NLO 0 200 400 600 800 1000 E1 [GeV] 0 1 2 POWHEG fixed-order NLO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='06 1 σNLO dσNLO dMt¯t [GeV−1] POWHEG events fixed-order NLO 300 400 500 600 700 800 900 1000 Mt¯t [GeV] 0 1 2 POWHEG fixed-order NLO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='030 1 σNLO dσNLO dφ1 POWHEG events fixed-order NLO −4 −3 −2 −1 0 1 2 3 4 φ1 0 1 2 POWHEG fixed-order NLO FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 7 but for the energy, Mt¯t- and φ1-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 10 [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Nason, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Dawson, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Ellis, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' B 303, 607 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [2] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Nason, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Dawson, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Ellis, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' B 327, 49 (1989), [Erratum: Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='B 335, 260–260 (1990)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [3] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Beenakker, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Kuijf, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' van Neerven, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Smith, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' D 40, 54 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [4] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Beenakker, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' van Neerven, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Meng, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Schuler, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Smith, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' B 351, 507 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [5] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Bernreuther, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Brandenburg, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Si, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Uwer, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' B 690, 81 (2004), arXiv:hep-ph/0403035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [6] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Melnikov and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Schulze, JHEP 08, 049 (2009), arXiv:0907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='3090 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Czakon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Fiedler, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Mitov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 110, 252004 (2013), arXiv:1303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='6254 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Czakon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Heymes, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Mitov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 116, 082003 (2016), arXiv:1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='00549 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Czakon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Fiedler, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Heymes, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Mitov, JHEP 05, 034 (2016), arXiv:1601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='05375 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Beneke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Czakon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Falgari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Mitov, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Schwinn, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' B690, 483 (2010), arXiv:0911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5166 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Czakon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Mitov, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Sterman, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' D80, 074017 (2009), arXiv:0907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='1790 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Beneke, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Falgari, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Klein, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Schwinn, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' B855, 695 (2012), arXiv:1109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='1536 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Cacciari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Czakon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Mangano, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Mitov, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Nason, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' B710, 612 (2012), arXiv:1111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5869 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [14] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Kidonakis, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 45, 714 (2014), arXiv:1210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='7813 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Ferroglia, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Pecjak, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Yang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' D86, 034010 (2012), arXiv:1205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='3662 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Ferroglia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Marzani, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Pecjak, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Yang, JHEP 01, 028 (2014), arXiv:1310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='3836 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Czakon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Ferroglia, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Heymes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Mitov, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Pecjak, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Scott, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Wang, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Yang, JHEP 05, 149 (2018), arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='07623 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [18] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Beenakker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Denner, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Hollik, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Mertig, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Sack, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Wackeroth, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' B411, 343 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [19] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Bernreuther, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Fuecker, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Si, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' D74, 113005 (2006), arXiv:hep-ph/0610334 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Kühn, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Scharf, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Uwer, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' C51, 37 (2007), arXiv:hep-ph/0610335 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [21] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Moretti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Nolten, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Ross, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' B639, 513 (2006), [Erratum: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='B660,607(2008)], arXiv:hep- ph/0603083 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [22] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Pagani, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Tsinikos, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Zaro, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' C76, 479 (2016), arXiv:1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='01915 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [23] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Martini and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Uwer, JHEP 09, 083 (2015), arXiv:1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='08798 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [24] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Martini and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Uwer, (2017), arXiv:1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='04527 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Kraus, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Martini, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Uwer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' D 100, 076010 (2019), arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='08008 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [26] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Kraus, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Martini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Peitzsch, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Uwer, (2019), arXiv:1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='09100 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [27] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Nason, JHEP 11, 040 (2004), arXiv:hep-ph/0409146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [28] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Sjostrand, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Mrenna, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Skands, JHEP 05, 026 (2006), arXiv:hep-ph/0603175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [29] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Frixione, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Nason, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Oleari, JHEP 11, 070 (2007), arXiv:0709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='2092 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [30] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Frixione, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Nason, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Ridolfi, JHEP 09, 126 (2007), arXiv:0707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='3088 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [31] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Alioli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Nason, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Oleari, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Re, JHEP 06, 043 (2010), arXiv:1002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='2581 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [32] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Giele, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Glover, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Kosower, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' B403, 633 (1993), arXiv:hep-ph/9302225 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [33] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Badger, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Sattler, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Yundin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' D 83, 074020 (2011), arXiv:1101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='5947 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [34] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Kinoshita, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 3, 650 (1962).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [35] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Lee and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Nauenberg, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 133, B1549 (1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [36] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Aliev, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Lacker, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Langenfeld, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Moch, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Uwer, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=', Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 182, 1034 (2011), arXiv:1007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='1327 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [37] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Alwall, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Frederix, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Frixione, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Hirschi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Maltoni, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Mattelaer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Shao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Stelzer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Torrielli, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Zaro, JHEP 07, 079 (2014), arXiv:1405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='0301 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [38] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' von Neumann, in Monte Carlo Method, National Bureau of Standards Applied Mathematics Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 12, edited by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Householder, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Forsythe, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Germond (US Gov- ernment Printing Office, Washington, DC, 1951) Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 13, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' 36–38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [39] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Butter, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Heimel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Martini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Peitzsch, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Plehn, (2022), arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='00019 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [40] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Catani and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Seymour, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' B485, 291 (1997), arXiv:hep-ph/9605323 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' [41] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Catani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Dittmaier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Seymour, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' Trocsanyi, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} +page_content=' B627, 189 (2002), arXiv:hep-ph/0201036 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E1T4oBgHgl3EQflAR-/content/2301.03280v1.pdf'} diff --git a/0NE1T4oBgHgl3EQfRQO7/content/tmp_files/2301.03051v1.pdf.txt b/0NE1T4oBgHgl3EQfRQO7/content/tmp_files/2301.03051v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c673ba6ed22c3f0302cdf41d7f057a78b543f65f --- /dev/null +++ b/0NE1T4oBgHgl3EQfRQO7/content/tmp_files/2301.03051v1.pdf.txt @@ -0,0 +1,1086 @@ +arXiv:2301.03051v1 [math.RA] 8 Jan 2023 +FUNCTORS BETWEEN REPRESENTATION CATEGORIES. +UNIVERSAL MODULES +A. L. AGORE +Abstract. Let g and h be two Lie algebras with h finite dimensional and consider +A = A(h, g) to be the corresponding universal algebra as introduced in [4]. Given an +A-module U and a Lie h-module V we show that U ⊗ V can be naturally endowed +with a Lie g-module structure. This gives rise to a functor between the category of Lie +h-modules and the category of Lie g-modules and, respectively, to a functor between +the category of A-modules and the category of Lie g-modules. +Under some finite +dimensionality assumptions, we prove that the two functors admit left adjoints which +leads to the construction of universal A-modules and universal Lie h-modules as the +representation theoretic counterparts of Manin-Tambara’s universal coacting objects +[11, 16]. +Introduction +The universal coacting bialgebra/Hopf algebra on a finite dimensional (graded) asso- +ciative algebra originates in the work of Yu. I. Manin ([11]). The importance of this +construction became obvious mostly due to its interaction with non-commutative geom- +etry where it is seen as some sort of symmetry group (see [13] for more details on this +view point). The non-graded version of this construction appeared a few years later in a +paper by D. Tambara ([16]). However, as remarked in [16], the universal coacting bialge- +bra is in fact the dual of the so-called universal measuring bialgebra introduced by M.E. +Sweedler in [15]. We should note that, unlike Manin-Tambara’s construction, Sweedler’s +universal measuring bialgebra/Hopf algebra exists even in the infinite-dimensional case. +In recent years, universal (co)acting objects have been considered in various settings +and for different purposes. For instance, [8] extends Sweedler’s construction to monoids +in a braided monoidal category. On the other hand, the Manin-Tambara construction +was introduced in the setting of Poisson algebras ([2]), finite index-subfactors ([6]), su- +perpotential algebras ([7]), polynomial algebras ([14]), bialgebroids ([5]) or Lie/Leibniz +algebras ([4]). The corresponding universal coacting bialgebras/Hopf algebras, which in +certain cases carry some extra structure (e.g. a Poisson Hopf algebra structure as in [2]), +seem to play a prominent role in solving other seemingly unrelated problems such as +the classification of gradings on various kinds of algebras ([4, 12]), the description of the +automorphisms group of certain algebraic structures ([4]) and even in quantum Galois +2010 Mathematics Subject Classification. 16D90, 16T05, 17A32, 17B10. +Key words and phrases. universal module. +This work was supported by a grant of Romanian Ministry of Research, Innovation and Digitization, +CNCS/CCCDI – UEFISCDI, project number PN-III-P4-ID-PCE-2020-0458, within PNCDI III. +1 + +2 +A. L. AGORE +theory ([6]). Another related universal (co)acting construction was considered in [3] as +the Hopf algebraic analogue of the universal group of a grading and its connections to +the problem of classifying Hopf algebra coactions have been highlighted. +One of the most general constructions of universal (co)acting bialgebras/Hopf algebras, +performed in the setting of Ω-algebras, was introduced in [1] together with generalized +duality results. Necessary and sufficient conditions for the existence of universal coacting +bialgebras/Hopf algebras are provided, explaining in this general setting the need for +assuming finite dimensionality in both Manin and Tambara’s papers. +It is worth to point out that both Sweedler and Manin-Tambara’s constructions have +a categorical interpretation. More precisely, for Tambara’s construction one considers +the left adjoint, say a(A, −), of the tensor product endofunctor A ⊗ − on the category +of k-algebras, where A is a finite dimensional associative algebra. Tambara’s universal +coacting bialgebra is precisely a(A, A) which turns out to be naturally endowed with a +bialgebra structure. Similarly, for an arbitrary associative algebra A, it can be proved +that the contravariant functor Hom(−, A) taking coalgebras to (convolution) algebras +has a right adjoint which hereafter we denote by M(A, −). As before, Sweedler’s uni- +versal measuring bialgebra is exactly M(A, A) which again has a bialgebra structure. +In this paper we deal with the representation theoretic version of Manin-Tambara’s con- +struction in the Lie algebra setting. Our approach is a categorical one. More precisely, +given two fixed Lie algebras g and h, with h finite dimensional, and the corresponding +universal algebra A = A(h, g) (see[4]), we first show that the tensor product between +an A-module U and a Lie h-module V can be endowed with a Lie g-module structure +(Theorem 2.1). As a consequence, we are able to construct two ”tensor product” func- +tors between the categories of Lie modules over h and g and respectively between the +category of A-modules and the category of Lie g-modules. Under the appropriate finite +dimensionality assumptions, the two functors mentioned above are proved to admit left +adjoints. These left adjoints are given precisely by what we have called the universal +Lie h-module and the universal A-module, respectively. The two universal modules are +introduced in a constructive manner in Theorem 2.4 and Theorem 2.10. These are the +counterparts for Lie and associative representations of Manin-Tambara’s constructions. +Furthermore, the two aforementioned pairs of adjoint functors allow us to travel both +ways between the representation categories of different algebraic structures, such as Lie +and associative algebras, and to transfer certain properties which are usually preserved +by left/right adjoints. +1. Preliminaries +This section will be used mostly as an opportunity to fix some notation and to provide +certain useful references. Let us start with a few words on notation. +1.1. Notational conventions. All vector spaces, (bi)linear maps, unadorned tensor +products, Lie or associative algebras, bialgebras and so on are over an arbitrary com- +mutative field k. All (co)associative (co)algebras are assumed to be (co)unital. +The +notation employed for coalgebras is standard: ∆ stands for the comultiplication and ε + +UNIVERSAL MODULES +3 +for the counit. We use Sweedler’s notation with implied summation for both coalge- +bras (resp. bialgebras), as in ∆(c) = c(1) ⊗ c2, and for comodule structures: a right +C-comodule structure ρ on a vector space V will be denoted by ρ(v) = v(0) ⊗v(1). When +we need to be precise, the structures involved will be adorned. δij denotes Kronecker’s +symbol while IdX stands for the identity map on the set X. +In the sequel, k[Xsi |s = 1, · · · , n, i ∈ I] denotes the usual polynomial algebra on vari- +ables Xsi. We shall denote by Liek and ComAlgk the categories of Lie and commutative +associative algebras, respectively. Given an associative algebra A and a Lie algebra g +we denote by AM and gLM the categories of left A-modules and left Lie g-modules, re- +spectively. Recall that a (left) Lie g-module is a vector space V equipped with a bilinear +map ⇀: g × V → V such that for all x, y ∈ g and v ∈ V we have: +[x, y] ⇀ v = x ⇀ (y ⇀ v) − y ⇀ (x ⇀ v). +Throughout the paper, g and h will denote two arbitrary Lie algebras with h finite +dimensional. Let {fi | i ∈ I} and {e1, · · · , en} be two fixed basis in g and h, respectively. +We consider {τ s +i,j | i, j, s = 1, · · · , n} to be the structure constants of h, i.e. for any i, +j = 1, · · · , n we have: +[ei, ej]h = +n +� +s=1 +τ s +i,j es. +(1) +Similarly, for any i, j ∈ I, let Bi,j ⊆ I be a finite subset of I such that for any i, j ∈ I +we have: +[fi, fj]g = +� +u∈Bi,j +βu +i,j fu. +(2) +1.2. The universal algebra of h and g. We recall briefly, for further use, the con- +struction of the universal commutative algebra A(h, g) of two given Lie algebras h and +g (recall that h is always assumed to be finite dimensional). It was first introduced in +[4] in the more general setting of Leibniz algebras as the counterpart of Tambara’s con- +struction ([16]). We restrict here to the Lie algebra version of the construction which +can be summarized as follows. We have: +A(h, g) := k[Xsi |s = 1, · · · , n, i ∈ I]/J +(3) +where J is the ideal generated by all polynomials of the form +P (h, g) +(a,i,j) := +� +u∈Bi,j +βu +i,j Xau − +n +� +s,t=1 +τ a +s,t XsiXtj, +for all a = 1, · · · , n and i, j ∈ I. +(4) +When working in the universal algebra A(h, g), we denote by xsi := � +Xsi the class of Xsi. +Consequently, the following relations hold in A(h, g): +� +u∈Bi,j +βu +i,j xau = +n +� +s,t=1 +τ a +s,t xsixtj, +for all a = 1, · · · , n, and i, j ∈ I. +(5) +When the (finite dimensional) Lie algebra h is fixed, the universal algebra construction +gives rise to a functor A(h, −): Liek → ComAlgk which turns out to be the left adjoint + +4 +A. L. AGORE +of the tensor product h ⊗ −: ComAlgk → Liek (see [4, Theorem 2.1]), where for any +commutative algebra X the tensor product h ⊗ X is endowed with the current Lie +algebra structure. In order to avoid dealing with cumbersome notation, when there is no +fear of confusion, we denote A = A(h, g). Furthermore, If h = g, then the corresponding +universal algebra A(h, h) will be denoted simply by B. The notation is meant to highlight +the fact that B is a bialgebra; in fact, it admits a unique bialgebra structure such that h +becomes a right B-comodule with respect to ηh : h → h⊗B where η: 1Liek → h⊗A(h, −) +denotes the unit of the adjunction between A(h, −) and h ⊗ −. +More precisely, the +comultiplication and the counit on B are given for any i, j = 1, · · · , n by +∆(xij) = +n +� +s=1 +xis ⊗ xsj +and +ε(xij) = δi,j1k +(6) +For basic categorical concepts we refer the reader to [10] and for unexplained notions +pertaining to Lie and Hopf algebras to [9] and [15], respectively. +2. Universal modules +Our first important result provides a way of defining a Lie g-module structure on the +tensor product between a Lie h-module and an A-module. +Theorem 2.1. Let (U, ↷) ∈ hLM be a Lie h-module and (V, ·) ∈ AM an A-module. +Then (U ⊗ V, ⇀) ∈ gLM is a Lie g-module where the action of g on U ⊗ V is given for +all i ∈ I, l ∈ U and t ∈ V by: +fi ⇀ (l ⊗ t) = +n +� +j=1 +(ej ↷ l) ⊗ (xji · t) +(7) +Proof. Indeed, having in mind that (U, ↷) is a Lie module and A = A(h, g) is a com- +mutative algebra, we have: +[fi, fj] ⇀ (l ⊗ t) +(2) += +� +u∈Bi,j +βu +i,j fu ⇀ (l ⊗ t) +(7) += +� +u∈Vi,j,r=1,n +βu +i,j (er ↷ l) ⊗ (xru · t) += +� +r=1,n +(er ↷ l) ⊗ +� � +u∈Bi,j +βu +i,j xru +� +·t +(5) += +� +s,p,r=1,n +τ r +s,p (er ↷ l) ⊗ (xsixpj) · t += +� +s,p=1,n +� n +� +r=1 +τ r +s,p er +� +↷ l ⊗ (xsixpj) · t +(1) += +� +s,p=1,n +[es, ep] ↷ l ⊗ (xsixpj) · t += +� +s,p=1,n +es ↷ (ep ↷ l) ⊗ xsi · (xpj · t) − +� +s,p=1,n +ep ↷ (es ↷ l) ⊗ xpj · (xsi · t) +(7) += fi ⇀ +n +� +p=1 +(ep ↷ l) ⊗ (xpj · t) − fj ⇀ +n +� +s=1 +(es ↷ l) ⊗ (xsi · t) +(7) += fi ⇀ +� +fj ⇀ (l ⊗ t) +� +− fj ⇀ +� +fi ⇀ (l ⊗ t) +� + +UNIVERSAL MODULES +5 +for all i, j ∈ I and l ∈ U, t ∈ V , i.e. (U ⊗ V, ⇀) is a left Lie g-module. +□ +Inspired by Theorem 2.1 we can consider two types of universal modules. +2.1. The universal A-module. The first such universal module is associated with a +Lie h-module and a Lie g-module as follows: +Definition 2.2. Given a Lie h-module U and a Lie g-module Z, the universal A-module +of U and Z is a pair +� +U(U, Z), ρU(U, Z) +� +consisting of an A-module U(U, Z) and a mor- +phism of Lie g-modules ρU(U, Z) : Z → U ⊗ U(U, Z) such that for any other pair (X, f) +consisting of an A-module X and a morphism of Lie g-modules f : Z → U ⊗X, there ex- +ists a unique morphism of A-modules g: U(U, Z) → X such that the following diagram +is commutative: +Z +ρU(U, Z) +� +f +�❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +U ⊗ U(U, Z) +IdU⊗g +� +U ⊗ X +(8) +In other words, the above definition is saying that, when it exists, the universal A-module +of U and Z is in fact the initial object of the category whose objects are pairs (X, f) +consisting of an A-module X and a morphism of Lie g-modules f : Z → U ⊗ X, while +morphisms between two such objects (X, f) and (X′, f ′) are defined to be A-module +maps g: X → X′ satisfying (IdU ⊗ g) ◦ f = f ′. +As direct consequences of the above definition, we obtain the following: +Corollary 2.3. Let U be a Lie h-module. Then, for all Lie g-modules Z and all A- +modules X, we have a bijective correspondence between: +(1) Lie g-module maps f : Z → U ⊗ X; +(2) A-module maps g: U(U, Z) → X. +Under the appropiate finite-dimensionality assumptions required for all Manin-Tambara +type constructions, the universal A-module introduced in Definition 2.2 exists: +Theorem 2.4. If U is a finite dimensional Lie h-module then the universal A -module +of U and any other Lie g-module Z exists. +Proof. Let {u1, · · · , um}, m ∈ N∗, be a k-basis of the Lie module U and denote by ωt +ij ∈ k +the structure constants of U with respect to its Lie h-module structure ↷, i.e. for all +i = 1, · · · , n, j = 1, · · · , m we have: +ei ↷ uj = +m +� +s=1 +ωs +i,j us +(9) +Furthermore, consider {zr | r ∈ J} to be a k-basis for the arbitrary Lie g-module Z and +if ↬ denotes its Lie module structure, then for all j ∈ I and r ∈ J we can find a finite + +6 +A. L. AGORE +subset Tj,r of J such that: +fj ↬ zr = +� +l∈Tj,r +ηl +j,r zl +(10) +where ηl +j,r ∈ k for all j ∈ I, r ∈ J, and l ∈ Tj,r. +Consider now T (U, Z) to be the free A-module on the set {Yij | i = 1, · · · , m, j ∈ J} +and denote by U(U, Z) the quotient of T (U, Z) by its A-submodule generated by the +following elements: +� +p∈Tj,i +ηp +j,i Ysp − +m +� +t=1 +n +� +r=1 +ωs +r,t xrj • Yti +(11) +for all s = 1, · · · , m, i ∈ J and j ∈ I, where • denotes the A-module action on T (U, Z). +Denoting ytj := � +Ytj, where � +Ytj stands for the equivalence class of Ytj in the quotient +module U(U, Z), it follows that the relations below hold in U(U, Z): +� +p∈Tj,i +ηp +j,i ysp = +m +� +t=1 +n +� +r=1 +ωs +r,t xrj • yti +(12) +for all s = 1, · · · , m, i ∈ J and j ∈ I. +Furthermore, we can define a morphism of Lie g-modules ρU(U, Z): Z → U ⊗ U(U, Z) as +follows: +ρU(U, Z)(zr) := +m +� +s=1 +us ⊗ ysr, +for all r ∈ J. +(13) +It follows now that for all j ∈ I and i ∈ J we have: +ρU(U, Z)(fj ↬ zi) +(10) += ρU(U,Z) +� � +p∈Tj,i +ηp +ji zp +� += +� +p∈Tj,i +m +� +s=1 +ηp +ji us ⊗ ysp = +m +� +s=1 +� +us ⊗ +� +p∈Tj,i +ηp +ji ysp +� +(12) += +m +� +s,t=1 +n +� +r=1 +ωs +r,t us ⊗ xrj • yti = +m +� +t=1 +n +� +r=1 +� m +� +s=1 +ωs +r,t us +� +⊗ xrj • yti +(9) += +m +� +t=1 +n +� +r=1 +er ↷ ut ⊗ xrj • yti +(7) += +m +� +t=1 +fj ⇀ (ut ⊗ yti) = fj ⇀ +m +� +t=1 +ut ⊗ yti +(13) += fj ⇀ ρU(U, Z)(zi) +which shows that ρU(U, Z) is indeed a Lie g-modules map. +We will show that the pair +� +U(U, Z), ρU(U, Z) +� +constructed above is in fact the universal +A-module of U and Z. To this end, consider a pair (X, f) consisting of an A-module X +and a morphism of Lie g-modules f : Z → U ⊗ X. Let {wsr | s = 1, · · · , m, r ∈ J} be a +family of elements of X such that for all r ∈ J we have: +g(zr) = +m +� +s=1 +us ⊗ wsr +(14) + +UNIVERSAL MODULES +7 +Furthermore, as g: Z → U ⊗ X is a Lie g-modules map, a straightforward computation +shows that the following compatibilities hold for all s = 1, · · · , m, i ∈ J and j ∈ I: +� +p∈Tj,i +ηp +j,i wsp = +m +� +t=1 +n +� +r=1 +ωs +r,t xrj · wti +(15) +where · denotes the A-module action on X. +The universal property of the free module yields a unique A-module map g: T (U, Z) → X +such that g(Ysr) = wsr, for all s = 1, · · · , m and r ∈ J. Moreover, Ker(g) contains the A- +submodule of T (U, Z) generated by the elements listed in (11). Indeed, as g : U(U, Z) → +X is a morphism of A-modules we have: +g +� � +p∈Tj,i +ηp +j,i Ysp − +m +� +t=1 +n +� +r=1 +ωs +r,t xrj • Yti +� += +� +p∈Tj,i +ηp +j,i wsp − +m +� +t=1 +n +� +r=1 +ωs +r,t xrj · wti +(15) += 0 +for all s = 1, · · · , m, i ∈ J and j ∈ I. This shows that there exists a unique A-modules +map g: U(U, Z) → X such that g(ysr) = zsr, for all s = 1, · · · , m and r ∈ J. This +implies that for all r ∈ J we have: +� +IdU ⊗ g +� +◦ ρU(U, Z)(zr) = +� +IdU ⊗ g +�� m +� +s=1 +us ⊗ ysr +� += +m +� +s=1 +us ⊗ wsr +(14) += g(zr) +which means precisely that diagram (8) is commutative. Moreover, g is obviously the +unique A-modules map with this property and the proof is now finished. +□ +The case g = h. Particularizing the results of Section 2 for g = h, where h is the finite +dimensional Lie algebra defined in (1), leads to the following interesting consequences. +According to the discussion in Preliminaries, the universal algebra A(h, h) denoted by B +is in this case a bialgebra with coalgebra structure depicted in (6). This allows us to see +the tensor product U(U, Z) ⊗ U(U, Z) as well as the base field k as B-modules via the +comultiplication and the counit of B as follows: +xij ∗ (y ⊗ t) = +n +� +t=1 +xit • y ⊗ xtj • t +(16) +xij · α = δijα +(17) +for all xij ∈ B, y, t ∈ U(U, Z) and α ∈ k, where • denotes the B-module strucuture on +U(U, Z) as in the proof of Theorem 2.4. +First we show that if U is a finite dimensional Lie h-module as considered in (9), then +the B-module U(U, U) denoted by U(U) admits a coalgebra structure with respect to +which +� +U, ρU(U) +� +becomes a right U(U)-comodule. +Proposition 2.5. Let U be a finite dimensional Lie h-module. There exists a unique +coalgebra structure on U(U) such that +� +U, ρU(U) +� +becomes a right U(U)-comodule. +Proof. In particular both U(U) ⊗ U(U) and k are B-modules via the formulas (16) and +(17) respectively. Therefore, U ⊗ U(U) ⊗ U(U) and U ⊗ k are Lie h-modules via (7). +Furthermore, it can be easily checked that the maps +� +ρU(U) ⊗ IdU(U) +� +◦ρU(U) : U → U ⊗ + +8 +A. L. AGORE +U(U) ⊗ U(U) and canU : U → U ⊗ k are morphisms of Lie h-modules, where canU : U → +U ⊗ k is the canonical isomorphism. Now Definition 2.2 yields a unique B-modules map +∆: U(U) → U(U) ⊗ U(U) such that the following diagram is commutative: +U +ρU(U) +� +� +ρU(U)⊗IdU(U) +� +◦ρU(U) +�❆ +❆ +❆ +❆ +❆ +❆ +❆ +❆ +❆ +❆ +❆ +❆ +❆ +❆ +❆ +❆ +❆ +U ⊗ U(U) +IdU ⊗∆ +� +U ⊗ U(U) ⊗ U(U) +Similarly, we obtain a unique B-modules map ε: U(U) → k such that the following +diagram is commutative: +U +ρU(U) � +canU +�■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +U ⊗ U(U) +IdU⊗ε +� +U ⊗ k +A straightforward computation shows that the commutativity of the two diagrams above +imply that ∆ and ε take the following form for all l, t = 1, · · · , m: +∆(ylt) = +m +� +s=1 +yls ⊗ yst, +ε(ylt) = δlt1k. +It is now obvious that +� +U(U), ∆, ε +� +form a coalgebra. Finally, by the commutativity of +the two diagrams above we obtain that +� +U, ρU(U) +� +is a right U(U)-comodule. +□ +Remark 2.6. It is worth pointing out that with the coalgebra structure introduced +above, U(U) becomes a B-module coalgebra. Indeed, having in mind that both ∆ and ε +are B-module maps, we have: +∆(xab • ylt) = xab ∗ ∆(ylt) = xab ∗ +� m +� +s=1 +yls ⊗ yst +�(16) += +n +� +c=1 +m +� +s=1 +xac • yls ⊗ xcb • yst += (xab)(1) • (ylt)(1) ⊗ (xab)(2) • (ylt)(2) +and +ε(xab • ylt) = xab · ε(ylt) +(17) += δab ε(ylt) = ε(xab) ε(ylt). +This shows that • is a coalgebra map, as desired. +It turns out that the pair +� +U(U), ρU(U) +� +is universal in the following way: +Proposition 2.7. For any coalgebra X with a B-module structure and any Lie h-module +morphism ψ: U → U ⊗X which makes U into a right X-comodule, there exists a unique + +UNIVERSAL MODULES +9 +B-modules and coalgebra morphism θ: U(U) → X such that the following diagram is +commutative: +U +ρU(U) � +ψ +�■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +U ⊗ U(U) +IdU ⊗θ +� +U ⊗ X +Proof. In light of Definition 2.2, such a unique A-modules map θ exists. We are left to +show that θ is also a coalgebra map. From the proof of Theorem 2.4 we know that θ is +defined for all l, t = 1, · · · , m by θ(ylt) = zlt where zlt are elements of X such that for all +r = 1, · · · , m we have ψ(ur) = �m +s=1 us ⊗ zsr. As (U, ψ) is a right comodule, we obtain: +∆(zlt) = +m +� +s=1 +zls ⊗ zst, +ε(zlt) = δlt1k. +To this end, we have: +∆ +� +θ(ylt) +� += ∆(zlt) = +m +� +s=1 +zls ⊗ zst = +m +� +s=1 +θ(yls) ⊗ θ(yst) = (θ ⊗ θ) ◦ ∆(ylt) +Similarly one can check that ε◦θ = ε which shows that θ is indeed a coalgebra map. +□ +2.2. The universal h-module. The second type of universal module one can consider +is the following: +Definition 2.8. Given an A-module V and a Lie g-module W, the universal Lie h- +module of V and W is a pair +� +V(V, W), τV(V, W ) +� +consisting of a Lie h-module V(V, W) +and a morphism of Lie g-modules τV(V, W ): W → V(V, W)⊗V such that for any other pair +(Y, f) consisting of a Lie h-module Y and a morphism of Lie g-modules f : W → Y ⊗ V , +there exists a unique morphism of Lie h-modules g: V(V, W) → Y such that the following +diagram is commutative: +W +τV(V, W ) +� +f +�❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +V(V, W) ⊗ V +g⊗IdV +� +Y ⊗ V +(18) +The universal Lie h-module of V and W, when it exists, can again be seen as the initial +object of the category whose objects are pairs (Y, f) consisting of a Lie h-module Y +and a morphism of Lie g-modules f : W → Y ⊗ V , while morphisms between two such +objects (Y, f) and (Y ′, f ′) are defined to be Lie h-module maps g: Y → Y ′ satisfying +(g ⊗ IdV ) ◦ f = f ′. +Corollary 2.9. Let V be an A-module. Then, for all Lie g-modules W and all Lie +h-modules Y , we have a bijective correspondence between: +(1) Lie g-module maps f : W → Y ⊗ V ; +(2) Lie h-module maps g: V(V, W) → Y . + +10 +A. L. AGORE +The universal h-module introduced in Definition 2.8 also exists provided that the A- +module V is finite dimensional. +Theorem 2.10. If V is a finite dimensional A-module then the universal Lie h-module +of V and any other Lie g-module W exists. +Proof. As this proof is somewhat similar in spirit with the one of Theorem 2.4, we will be +brief and provide only the main ingredients required for the construction of the universal +Lie h-module. +Let {v1, · · · , vl}, l ∈ N∗, be a k-basis of the finite dimensional A-module V and denote +by γt +r,i,j ∈ k the structure constants of V with respect to its A-module structure ·, i.e. +for all r = 1, · · · , n, i ∈ I and j = 1, · · · , l we have: +xri · vj = +l +� +s=1 +γs +r,i,j vs +(19) +Consider {wr | r ∈ T} to be a k-basis for W and if ⊲ denotes its Lie g-module structure, +then for all j ∈ I and r ∈ T we can find a finite subset Sj,r of T such that: +fj ⊲ wr = +� +p∈Sj,r +σp +j,r wp +(20) +where σp +j,r ∈ k for all j ∈ I, r ∈ T, and p ∈ Sj,r. +Now let S(V, W) be the free Lie h-module on the set {Yri | r ∈ T, i = 1, · · · , l} and +denote by V(V, W) the quotient of S(V, W) by its Lie h-submodule generated by the +following elements: +� +p∈Sj,r +σp +j,r Yps − +l +� +k=1 +n +� +p=1 +γs +p,j,k ep ◮ Yrk +(21) +for all s = 1, · · · , l, r ∈ T and j ∈ I, where ◮ denotes the h-module action on S(V, W). +By denoting yri := � +Yri, where � +Yri stands for the equivalence class of Yri in the quotient +module V(V, W), it follows that the relations below hold in V(V, W): +� +p∈Sj,r +σp +j,r yps = +l +� +k=1 +n +� +t=1 +γs +t,j,k et ◮ yrk +(22) +for all s = 1, · · · , l, r ∈ T and j ∈ I. +It can now be easily seen, as in the proof of Theorem 2.4, that the pair (V(V, W), τV(V, W )) +is the universal Lie h-module of V and W, where τV(V, W ): W → V(V, W) ⊗ V is the +morphism of Lie g-modules defined for all r ∈ T as follows: +τV(V, W )(wr) := +l +� +s=1 +yrs ⊗ vs. +(23) +□ + +UNIVERSAL MODULES +11 +3. Functors between module categories +In this section we show that the two universal module constructions previously introduced +are functorial and, moreover, if certain conditions are fulfilled the corresponding functors +admit right adjoints. We start, however, by stating the following easy consequence of +Theorem 2.1: +Proposition 3.1. Let (U, ↷) ∈ hLM and (V, ·) ∈ AM. Then: +1) We have a functor U ⊗ −: AM → gLM from the category of A-modules to the +category of Lie g-modules; +2) We have a functor − ⊗ V : hLM → gLM between the categories of Lie modules +over h and g respectively. +Proof. In light of Theorem 2.1, we are only left to show that morphisms behave well with +respect to the corresponding associative or Lie module structures. We will treat only the +first statement and leave the second one to the reader. To this end, consider (V, ·) and +(V ′, •) two A-modules, ⇀ and ⇀′ the corresponding induced Lie g-module actions via +(7) and g: V → V ′ a morphism in AM . Then, for all i ∈ I, l ∈ U and t ∈ V we have: +(IdU ⊗ g) +� +fi ⇀ (l ⊗ t) +�(7) += +n +� +j=1 +(ej ↷ l) ⊗ g(xji · t) = +n +� +j=1 +(ej ↷ l) ⊗ xji • g(t) +(7) += fi ⇀′ � +l ⊗ g(t) +� +□ +We consider now the universal module functors: +Theorem 3.2. Let U be a finite dimensional Lie h-module and V a finite dimensional +A-module. +(1) There exists a functor UU : gLM → AM defined as follows for all Lie g-modules +X, Y and all morphisms f : X → Y in gLM: +UU(X) = U(U, X), +UU(f) = f +where f : U(U, X) → U(U, Y ) is the unique A-modules morphism which makes +the following diagram commutative: +X +ρU(U, X) +� +ρU(U, Y )◦f +�◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +U ⊗ U(U, X) +IdU⊗f +� +U ⊗ U(U, Y ) +(24) +(2) There exists a functor VV : gLM → hLM defined as follows for all Lie g-modules +X, Y and all morphisms f : X → Y in gLM: +VV (X) = V(V, X), +VV (f) = f + +12 +A. L. AGORE +where f : V(V, X) → V(V, Y ) is the unique morphism of Lie h-modules which +makes the following diagram commutative: +X +τV(V, X) +� +τV(V, Y )◦f +�◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +V(V, X) ⊗ V +f⊗IdV +� +V(V, Y ) ⊗ V +(25) +Proof. As the result follows in a straightforward manner by a standard category the- +ory argument, we only sketch the proof of the first assertion. Indeed, if f = IdX then +IdU(U, X) is obviously the unique A-modules morphism which makes diagram (24) com- +mute and therefore UU(IdX) = IdU(U, X). Moreover, if f : X → Y and g: Y → W are two +morphisms in gLM, then g ◦f : U(U, X) → U(U, W) is obviously the unique A-modules +morphism which makes the following diagram commutative: +Z +ρU(U, X) +� +ρU(U, W )◦g◦f +�◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +U ⊗ U(U, X) +IdU⊗ +� +g◦f +� +� +U ⊗ U(U, W) +and we can conclude that UU(g ◦ f) = UU(g) ◦ UU(f), as desired. +□ +Under the appropriate finite-dimensionality assumptions, the functors constructed in +Proposition 3.1 are right adjoints to the universal module functors: +Theorem 3.3. Let (U, ↷) be a finite dimensional Lie h-module and (V, ·) a finite di- +mensional A-module. Then: +1) The following functors form an adjunction: +UU : gLM → AM, +U ⊗ −: AM → gLM; +2) Similarly, the following functors also form an adjunction: +VV : gLM → hLM, +− ⊗ V : hLM → gLM. +Proof. 1) As pointed out in Corollary 2.3, for all Lie g-modules Z and all A-modules +X, there is a bijection between HomAM +� +UU(Z), X +� +and HomgLM (Z, U ⊗ X) given as +follows for all morphisms of A-modules θ: UU(Z) → X: +ΓZ,X : HomAM (UU(Z), X) → HomgLM (Z, U ⊗ X), +ΓZ,X(θ) = (IdU ⊗ θ) ◦ ρU(U, Z). +The desired conclusion now follows by showing that the above bijection is natural in +both variables. This can be easily proved by a straightforward diagram chase and is left +to the reader. +2) Using now Corollary 2.9, for all Lie g-modules W and all Lie h-modules Z, we obtain +a bijection between HomhLM +� +VV (W), Z +� +and HomgLM (W, Z ⊗ V ) defined as follows + +UNIVERSAL MODULES +13 +for all morphisms of Lie h-modules θ: VV (W) → Z: +ΓW,Z : HomhLM +� +VV (W), Z +� +→ HomgLM (W, Z ⊗ V ), +ΓW,Z(θ) = (θ ⊗ IdV ) ◦ ρV(V, W ). +□ +In particular, the two pairs of adjoint functors allow us to travel both ways between the +representation categories of the two (arbitrary) Lie algebras h and g and respectively +between the representation category of the associative algebra A and the representation +category of the Lie algebra g. +Example 3.4. Let ρi : g ⊗ Wi → Wi be representations of g, where i = 1, 2. By the +colimit preservation property of left adjoints we can easily conclude that for any finite +dimensional Lie h-module U, UU(W1) ⊕ UU(W2) is the direct sum of the A-modules +UU(W1) and UU(W2). +Similarly, for any finite dimensional A-module V , VV (W1) ⊕ +UU(W2) is the direct sum of the Lie h-modules UU(W1) and UU(W2). This can be easily +extended to an arbitrary family of representations. +References +[1] Agore, A.L., Gordienko, A.S., Vercruysse, J. - V -universal Hopf algebras (co)acting on Ω-algebras, +Commun. Contemp. Math. 25 (2023), 2150095. +[2] Agore, A.L. - Universal coacting Poisson Hopf algebras, Manuscripta Math. 165 (2021), 255–268. +[3] Agore, A.L., Gordienko, A.S., Vercruysse, J. - Equivalences of (co)module algebra structures over +Hopf algebras, J. Noncommut. Geom., 15 (2021), 951–993. +[4] Agore, A.L., Militaru, G. - A new invariant for finite dimensional Leibniz/Lie algebras, J. Algebra +562 (2020), 390–409. +[5] Ardizzoni, A., El Kaoutit, L., Menini, C. - Categories of comodules and chain complexes of modules, +Internat. J. Math. 23 (2012), 1250109 +[6] Bhattacharjee, S., Chirvˇasitu, A., Goswami, D. - Quantum Galois groups of subfactors, Internat. J. +Math. 33 (2022), 2250013. +[7] Chirvˇasitu, A., Walton, C., Wang, X. - On quantum groups associated to a pair of preregular forms, +J. Noncommut. Geom. bf 13 (2019), 115—159. +[8] Hyland, M., Lopez Franco, I., Vasilakopoulou, C. - Hopf measuring comonoids and enrichment, +Proc. Lond. Math. Soc. 115 (2017), 1118—1148. +[9] Jacobson, N. – Lie algebras, Dover Publications, NY, 1962. +[10] Mac Lane, S. - Categories for the Working Mathematician, Graduate Texts in Mathematics 5, +Springer, 1998. +[11] Manin, Yu. I. - Quantum groups and noncommutative geometry, Universite de Montreal, Centre de +Recherches Mathematiques, Montreal, QC, 1988. +[12] Militaru, G. - The automorphisms group and the classification of gradings of finite dimensional +associative algebras, Results Math. 77 (2022). +[13] Raedschelders, T., Van den Bergh, M. - The Manin Hopf algebra of a Koszul Artin-Schelter regular +algebra is quasi-hereditary, Adv. Math. 305 (2017), 601-–660. +[14] Rodrıiguez-Romo, S., Taft, E. - Some quantum-like Hopf algebras which remain noncommutative +when q = 1, Lett. Math. Phys. 61(2002), 41-–50. +[15] Sweedler, M.E. - Hopf Algebras, Benjamin New York, 1969. +[16] Tambara, D. - The coendomorphism bialgebra of an algebra. J. Fac. Sci. Univ. Tokyo Math. 37 +(1990), 425–456. + +14 +A. L. AGORE +Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium +Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 +Bucharest, Romania +Email address: ana.agore@gmail.com + diff --git a/0NE1T4oBgHgl3EQfRQO7/content/tmp_files/load_file.txt b/0NE1T4oBgHgl3EQfRQO7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..befd801aec34f428d7a93964912d9ac6a6a4a56b --- /dev/null +++ b/0NE1T4oBgHgl3EQfRQO7/content/tmp_files/load_file.txt @@ -0,0 +1,434 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf,len=433 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='03051v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='RA] 8 Jan 2023 FUNCTORS BETWEEN REPRESENTATION CATEGORIES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' UNIVERSAL MODULES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' AGORE Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Let g and h be two Lie algebras with h finite dimensional and consider A = A(h, g) to be the corresponding universal algebra as introduced in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Given an A-module U and a Lie h-module V we show that U ⊗ V can be naturally endowed with a Lie g-module structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' This gives rise to a functor between the category of Lie h-modules and the category of Lie g-modules and, respectively, to a functor between the category of A-modules and the category of Lie g-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Under some finite dimensionality assumptions, we prove that the two functors admit left adjoints which leads to the construction of universal A-modules and universal Lie h-modules as the representation theoretic counterparts of Manin-Tambara’s universal coacting objects [11, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Introduction The universal coacting bialgebra/Hopf algebra on a finite dimensional (graded) asso- ciative algebra originates in the work of Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Manin ([11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' The importance of this construction became obvious mostly due to its interaction with non-commutative geom- etry where it is seen as some sort of symmetry group (see [13] for more details on this view point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' The non-graded version of this construction appeared a few years later in a paper by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Tambara ([16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' However, as remarked in [16], the universal coacting bialge- bra is in fact the dual of the so-called universal measuring bialgebra introduced by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Sweedler in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' We should note that, unlike Manin-Tambara’s construction, Sweedler’s universal measuring bialgebra/Hopf algebra exists even in the infinite-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' In recent years, universal (co)acting objects have been considered in various settings and for different purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' For instance, [8] extends Sweedler’s construction to monoids in a braided monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' On the other hand, the Manin-Tambara construction was introduced in the setting of Poisson algebras ([2]), finite index-subfactors ([6]), su- perpotential algebras ([7]), polynomial algebras ([14]), bialgebroids ([5]) or Lie/Leibniz algebras ([4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' The corresponding universal coacting bialgebras/Hopf algebras, which in certain cases carry some extra structure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' a Poisson Hopf algebra structure as in [2]), seem to play a prominent role in solving other seemingly unrelated problems such as the classification of gradings on various kinds of algebras ([4, 12]), the description of the automorphisms group of certain algebraic structures ([4]) and even in quantum Galois 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' 16D90, 16T05, 17A32, 17B10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' universal module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' This work was supported by a grant of Romanian Ministry of Research, Innovation and Digitization, CNCS/CCCDI – UEFISCDI, project number PN-III-P4-ID-PCE-2020-0458, within PNCDI III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' 1 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' AGORE theory ([6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Another related universal (co)acting construction was considered in [3] as the Hopf algebraic analogue of the universal group of a grading and its connections to the problem of classifying Hopf algebra coactions have been highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' One of the most general constructions of universal (co)acting bialgebras/Hopf algebras, performed in the setting of Ω-algebras, was introduced in [1] together with generalized duality results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Necessary and sufficient conditions for the existence of universal coacting bialgebras/Hopf algebras are provided, explaining in this general setting the need for assuming finite dimensionality in both Manin and Tambara’s papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' It is worth to point out that both Sweedler and Manin-Tambara’s constructions have a categorical interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' More precisely, for Tambara’s construction one considers the left adjoint, say a(A, −), of the tensor product endofunctor A ⊗ − on the category of k-algebras, where A is a finite dimensional associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Tambara’s universal coacting bialgebra is precisely a(A, A) which turns out to be naturally endowed with a bialgebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Similarly, for an arbitrary associative algebra A, it can be proved that the contravariant functor Hom(−, A) taking coalgebras to (convolution) algebras has a right adjoint which hereafter we denote by M(A, −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' As before, Sweedler’s uni- versal measuring bialgebra is exactly M(A, A) which again has a bialgebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' In this paper we deal with the representation theoretic version of Manin-Tambara’s con- struction in the Lie algebra setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Our approach is a categorical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' More precisely, given two fixed Lie algebras g and h, with h finite dimensional, and the corresponding universal algebra A = A(h, g) (see[4]), we first show that the tensor product between an A-module U and a Lie h-module V can be endowed with a Lie g-module structure (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' As a consequence, we are able to construct two ”tensor product” func- tors between the categories of Lie modules over h and g and respectively between the category of A-modules and the category of Lie g-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Under the appropriate finite dimensionality assumptions, the two functors mentioned above are proved to admit left adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' These left adjoints are given precisely by what we have called the universal Lie h-module and the universal A-module, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' The two universal modules are introduced in a constructive manner in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='4 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' These are the counterparts for Lie and associative representations of Manin-Tambara’s constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Furthermore, the two aforementioned pairs of adjoint functors allow us to travel both ways between the representation categories of different algebraic structures, such as Lie and associative algebras, and to transfer certain properties which are usually preserved by left/right adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Preliminaries This section will be used mostly as an opportunity to fix some notation and to provide certain useful references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Let us start with a few words on notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Notational conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' All vector spaces, (bi)linear maps, unadorned tensor products, Lie or associative algebras, bialgebras and so on are over an arbitrary com- mutative field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' All (co)associative (co)algebras are assumed to be (co)unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' The notation employed for coalgebras is standard: ∆ stands for the comultiplication and ε UNIVERSAL MODULES 3 for the counit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' We use Sweedler’s notation with implied summation for both coalge- bras (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' bialgebras), as in ∆(c) = c(1) ⊗ c2, and for comodule structures: a right C-comodule structure ρ on a vector space V will be denoted by ρ(v) = v(0) ⊗v(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' When we need to be precise, the structures involved will be adorned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' δij denotes Kronecker’s symbol while IdX stands for the identity map on the set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' In the sequel, k[Xsi |s = 1, · · · , n, i ∈ I] denotes the usual polynomial algebra on vari- ables Xsi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' We shall denote by Liek and ComAlgk the categories of Lie and commutative associative algebras, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Given an associative algebra A and a Lie algebra g we denote by AM and gLM the categories of left A-modules and left Lie g-modules, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Recall that a (left) Lie g-module is a vector space V equipped with a bilinear map ⇀: g × V → V such that for all x, y ∈ g and v ∈ V we have: [x, y] ⇀ v = x ⇀ (y ⇀ v) − y ⇀ (x ⇀ v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Throughout the paper, g and h will denote two arbitrary Lie algebras with h finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Let {fi | i ∈ I} and {e1, · · · , en} be two fixed basis in g and h, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' We consider {τ s i,j | i, j, s = 1, · · · , n} to be the structure constants of h, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' for any i, j = 1, · · · , n we have: [ei, ej]h = n � s=1 τ s i,j es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' (1) Similarly, for any i, j ∈ I, let Bi,j ⊆ I be a finite subset of I such that for any i, j ∈ I we have: [fi, fj]g = � u∈Bi,j βu i,j fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' (2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' The universal algebra of h and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' We recall briefly, for further use, the con- struction of the universal commutative algebra A(h, g) of two given Lie algebras h and g (recall that h is always assumed to be finite dimensional).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' It was first introduced in [4] in the more general setting of Leibniz algebras as the counterpart of Tambara’s con- struction ([16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' We restrict here to the Lie algebra version of the construction which can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' We have: A(h, g) := k[Xsi |s = 1, · · · , n, i ∈ I]/J (3) where J is the ideal generated by all polynomials of the form P (h, g) (a,i,j) := � u∈Bi,j βu i,j Xau − n � s,t=1 τ a s,t XsiXtj, for all a = 1, · · · , n and i, j ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' (4) When working in the universal algebra A(h, g), we denote by xsi := � Xsi the class of Xsi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Consequently, the following relations hold in A(h, g): � u∈Bi,j βu i,j xau = n � s,t=1 τ a s,t xsixtj, for all a = 1, · · · , n, and i, j ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' (5) When the (finite dimensional) Lie algebra h is fixed, the universal algebra construction gives rise to a functor A(h, −): Liek → ComAlgk which turns out to be the left adjoint 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' AGORE of the tensor product h ⊗ −: ComAlgk → Liek (see [4, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='1]), where for any commutative algebra X the tensor product h ⊗ X is endowed with the current Lie algebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' In order to avoid dealing with cumbersome notation, when there is no fear of confusion, we denote A = A(h, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Furthermore, If h = g, then the corresponding universal algebra A(h, h) will be denoted simply by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' The notation is meant to highlight the fact that B is a bialgebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' in fact, it admits a unique bialgebra structure such that h becomes a right B-comodule with respect to ηh : h → h⊗B where η: 1Liek → h⊗A(h, −) denotes the unit of the adjunction between A(h, −) and h ⊗ −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' More precisely, the comultiplication and the counit on B are given for any i, j = 1, · · · , n by ∆(xij) = n � s=1 xis ⊗ xsj and ε(xij) = δi,j1k (6) For basic categorical concepts we refer the reader to [10] and for unexplained notions pertaining to Lie and Hopf algebras to [9] and [15], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Universal modules Our first important result provides a way of defining a Lie g-module structure on the tensor product between a Lie h-module and an A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Let (U, ↷) ∈ hLM be a Lie h-module and (V, ·) ∈ AM an A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Then (U ⊗ V, ⇀) ∈ gLM is a Lie g-module where the action of g on U ⊗ V is given for all i ∈ I, l ∈ U and t ∈ V by: fi ⇀ (l ⊗ t) = n � j=1 (ej ↷ l) ⊗ (xji · t) (7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' having in mind that (U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' ↷) is a Lie module and A = A(h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' g) is a com- mutative algebra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' we have: [fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' fj] ⇀ (l ⊗ t) (2) = � u∈Bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='j βu i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='j fu ⇀ (l ⊗ t) (7) = � u∈Vi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='r=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='n βu i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='j (er ↷ l) ⊗ (xru · t) = � r=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='n (er ↷ l) ⊗ � � u∈Bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='j βu i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='j xru � t (5) = � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='r=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='n τ r s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='p (er ↷ l) ⊗ (xsixpj) · t = � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='p=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='n � n � r=1 τ r s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='p er � ↷ l ⊗ (xsixpj) · t (1) = � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='p=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='n [es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' ep] ↷ l ⊗ (xsixpj) · t = � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='p=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='n es ↷ (ep ↷ l) ⊗ xsi · (xpj · t) − � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='p=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='n ep ↷ (es ↷ l) ⊗ xpj · (xsi · t) (7) = fi ⇀ n � p=1 (ep ↷ l) ⊗ (xpj · t) − fj ⇀ n � s=1 (es ↷ l) ⊗ (xsi · t) (7) = fi ⇀ � fj ⇀ (l ⊗ t) � − fj ⇀ � fi ⇀ (l ⊗ t) � UNIVERSAL MODULES 5 for all i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' j ∈ I and l ∈ U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' t ∈ V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' (U ⊗ V, ⇀) is a left Lie g-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' □ Inspired by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='1 we can consider two types of universal modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' The universal A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' The first such universal module is associated with a Lie h-module and a Lie g-module as follows: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Given a Lie h-module U and a Lie g-module Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' the universal A-module of U and Z is a pair � U(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' ρU(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Z) � consisting of an A-module U(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Z) and a mor- phism of Lie g-modules ρU(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Z) : Z → U ⊗ U(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Z) such that for any other pair (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' f) consisting of an A-module X and a morphism of Lie g-modules f : Z → U ⊗X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' there ex- ists a unique morphism of A-modules g: U(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Z) → X such that the following diagram is commutative: Z ρU(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Z) � f �❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ U ⊗ U(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Z) IdU⊗g � U ⊗ X (8) In other words,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' the above definition is saying that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' when it exists,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' the universal A-module of U and Z is in fact the initial object of the category whose objects are pairs (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' f) consisting of an A-module X and a morphism of Lie g-modules f : Z → U ⊗ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' while morphisms between two such objects (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' f) and (X′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' f ′) are defined to be A-module maps g: X → X′ satisfying (IdU ⊗ g) ◦ f = f ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' As direct consequences of the above definition, we obtain the following: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Let U be a Lie h-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Then, for all Lie g-modules Z and all A- modules X, we have a bijective correspondence between: (1) Lie g-module maps f : Z → U ⊗ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' (2) A-module maps g: U(U, Z) → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Under the appropiate finite-dimensionality assumptions required for all Manin-Tambara type constructions, the universal A-module introduced in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='2 exists: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' If U is a finite dimensional Lie h-module then the universal A -module of U and any other Lie g-module Z exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Let {u1, · · · , um}, m ∈ N∗, be a k-basis of the Lie module U and denote by ωt ij ∈ k the structure constants of U with respect to its Lie h-module structure ↷, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' for all i = 1, · · · , n, j = 1, · · · , m we have: ei ↷ uj = m � s=1 ωs i,j us (9) Furthermore, consider {zr | r ∈ J} to be a k-basis for the arbitrary Lie g-module Z and if ↬ denotes its Lie module structure, then for all j ∈ I and r ∈ J we can find a finite 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' AGORE subset Tj,r of J such that: fj ↬ zr = � l∈Tj,r ηl j,r zl (10) where ηl j,r ∈ k for all j ∈ I, r ∈ J, and l ∈ Tj,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Consider now T (U, Z) to be the free A-module on the set {Yij | i = 1, · · · , m, j ∈ J} and denote by U(U, Z) the quotient of T (U, Z) by its A-submodule generated by the following elements: � p∈Tj,i ηp j,i Ysp − m � t=1 n � r=1 ωs r,t xrj • Yti (11) for all s = 1, · · · , m, i ∈ J and j ∈ I, where • denotes the A-module action on T (U, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Denoting ytj := � Ytj, where � Ytj stands for the equivalence class of Ytj in the quotient module U(U, Z), it follows that the relations below hold in U(U, Z): � p∈Tj,i ηp j,i ysp = m � t=1 n � r=1 ωs r,t xrj • yti (12) for all s = 1, · · · , m, i ∈ J and j ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Furthermore, we can define a morphism of Lie g-modules ρU(U, Z): Z → U ⊗ U(U, Z) as follows: ρU(U, Z)(zr) := m � s=1 us ⊗ ysr, for all r ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' (13) It follows now that for all j ∈ I and i ∈ J we have: ρU(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Z)(fj ↬ zi) (10) = ρU(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='Z) � � p∈Tj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='i ηp ji zp � = � p∈Tj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='i m � s=1 ηp ji us ⊗ ysp = m � s=1 � us ⊗ � p∈Tj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='i ηp ji ysp � (12) = m � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='t=1 n � r=1 ωs r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='t us ⊗ xrj • yti = m � t=1 n � r=1 � m � s=1 ωs r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='t us � ⊗ xrj • yti (9) = m � t=1 n � r=1 er ↷ ut ⊗ xrj • yti (7) = m � t=1 fj ⇀ (ut ⊗ yti) = fj ⇀ m � t=1 ut ⊗ yti (13) = fj ⇀ ρU(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Z)(zi) which shows that ρU(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Z) is indeed a Lie g-modules map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' We will show that the pair � U(U, Z), ρU(U, Z) � constructed above is in fact the universal A-module of U and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' To this end, consider a pair (X, f) consisting of an A-module X and a morphism of Lie g-modules f : Z → U ⊗ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Let {wsr | s = 1, · · · , m, r ∈ J} be a family of elements of X such that for all r ∈ J we have: g(zr) = m � s=1 us ⊗ wsr (14) UNIVERSAL MODULES 7 Furthermore, as g: Z → U ⊗ X is a Lie g-modules map, a straightforward computation shows that the following compatibilities hold for all s = 1, · · · , m, i ∈ J and j ∈ I: � p∈Tj,i ηp j,i wsp = m � t=1 n � r=1 ωs r,t xrj · wti (15) where · denotes the A-module action on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' The universal property of the free module yields a unique A-module map g: T (U, Z) → X such that g(Ysr) = wsr, for all s = 1, · · · , m and r ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Moreover, Ker(g) contains the A- submodule of T (U, Z) generated by the elements listed in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Indeed, as g : U(U, Z) → X is a morphism of A-modules we have: g � � p∈Tj,i ηp j,i Ysp − m � t=1 n � r=1 ωs r,t xrj • Yti � = � p∈Tj,i ηp j,i wsp − m � t=1 n � r=1 ωs r,t xrj · wti (15) = 0 for all s = 1, · · · , m, i ∈ J and j ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' This shows that there exists a unique A-modules map g: U(U, Z) → X such that g(ysr) = zsr, for all s = 1, · · · , m and r ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' This implies that for all r ∈ J we have: � IdU ⊗ g � ρU(U, Z)(zr) = � IdU ⊗ g �� m � s=1 us ⊗ ysr � = m � s=1 us ⊗ wsr (14) = g(zr) which means precisely that diagram (8) is commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Moreover, g is obviously the unique A-modules map with this property and the proof is now finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' □ The case g = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Particularizing the results of Section 2 for g = h, where h is the finite dimensional Lie algebra defined in (1), leads to the following interesting consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' According to the discussion in Preliminaries, the universal algebra A(h, h) denoted by B is in this case a bialgebra with coalgebra structure depicted in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' This allows us to see the tensor product U(U, Z) ⊗ U(U, Z) as well as the base field k as B-modules via the comultiplication and the counit of B as follows: xij ∗ (y ⊗ t) = n � t=1 xit • y ⊗ xtj • t (16) xij · α = δijα (17) for all xij ∈ B, y, t ∈ U(U, Z) and α ∈ k, where • denotes the B-module strucuture on U(U, Z) as in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' First we show that if U is a finite dimensional Lie h-module as considered in (9), then the B-module U(U, U) denoted by U(U) admits a coalgebra structure with respect to which � U, ρU(U) � becomes a right U(U)-comodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Let U be a finite dimensional Lie h-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' There exists a unique coalgebra structure on U(U) such that � U, ρU(U) � becomes a right U(U)-comodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' In particular both U(U) ⊗ U(U) and k are B-modules via the formulas (16) and (17) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Therefore, U ⊗ U(U) ⊗ U(U) and U ⊗ k are Lie h-modules via (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Furthermore, it can be easily checked that the maps � ρU(U) ⊗ IdU(U) � ρU(U) : U → U ⊗ 8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' AGORE U(U) ⊗ U(U) and canU : U → U ⊗ k are morphisms of Lie h-modules, where canU : U → U ⊗ k is the canonical isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Now Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='2 yields a unique B-modules map ∆: U(U) → U(U) ⊗ U(U) such that the following diagram is commutative: U ρU(U) � � ρU(U)⊗IdU(U) � ρU(U) �❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ U ⊗ U(U) IdU ⊗∆ � U ⊗ U(U) ⊗ U(U) Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' we obtain a unique B-modules map ε: U(U) → k such that the following diagram is commutative: U ρU(U) � canU �■ ■ ■ ■ ■ ■ ■ ■ ■ ■ U ⊗ U(U) IdU⊗ε � U ⊗ k A straightforward computation shows that the commutativity of the two diagrams above imply that ∆ and ε take the following form for all l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' t = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' m: ∆(ylt) = m � s=1 yls ⊗ yst,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' ε(ylt) = δlt1k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' It is now obvious that � U(U), ∆, ε � form a coalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Finally, by the commutativity of the two diagrams above we obtain that � U, ρU(U) � is a right U(U)-comodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' It is worth pointing out that with the coalgebra structure introduced above, U(U) becomes a B-module coalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Indeed, having in mind that both ∆ and ε are B-module maps, we have: ∆(xab • ylt) = xab ∗ ∆(ylt) = xab ∗ � m � s=1 yls ⊗ yst �(16) = n � c=1 m � s=1 xac • yls ⊗ xcb • yst = (xab)(1) • (ylt)(1) ⊗ (xab)(2) • (ylt)(2) and ε(xab • ylt) = xab · ε(ylt) (17) = δab ε(ylt) = ε(xab) ε(ylt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' This shows that • is a coalgebra map, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' It turns out that the pair � U(U), ρU(U) � is universal in the following way: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' For any coalgebra X with a B-module structure and any Lie h-module morphism ψ: U → U ⊗X which makes U into a right X-comodule, there exists a unique UNIVERSAL MODULES 9 B-modules and coalgebra morphism θ: U(U) → X such that the following diagram is commutative: U ρU(U) � ψ �■ ■ ■ ■ ■ ■ ■ ■ ■ ■ U ⊗ U(U) IdU ⊗θ � U ⊗ X Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' In light of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='2, such a unique A-modules map θ exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' We are left to show that θ is also a coalgebra map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' From the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='4 we know that θ is defined for all l, t = 1, · · · , m by θ(ylt) = zlt where zlt are elements of X such that for all r = 1, · · · , m we have ψ(ur) = �m s=1 us ⊗ zsr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' As (U, ψ) is a right comodule, we obtain: ∆(zlt) = m � s=1 zls ⊗ zst, ε(zlt) = δlt1k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' To this end, we have: ∆ � θ(ylt) � = ∆(zlt) = m � s=1 zls ⊗ zst = m � s=1 θ(yls) ⊗ θ(yst) = (θ ⊗ θ) ◦ ∆(ylt) Similarly one can check that ε◦θ = ε which shows that θ is indeed a coalgebra map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' The universal h-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' The second type of universal module one can consider is the following: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Given an A-module V and a Lie g-module W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' the universal Lie h- module of V and W is a pair � V(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' W),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' τV(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' W ) � consisting of a Lie h-module V(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' W) and a morphism of Lie g-modules τV(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' W ): W → V(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' W)⊗V such that for any other pair (Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' f) consisting of a Lie h-module Y and a morphism of Lie g-modules f : W → Y ⊗ V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' there exists a unique morphism of Lie h-modules g: V(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' W) → Y such that the following diagram is commutative: W τV(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' W ) � f �❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ V(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' W) ⊗ V g⊗IdV � Y ⊗ V (18) The universal Lie h-module of V and W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' when it exists,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' can again be seen as the initial object of the category whose objects are pairs (Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' f) consisting of a Lie h-module Y and a morphism of Lie g-modules f : W → Y ⊗ V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' while morphisms between two such objects (Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' f) and (Y ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' f ′) are defined to be Lie h-module maps g: Y → Y ′ satisfying (g ⊗ IdV ) ◦ f = f ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Let V be an A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Then, for all Lie g-modules W and all Lie h-modules Y , we have a bijective correspondence between: (1) Lie g-module maps f : W → Y ⊗ V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' (2) Lie h-module maps g: V(V, W) → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' AGORE The universal h-module introduced in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='8 also exists provided that the A- module V is finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' If V is a finite dimensional A-module then the universal Lie h-module of V and any other Lie g-module W exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' As this proof is somewhat similar in spirit with the one of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='4, we will be brief and provide only the main ingredients required for the construction of the universal Lie h-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Let {v1, · · · , vl}, l ∈ N∗, be a k-basis of the finite dimensional A-module V and denote by γt r,i,j ∈ k the structure constants of V with respect to its A-module structure ·, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' for all r = 1, · · · , n, i ∈ I and j = 1, · · · , l we have: xri · vj = l � s=1 γs r,i,j vs (19) Consider {wr | r ∈ T} to be a k-basis for W and if ⊲ denotes its Lie g-module structure, then for all j ∈ I and r ∈ T we can find a finite subset Sj,r of T such that: fj ⊲ wr = � p∈Sj,r σp j,r wp (20) where σp j,r ∈ k for all j ∈ I, r ∈ T, and p ∈ Sj,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Now let S(V, W) be the free Lie h-module on the set {Yri | r ∈ T, i = 1, · · · , l} and denote by V(V, W) the quotient of S(V, W) by its Lie h-submodule generated by the following elements: � p∈Sj,r σp j,r Yps − l � k=1 n � p=1 γs p,j,k ep ◮ Yrk (21) for all s = 1, · · · , l, r ∈ T and j ∈ I, where ◮ denotes the h-module action on S(V, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' By denoting yri := � Yri, where � Yri stands for the equivalence class of Yri in the quotient module V(V, W), it follows that the relations below hold in V(V, W): � p∈Sj,r σp j,r yps = l � k=1 n � t=1 γs t,j,k et ◮ yrk (22) for all s = 1, · · · , l, r ∈ T and j ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' It can now be easily seen, as in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='4, that the pair (V(V, W), τV(V, W )) is the universal Lie h-module of V and W, where τV(V, W ): W → V(V, W) ⊗ V is the morphism of Lie g-modules defined for all r ∈ T as follows: τV(V, W )(wr) := l � s=1 yrs ⊗ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' (23) □ UNIVERSAL MODULES 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Functors between module categories In this section we show that the two universal module constructions previously introduced are functorial and, moreover, if certain conditions are fulfilled the corresponding functors admit right adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' We start, however, by stating the following easy consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='1: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Let (U, ↷) ∈ hLM and (V, ·) ∈ AM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Then: 1) We have a functor U ⊗ −: AM → gLM from the category of A-modules to the category of Lie g-modules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' 2) We have a functor − ⊗ V : hLM → gLM between the categories of Lie modules over h and g respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' In light of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='1, we are only left to show that morphisms behave well with respect to the corresponding associative or Lie module structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' We will treat only the first statement and leave the second one to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' To this end, consider (V, ·) and (V ′, •) two A-modules, ⇀ and ⇀′ the corresponding induced Lie g-module actions via (7) and g: V → V ′ a morphism in AM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Then, for all i ∈ I, l ∈ U and t ∈ V we have: (IdU ⊗ g) � fi ⇀ (l ⊗ t) �(7) = n � j=1 (ej ↷ l) ⊗ g(xji · t) = n � j=1 (ej ↷ l) ⊗ xji • g(t) (7) = fi ⇀′ � l ⊗ g(t) � □ We consider now the universal module functors: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Let U be a finite dimensional Lie h-module and V a finite dimensional A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' (1) There exists a functor UU : gLM → AM defined as follows for all Lie g-modules X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Y and all morphisms f : X → Y in gLM: UU(X) = U(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' UU(f) = f where f : U(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' X) → U(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Y ) is the unique A-modules morphism which makes the following diagram commutative: X ρU(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' X) � ρU(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Y )◦f �◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ U ⊗ U(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' X) IdU⊗f � U ⊗ U(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Y ) (24) (2) There exists a functor VV : gLM → hLM defined as follows for all Lie g-modules X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Y and all morphisms f : X → Y in gLM: VV (X) = V(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' VV (f) = f 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' AGORE where f : V(V, X) → V(V, Y ) is the unique morphism of Lie h-modules which makes the following diagram commutative: X τV(V, X) � τV(V, Y )◦f �◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ V(V, X) ⊗ V f⊗IdV � V(V, Y ) ⊗ V (25) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' As the result follows in a straightforward manner by a standard category the- ory argument, we only sketch the proof of the first assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Indeed, if f = IdX then IdU(U, X) is obviously the unique A-modules morphism which makes diagram (24) com- mute and therefore UU(IdX) = IdU(U, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Moreover, if f : X → Y and g: Y → W are two morphisms in gLM, then g ◦f : U(U, X) → U(U, W) is obviously the unique A-modules morphism which makes the following diagram commutative: Z ρU(U, X) � ρU(U, W )◦g◦f �◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ U ⊗ U(U, X) IdU⊗ � g◦f � � U ⊗ U(U, W) and we can conclude that UU(g ◦ f) = UU(g) ◦ UU(f), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' □ Under the appropriate finite-dimensionality assumptions, the functors constructed in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='1 are right adjoints to the universal module functors: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Let (U, ↷) be a finite dimensional Lie h-module and (V, ·) a finite di- mensional A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Then: 1) The following functors form an adjunction: UU : gLM → AM, U ⊗ −: AM → gLM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' 2) Similarly, the following functors also form an adjunction: VV : gLM → hLM, − ⊗ V : hLM → gLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' 1) As pointed out in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='3, for all Lie g-modules Z and all A-modules X, there is a bijection between HomAM � UU(Z), X � and HomgLM (Z, U ⊗ X) given as follows for all morphisms of A-modules θ: UU(Z) → X: ΓZ,X : HomAM (UU(Z), X) → HomgLM (Z, U ⊗ X), ΓZ,X(θ) = (IdU ⊗ θ) ◦ ρU(U, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' The desired conclusion now follows by showing that the above bijection is natural in both variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' This can be easily proved by a straightforward diagram chase and is left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' 2) Using now Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='9, for all Lie g-modules W and all Lie h-modules Z, we obtain a bijection between HomhLM � VV (W), Z � and HomgLM (W, Z ⊗ V ) defined as follows UNIVERSAL MODULES 13 for all morphisms of Lie h-modules θ: VV (W) → Z: ΓW,Z : HomhLM � VV (W), Z � → HomgLM (W, Z ⊗ V ), ΓW,Z(θ) = (θ ⊗ IdV ) ◦ ρV(V, W ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' □ In particular, the two pairs of adjoint functors allow us to travel both ways between the representation categories of the two (arbitrary) Lie algebras h and g and respectively between the representation category of the associative algebra A and the representation category of the Lie algebra g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Let ρi : g ⊗ Wi → Wi be representations of g, where i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' By the colimit preservation property of left adjoints we can easily conclude that for any finite dimensional Lie h-module U, UU(W1) ⊕ UU(W2) is the direct sum of the A-modules UU(W1) and UU(W2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Similarly, for any finite dimensional A-module V , VV (W1) ⊕ UU(W2) is the direct sum of the Lie h-modules UU(W1) and UU(W2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' This can be easily extended to an arbitrary family of representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' References [1] Agore, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=', Gordienko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=', Vercruysse, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' - V -universal Hopf algebras (co)acting on Ω-algebras, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' 25 (2023), 2150095.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' [2] Agore, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' - Universal coacting Poisson Hopf algebras, Manuscripta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' 165 (2021), 255–268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' [3] Agore, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=', Gordienko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=', Vercruysse, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' - Equivalences of (co)module algebra structures over Hopf algebras, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Noncommut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=', 15 (2021), 951–993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' [4] Agore, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=', Militaru, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' - A new invariant for finite dimensional Leibniz/Lie algebras, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Algebra 562 (2020), 390–409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' [5] Ardizzoni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=', El Kaoutit, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=', Menini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' - Categories of comodules and chain complexes of modules, Internat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' 23 (2012), 1250109 [6] Bhattacharjee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=', Chirvˇasitu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=', Goswami, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' - Quantum Galois groups of subfactors, Internat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' 33 (2022), 2250013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' [7] Chirvˇasitu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=', Walton, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' - On quantum groups associated to a pair of preregular forms, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Noncommut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' bf 13 (2019), 115—159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' [8] Hyland, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=', Lopez Franco, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=', Vasilakopoulou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' - Hopf measuring comonoids and enrichment, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' 115 (2017), 1118—1148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' [9] Jacobson, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' – Lie algebras, Dover Publications, NY, 1962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' [10] Mac Lane, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' - Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' [11] Manin, Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' - Quantum groups and noncommutative geometry, Universite de Montreal, Centre de Recherches Mathematiques, Montreal, QC, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' [12] Militaru, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' - The automorphisms group and the classification of gradings of finite dimensional associative algebras, Results Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' 77 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' [13] Raedschelders, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=', Van den Bergh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' - The Manin Hopf algebra of a Koszul Artin-Schelter regular algebra is quasi-hereditary, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' 305 (2017), 601-–660.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' [14] Rodrıiguez-Romo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=', Taft, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' - Some quantum-like Hopf algebras which remain noncommutative when q = 1, Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' 61(2002), 41-–50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' [15] Sweedler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' - Hopf Algebras, Benjamin New York, 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' [16] Tambara, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' - The coendomorphism bialgebra of an algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Fac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Tokyo Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' 37 (1990), 425–456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' 14 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' AGORE Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium Simion Stoilow Institute of Mathematics of the Romanian Academy, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content=' Box 1-764, 014700 Bucharest, Romania Email address: ana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='agore@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'} diff --git a/1dAyT4oBgHgl3EQfPfaV/content/tmp_files/2301.00026v1.pdf.txt b/1dAyT4oBgHgl3EQfPfaV/content/tmp_files/2301.00026v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c42ce2457b42062cb4e397b8a40579b1231c8700 --- /dev/null +++ b/1dAyT4oBgHgl3EQfPfaV/content/tmp_files/2301.00026v1.pdf.txt @@ -0,0 +1,2071 @@ +Killing Horizons Decohere Quantum Superpositions +Daine L. Danielson,1, ∗ Gautam Satishchandran,2, 1, † and Robert M. Wald1, ‡ +1Enrico Fermi Institute and Department of Physics, +The University of Chicago, 933 East 56th Street, Chicago, Illinois 60637, USA +2Princeton Gravity Initiative, Princeton University, +Jadwin Hall, Washington Road, Princeton NJ 08544, USA +(Dated: January 3, 2023) +We recently showed that if a massive (or charged) body is put in a quantum spatial superposition, the +mere presence of a black hole in its vicinity will eventually decohere the superposition. In this paper +we show that, more generally, decoherence of stationary superpositions will occur in any spacetime +with a Killing horizon. This occurs because, in effect, the long-range field of the body is registered +on the Killing horizon which, we show, necessitates a flux of “soft horizon gravitons/photons” +through the horizon. The Killing horizon thereby harvests “which path” information of quantum +superpositions and will decohere any quantum superposition in a finite time. It is particularly +instructive to analyze the case of a uniformly accelerating body in a quantum superposition in +flat spacetime. As we show, from the Rindler perspective the superposition is decohered by “soft +gravitons/photons” that propagate through the Rindler horizon with negligible (Rindler) energy. +We show that this decoherence effect is distinct from—and larger than—the decoherence resulting +from the presence of Unruh radiation. We further show that from the inertial perspective, the +decoherence is due to the radiation of high frequency (inertial) gravitons/photons to null infinity. +(The notion of gravitons/photons that propagate through the Rindler horizon is the same notion +as that of gravitons/photons that propagate to null infinity.) We also analyze the decoherence of +a spatial superposition due to the presence of a cosmological horizon in de Sitter spacetime. We +provide estimates of the decoherence time for such quantum superpositions in both the Rindler and +cosmological cases. +1. +INTRODUCTION +Consider a stationary spacetime in which an experimen- +talist, Alice, is present. Alice’s lab is stationary, and she +has control of a charged or massive body (hereinafter re- +ferred to as a “particle”). She sends her particle through a +Stern-Gerlach apparatus or other device that puts her par- +ticle in a quantum superposition of two spatially separated +states1. She keeps these spatially separated components +stationary for a time T and then recombines them. Will +Alice be able to maintain the coherence of these compo- +nents, so that, when recombined, the final state of her +particle will be pure—or will decoherence have occurred, +so that the final state of her particle will be mixed? +Ordinarily, any decoherence effects will be dominated +by “environmental influences,” i.e., additional degrees +of freedom present in Alice’s lab that interact with her +particle. We assume that Alice has perfect control of her +laboratory and its environment so that there is no deco- +herence from ordinary environmental effects. However, +for a charged or massive particle, Alice cannot perfectly +control the electromagnetic or gravitational field, since +her particle acts as a source for these fields and some +∗ daine@uchicago.edu +† gautam.satish@princeton.edu +‡ rmwa@uchicago.edu +1 Quantum spatial superpositions of massive bodies have been of +recent interest in both theoretical as well as proposed experimental +probes of fundamental properties of quantum gravity, e.g., [1–13]. +radiation will be emitted during the portions of her ex- +periment where she separates and recombines her particle. +Nevertheless, in Minkowski spacetime, if her lab is sta- +tionary in the ordinary, inertial sense, she can perform +her experiment in a sufficiently adiabatic manner that +negligible decohering radiation is emitted. In principle, +she can keep the particle separated for an arbitrarily long +time T and still maintain coherence when the components +are recombined. +In a recent paper [14], we showed that the above situ- +ation changes dramatically if a black hole is present in +the spacetime—even though the experiment is carried +out entirely in the black hole’s exterior. In effect, a black +hole horizon harvests “which path” information about any +quantum superposition in its exterior, via the long-range +fields sourced by the superposed matter. We showed that +this results in the unavoidable radiation of entangling +“soft photons or gravitons” through the horizon that carry +the “which path” information into the black hole. Con- +sequently, the mere presence of the black hole implies a +fundamental rate of decoherence on the quantum super- +position2. Although the rate of decoherence will be small +if the black hole is far away, the coherence decays expo- +nentially in the time, T, that the spatial superposition +is maintained. Thus, in any spacetime with a black hole, +there will be essentially complete decoherence within a +2 In QED, this effect applies only to superpositions of charged particles. +However, since all matter sources gravity, the quantum gravitational +decoherence applies to all superpositions. +arXiv:2301.00026v1 [hep-th] 30 Dec 2022 + +2 +finite time3. +The purpose of this paper is to generalize the results of +[14] to spacetimes with Killing horizons, i.e., spacetimes +with a Killing vector field such that there is a null surface +to which the Killing field is normal (see, e.g., [15] for a +discussion of properties of Killing horizons). The event +horizon of a stationary black hole is a Killing horizon +[16–18], so spacetimes with Killing horizons encompass +the case of stationary spacetimes that contain black holes. +However, there are many cases of interest where Killing +horizons are present without the presence of black holes. +One such case is that of Minkowski spacetime, where +the Rindler horizon is a Killing horizon with respect to +the Lorentz boost Killing field. +Another such case is +de Sitter spacetime, where the cosmological horizon is a +Killing horizon. We will show that in these cases, a spatial +superposition that is kept stationary (with respect to the +symmetry generating the Killing horizon) will decohere +in a manner similar to the black hole case. We will also +provide an estimate of the maximum amount of time +during which coherence can be maintained. +The case of the Rindler horizon is particularly instruc- +tive. +The relevant symmetry here is that of Lorentz +boosts, so Alice’s lab will be “stationary” if it is uniformly +accelerating. Our analysis based upon radiation through +the Rindler horizon shows that decoherence of a uniformly +accelerating spatially separated superposition occurs be- +cause of the emission of “soft” (i.e., very low frequency) +gravitons or photons, where the frequency is defined rel- +ative to an affine parameter on the Rindler horizon. As +we shall show, the decoherence effect of this radiation of +soft gravitons or photons is distinct from the (smaller) +decoherence effect resulting from the presence of Unruh +radiation. To gain further insight, we also analyze the +decohering radiation in the electromagnetic case from the +inertial point of view, using the Liénard-Wiechert solution +to determine the radiation at future null infinity. As we +shall show, the decohering photons are of high frequency +at null infinity. +In sec. 2 we provide a general discussion of the deco- +herence of a quantum superposition due to radiation in a +stationary spacetime. In sec. 3 we consider the decoher- +ence of a uniformly accelerating superposition, analyzing +it from both the Rindler and Minkowski viewpoints. We +also show that this decoherence is distinct from (and larger +than) the decoherence effects due to the presence of Un- +ruh radiation. In sec. 4 we analyze the decoherence in de +Sitter spacetime associated with the cosmological horizon. +We will work in Planck units where G = c = ℏ = kB = 1 +and, in electromagnetic formulas, we also put ϵ0 = 1, but +we will restore these constants in our formulas that give +estimates for decoherence times. Lower case Latin indices +represent abstract spacetime indices. Upper case Latin in- +dices from the early alphabet correspond to spatial indices +3 This maximal coherence time for superpositions in the exterior can +be much smaller than the evaporation time of the black hole. +on horizons or null infinity. +2. +DECOHERENCE DUE TO RADIATION IN A +STATIONARY SPACETIME +In this section, we will give a general analysis of the +decoherence of a spatial superposition in a stationary +spacetime due to emission of radiation by the body. Our +analysis applies both to the decoherence of a charged +body due to emission of electromagnetic radiation and to +the decoherence of a gravitating body due to emission of +linearized gravitational radiation. The analyses of these +two cases are very closely parallel. +In order to avoid +repetition, we will analyze only the electromagnetic case +in detail, but near the end of this section, we will state the +corresponding results in the linearized gravitational case, +which can be obtained straightforwardly by replacing the +vector potential Aa with the perturbed metric hab, the +charge-current ja with the stress-energy Tab, etc. +Consider a charged particle4 in a stationary spacetime. +We assume that the particle is initially in a stationary +state. The particle is then put through a Stern-Gerlach (or +other) apparatus, resulting in it being in a superposition +state5 +|ψ⟩ = +1 +√ +2 (|ψ1⟩ + |ψ2⟩) +(2.1) +where |ψ1⟩ and |ψ2⟩ are normalized states that are spa- +tially separated after passing through the apparatus. The +particle is then recombined via a reversing Stern-Gerlach +(or other) apparatus and returns to a stationary state. +We are particularly interested in the case where, between +separation and recombination, |ψ1⟩ and |ψ2⟩ are kept +stationary for a long period of time, T, but we do not +make any such assumption in this section. We wish to +estimate how much decoherence due to emission of elec- +tromagnetic radiation will have occurred by the time of +recombination6. +4 As already indicated above, the “particle” need not be an elementary +particle but could be a “nanoparticle” or any other body whose only +relevant degree of freedom for our analysis is its center of mass. +5 For simplicity, we have assumed that we have a 50-50 superposition +of |ψ1⟩ and |ψ2⟩, but this assumption is not necessary. +6 The decoherence of Alice’s particle can be experimentally deter- +mined as follows. We assume that Alice’s particle initially has spin +in the positive x-direction and thus is in a 50-50 superposition of +z-spin after passing through the initial Stern-Gerlach apparatus. +After recombination, Alice measures the x-spin of her particle. If +coherence of the superposition eq. (2.1) has been maintained, then +(assuming that Alice has made appropriate corrections if there are +any phase differences between the paths) the spin will always be +found to be in the positive x-direction. On the other hand, if any +coherence has been lost, the particle will not be in a state of definite +spin, and the spin will sometimes be found to be in the negative +x-direction. By repeating the experiment many times, Alice can, in +principle, determine the decoherence to any desired accuracy. + +3 +A key assumption that we shall make is that the fluctu- +ations in the charge-current operator ja in the states |ψ1⟩ +and |ψ2⟩ are negligibly small over the scales of interest +so that we can treat the charge current in each of these +states as c-number sources in Maxwell’s equations, given +by ja +1 = ⟨ψ1|ja|ψ1⟩ and ja +2 = ⟨ψ2|ja|ψ2⟩, respectively. In +the initial and final stationary eras, |ψ1⟩ and |ψ2⟩ are +assumed to coincide spatially (though they may differ in +other characteristics, such as spin) so that ja +1 = ja +2 at very +early and very late times. +In order to proceed further, we must specify the initial +state of the electromagnetic field. Since, prior to going +through the Stern-Gerlach apparatus, the charge is as- +sumed to be stationary, at early times we may subtract +the “Coulomb field” Cin +a of the charge, i.e., at early times +we may consider the electromagnetic field observable +Ain +a = Aa − Cin +a 1 +(2.2) +where Cin +a is the (assumed to be unique) stationary clas- +sical solution to Maxwell’s equations with the early time +stationary charged particle source ja +1 = ja +2 and Aa is +the vector potential operator. We need not assume any +specific choice of gauge for Ain +a . Then Ain +a satisfies the +source-free Maxwell’s equations at early times, and we +may extend its definition to all times by requiring it to +satisfy the source-free Maxwell equations everywhere. +The initial state of the electromagnetic field may be +specified by giving the “radiation state” of Ain +a . +The +choice of this state depends on the physical situation being +considered. If the spacetime were globally stationary—i.e., +if the stationary Killing field were everywhere timelike, so, +in particular, there are no Killing horizons—it would be +natural to assume that the initial state of the radiation +is the stationary vacuum state, i.e., the ground state +relative to the time translations. For the case of a black +hole spacetime, it would be correspondingly natural to +assume that the initial state of the radiation is that of +the Unruh vacuum, since for a black hole formed by +gravitational collapse, the state of a quantum field is +expected to approach the Unruh vacuum after the black +hole has “settled down” to a stationary state. For the +case of Minkowski spacetime, we take the initial state +of the radiation to be the ordinary (inertial) Minkowski +vacuum. For de Sitter spacetime, we take the initial state +of the radiation to be the de Sitter invariant vacuum7 for +the electromagnetic field [20]. We denote the initial state +of the radiation in all of the above cases by |Ψ0⟩. +In each of the above cases, |Ψ0⟩ is a pure, quasi-free (i.e., +Gaussian) state. It follows (see, e.g., [22] or appendix A +of [15]) that we can construct a one-particle Hilbert space +Hin and corresponding Fock space F(Hin) wherein |Ψ0⟩ +plays the role of the vacuum state and the field operator +7 +A de Sitter invariant vacuum state does not exist for the massless +scalar field [19] but such a state does exist for the electromagnetic +field [20] and linearized gravitational field [21]. +Ain +a is represented on F(Hin) by +Ain +a (f a) = ia(Kσf) − ia†(Kσf). +(2.3) +Here f a a divergence-free8 test function, σf denotes the +advanced minus retarded solution to Maxwell’s equations +with source f a, and K : S → Hin denotes the map taking +the space S of classical solutions to their representatives +in the one-particle Hilbert space Hin. The commutator +of the creation and annihilation operators in eq. (2.3) is +given by +[a(Kσf), a†(Kσg)] = ⟨Kσf|Kσg⟩ 1. +(2.4) +where ⟨Kσf|Kσg⟩ is the inner product on Hin, which is +given by a natural generalization of the Klein-Gordon +inner product to electromagnetic fields. +For the case of a globally stationary spacetime in the +stationary vacuum state, Kσf corresponds to taking the +positive frequency part of σf with respect to the time +translations generating the stationary symmetry. For the +case of a stationary black hole in the Unruh vacuum state, +Kσf corresponds to taking the positive frequency part of +σf with respect to affine time on the past horizon and +with respect to Killing time at past null infinity. For +Minkowski spacetime in the inertial Minkowski vacuum, +Kσf corresponds to taking the positive frequency part +of σf with respect to inertial time translations. Equiv- +alently, Kσf, in this case, corresponds to the solution +obtained by taking the positive frequency part of the re- +striction of σf to any null hyperplane N (i.e., any Rindler +horizon) with respect to an affine parametrization of the +null geodesics generating N. For de Sitter spacetime in +the de Sitter invariant vacuum, Kσf corresponds to the +solution obtained by taking the positive frequency part +of the restriction of σf to any cosmological horizon with +respect to an affine parametrization of the null geodesics +generating that horizon. +Under the above assumption that the charge-current +of |ψ1⟩ and |ψ2⟩ can be treated as c-number sources, the +electromagnetic field Ai,a in the presence of the charge +in state |ψi⟩ for i = 1, 2 is given in terms of the source +free field Ain +a by [23] +Ai,a = Ain +a + Gret +a (jb +i )1 +(2.5) +where Gret +a (jb +i ) denotes the classical retarded solution for +source jb +i . In particular, since the field Ain +a is in state +|Ψ0⟩, the correlation functions of the electromagnetic field +8 Restriction of the smearing to divergence-free test functions is +necessary and sufficient to eliminate the gauge dependence of Ain +a +(see, e.g., P.101 of [22]). + +4 +Ai,a for |ψi⟩ are given by9 +⟨Ai,a1(x1) . . . Ai,an(xn)⟩ += ⟨Ψ0| +� +Ain +a1(x1) + Gret +a1 (jb +i )(x1)1) +� +. . . +� +Ain +an(xn) + Gret +an (jb +i )(xn)1) +� +|Ψ0⟩. +(2.6) +Equation (2.6) is valid at all times. +However, at +late times—i.e., to the future of any Cauchy surface Σ +corresponding to the time at which recombination has +occurred—we can again subtract off the common sta- +tionary Coulomb field, Cout +a +, of ja +1 = ja +2 to obtain the +source-free field10 Aout +i,a that describes the radiation at +late times for the states |ψi⟩, +Aout +i,a = Ai,a − Cout +a +1 . +(2.7) +By eq. (2.6), at late times, the correlation functions of +Aout +a +are given by +⟨Aout +i,a1(x1) . . . Aout +i,an(xn)⟩ += ⟨Ψ0| +� +Ain +a1(x1) + Ai,a1(x1)1) +� +. . . +� +Ain +an(xn) + Ai,an(xn)1) +� +|Ψ0⟩ +(2.8) +where +Ai,a = Gret +a (jb +i ) − Cout +a +. +(2.9) +Note that Ai,a is a classical solution of the source-free +Maxwell equations in the late-time region. +The correlation functions eq. (2.8) on any late-time +Cauchy surface are precisely those of the coherent state +|Ψi⟩ = e− 1 +2 ∥KAi∥2 exp +� +a†(KAi) +� +|Ψ0⟩ , +(2.10) +where the norm is that of the one-particle inner product +of eq. (2.4). Thus, the coherent state |Ψ1⟩ describes the +“out” radiation state corresponding to charged particle +state |ψ1⟩ and the coherent state |Ψ2⟩ describes the “out” +radiation state corresponding to charged particle state +|ψ2⟩. The joint “out” state, |Υ⟩, of the particle-radiation +system is given by +|Υ⟩ = +1 +√ +2 (|ψ1⟩ ⊗ |Ψ1⟩ + |ψ2⟩ ⊗ |Ψ2⟩) . +(2.11) +Therefore, the decoherence of |ψ1⟩ and |ψ2⟩ due to emis- +sion of electromagnetic radiation is given by +D = 1 − | ⟨Ψ1|Ψ2⟩ |. +(2.12) +9 It is understood that each of the xk variables should be smeared +with a divergence-free test vector field fa +k . +10Note that Ain +a +did not have a subscript “i” whereas Ai,a and +Aout +i,a do carry such subscripts. This is a consequence of the fact +that we are working in the “in” representation—i.e., the Heisenberg +representation on the Hilbert space F(Hin)—so Ain +a does not depend +on the sources, but the other fields do. +We wish to evaluate D. +By the general formula for the inner product of coherent +states, we have +| ⟨Ψ1|Ψ2⟩ | = exp +� +−1 +2||K(A1 − A2)||2 +� +. +(2.13) +Now, in the late-time era, A1,a−A2,a is just the difference +between the classical retarded solutions with sources ja +1 +and ja +2, +A1,a −A2,a = Gret +a (jb +1)−Gret +a (jb +2) = Gret +a (jb +1 −jb +2). (2.14) +Consider the coherent state associated with Gret +a (jb +1 − jb +2) +in the late-time era. We refer to photons in this state as +entangling photons. By the general properties of coherent +states, the expected number, ⟨N⟩, of entangling photons +is given by +⟨N⟩ ≡ ||K +� +Gret(j1 − j2) +� +||2. +(2.15) +Thus, we have +| ⟨Ψ1|Ψ2⟩ | = exp +� +−1 +2⟨N⟩ +� +(2.16) +so +D = 1 − | ⟨Ψ1|Ψ2⟩ | = 1 − exp +� +−1 +2⟨N⟩ +� +(2.17) +and we see that the necessary and sufficient condition for +significant decoherence (D ∼ 1) is ⟨N⟩ ≳ 1. +We summarize the results that we have obtained above +as follows. Under the assumptions we have made above, +in order to calculate the decoherence, D, of the particle +due to radiation, we carry out the following steps: +(1) We obtain the expected charge current, ja +1 and ja +2, +for the particle in states |ψ1⟩ and |ψ2⟩ of the super- +position. +(2) We +calculate +the +classical +retarded +solution, +Gret +a (jb +1 − jb +2) for the difference of these charge cur- +rents, which is a source-free solution at late times, +since ja +1 = ja +2 at late times. +(3) We calculate the one-particle state KGret(j1 − j2) +corresponding to Gret +a (jb +1 − jb +2) at late times. +In +the various cases, this corresponds to the follow- +ing: (i) For a globally stationary spacetime initially +in the stationary vacuum state, this one-particle +state is the positive frequency part of the solution +with respect to the time translations generating the +stationary symmetry. (ii) For the case of a station- +ary black hole initially in the Unruh vacuum, the +one-particle state is the positive frequency part of +the solution with respect to affine time on the past +horizon and with respect to Killing time at past +null infinity. (iii) For Minkowski spacetime initially +in the Minkowski vacuum, the one-particle state +is the positive frequency part of the solution with + +5 +respect to inertial time or, equivalently, the posi- +tive frequency part with respect to affine time on +any Rindler horizon. (iv) For de Sitter spacetime +initially in the de Sitter invariant vacuum, the one- +particle state is the positive frequency part of the +solution with respect to affine time on any cosmo- +logical horizon. +(4) We compute the squared norm, ∥K[Gret(j1 −j2)]∥2, +of this one-particle state at late times. This quan- +tity is equal to the expected number of entangling +photons, ⟨N⟩. The decoherence due to radiation is +then given by +D = 1 − exp +� +−1 +2∥K +� +Gret(j1 − j2) +� +∥2 +� +. +(2.18) +As previously stated, the above analysis extends +straightforwardly to the linearized gravitational case, +where the perturbed metric, hab, is treated as a linear +quantum field propagating in the background classical +stationary spacetime. To compute the decoherence due +to gravitational radiation in this case, we carry out the +above steps, replacing Aa by hab and the charge-current +ja by the stress-energy tensor Tab. The retarded solu- +tion Gret +a (jb) for Maxwell’s equations is replaced by the +retarded solution Gret +ab (Tcd) for the linearized Einstein +equation. The map K : S → Hin is again obtained as +in item (3) above and the inner product on Hin is again +given by a natural generalization of the Klein-Gordon +inner product to linearized gravitational fields. The de- +coherence due to gravitational radiation is then given by +the analog of eq. (2.18). +The above analysis applies for any motion of the compo- +nents of Alice’s superposition. We are primarily interested +in the case where, during a time interval T1, Alice puts +a particle of charge q (or mass m) into a spatial super- +position, where the distance between the components of +the particle wavefunction is d. She then keeps this super- +position stationary in her lab for a time T. Finally, she +recombines her particle over a time interval T2. +In Minkowski spacetime in the case where Alice’s lab is +inertial, Gret +a (jb +1 − jb +2) will be nonzero at null infinity only +at the retarded times corresponding to the time intervals +T1 and T2. A rough estimate of the number of entangling +photons was obtained in [3] using the Larmor formula for +radiation in these eras, which, in natural units, yields +⟨N⟩ ∼ +q2d2 +[min(T1, T2)]2 +(Minkowski, EM). +(2.19) +The corresponding result in the linearized gravitational +case is [3] +⟨N⟩ ∼ +m2d4 +[min(T1, T2)]4 +(Minkowski, GR). +(2.20) +Therefore, if Alice recombines her particle sufficiently +slowly that T1, T2 ≫ qd in the electromagnetic case or +T1, T2 ≫ md2 in the gravitational case, then she can main- +tain the quantum coherence of her particle. In particular, +Alice can keep the components of her particle separated +for as long a time T as she likes without destruction of +the coherence. +As shown in [14], the situation is quite different if a +black hole is present. In the electromagnetic case, even +if T1, T2 ≫ qd so that a negligible number of entangling +photons is emitted to infinity, there will be entangling +radiation emitted into the black hole. For large T, the +number of entangling photons increases with T as11 +⟨N⟩ ∼ M 3q2d2 +D6 +T +(black hole, EM) +(2.21) +where M is the mass of the black hole, D is the proper +distance of Alice’s lab from the horizon of the black hole, +and we assume that D ≳ M. The corresponding result +in the linearized gravitational case is +⟨N⟩ ∼ M 5m2d4 +D10 +T +(black hole, GR). +(2.22) +Thus, the coherence of Alice’s particle will always be +destroyed within a finite time. +In the next two sections, we will apply the above anal- +ysis to the cases of Rindler spacetime and de Sitter space- +time. Although we will explicitly analyze only the Rindler +and de Sitter cases, it will be clear from our analysis of the +next two sections—as well as our analysis in [14]—that it +can be applied to any Killing horizon, provided only that +the initial “vacuum state” |Ψ0⟩ of the electromagnetic +and/or linearized gravitational field corresponds to one- +particle states that are positive frequency with respect to +affine time on the future Killing horizon. +3. +RINDLER HORIZONS DECOHERE +QUANTUM SUPERPOSITIONS +We now consider the case of Minkowski spacetime with +Alice’s lab uniformly accelerating with acceleration a. +Specifically, we take Alice’s lab to follow the orbit +t = 1 +a sinh(aτ), +z = 1 +a cosh(aτ) +(3.1) +of the boost Killing field +ba = a +� +z +� ∂ +∂t +�a ++ t +� ∂ +∂z +�a� +. +(3.2) +Here we have normalized ba such that baba = −1 on +the worldline of Alice’s laboratory. Thus, ba is the four- +velocity of Alice’s laboratory and τ is the proper time in +11In the analysis of [14], we used the fact that the Unruh vacuum is +well approximated by the Hartle-Hawking vacuum at low frequencies +near the horizon of the black hole. + +6 +her lab. We introduce the null coordinates +U ≡ t − z, +V ≡ t + z +(3.3) +and the corresponding vector fields +na ≡ (∂/∂V )a, +ℓa ≡ (∂/∂U)a, +(3.4) +which are globally defined, future-directed null vector +fields that satisfy ℓana = −1. In terms of these coordi- +nates, the Minkowski spacetime metric is +η = −dUdV + dx2 + dy2 +(3.5) +and the boost vector field is given by +ba = a +� +− Uℓa + V na� +. +(3.6) +The boost Killing field is null on the two “Rindler hori- +zons,” i.e., the two null planes U = 0 and V = 0, which +divide Minkowski spacetime into four wedges. The orbits +of the boost Killing field are future-directed and time- +like within the “right Rindler wedge” WR which is the +region U < 0 and V > 0. Thus, the “right Rindler wedge” +WR—where Alice performs her experiment—is a static, +globally hyperbolic spacetime where the notion of “time +translations” is defined by Lorentz boosts. +We refer to the null surface U = 0 as the future Rindler +horizon and denote it as H + +R . On the region V > 0 of +H + +R , it is useful to introduce the coordinate v by +V = V0eav +(3.7) +where V0 is an arbitrary constant. Then, for V > 0 on +H + +R , we have +ba�� +HR+ = aV +� ∂ +∂V +�a���� +HR+ += +� ∂ +∂v +�a���� +HR+ +. +(3.8) +Since (∂/∂V )a on the horizon is tangent to the affinely +parameterized null geodesic generators of H + +R , we refer +to V as the “affine time” on H + +R , whereas we refer to v +as the “boost Killing time” on H + +R . +1. +Decoherence Due to Radiation of Soft +Photons/Gravitons Through the Rindler Horizon +We are now in position to apply the results of sec. 2 +to the Rindler case. We will first analyze the electromag- +netic case and then give the corresponding results in the +gravitational case. +We assume that the electromagnetic field is initially +in the Minkowski vacuum state. We assume that Alice +possesses a charged particle that is initially stationary +(with respect to the boost Killing field) in her (uniformly +accelerating) lab. She then creates a quantum spatial +superposition which is held stationary (with respect to +the boost Killing field) for a proper time T and is then +recombined. We wish to know the degree of decoherence +of Alice’s particle due to emission of radiation. We may +directly apply the analysis of sec. 2 to answer this question. +The future Rindler horizon H + +R (U = 0) does not meet +the technical requirements of being a Cauchy surface for +Minkowski spacetime, since there are inextendible time- +like curves that remain in the past of H + +R as well as +inextendible timelike curves that lie in the future of H + +R . +However, as argued in [24], it is effectively a Cauchy sur- +face for determining evolution of solutions to the wave +equation. This is most easily seen in the conformally +completed spacetime, where H + +R is the past light cone of +a point p ∈ I + except for the single generator that lies +on I + and it also is the future light cone of a point on +p′ ∈ I − except for the single generator that lies on I −. +Data on the full past light cone of p would determine a +solution to the past of H + +R and data on the full future +light cone of p′ would determine a solution to the future +of H + +R , thereby determining a solution everywhere in +Minkowski spacetime. However, for solutions with ap- +propriate decay, the data on the missing null geodesic +generators of I + and I − can be determined by conti- +nuity from the data on H + +R . Consequently, data on H + +R +suffices to uniquely characterize solutions with appropri- +ate decay. Consequently, the “out” states |Ψ1⟩ and |Ψ2⟩ +of the radiation are completely determined by data on +H + +R . Note that this contrasts sharply with the black hole +case, where one would need data on both the future event +horizon and future null infinity to characterize the “out” +state of radiation. +The decoherence of Alice’s particle due to radiation is +given by eq. (2.17). In order to evaluate this, we first +consider a classical point charge of charge q in the “right +Rindler wedge” WR that is stationary with respect to the +boost Killing field and lies at proper distance D from the +bifurcation surface of the Rindler horizon. Such a charge +will be uniformly accelerating with acceleration a given +by +a = 1 +D . +(3.9) +The explicit solution for such a stationary charge in the +Rindler wedge has long been known [25–30]. The only +nonvanishing component of the electromagnetic field in +the region V > 0 of H + +R is +EU ≡ Fabℓanb = +2a2q +π(1 + a2ρ2)2 +(3.10) +where ρ2 ≡ x2 + y2. Electromagnetic radiation through +the Rindler horizon is described by the pullback, EA, of +the electric field Ea = Fabnb to H + +R , where the capital +Latin indices from the early alphabet denote spatial com- +ponents in the x and y directions. Since EA = 0 on the +horizon for a uniformly accelerated charge, one may say +that a charge held stationary in Alice’s lab does not pro- +duce any radiation as determined on H + +R —even though +a uniformly accelerated charge radiates (inertial) energy + +7 +to future null infinity12. +Now consider the case where the point charge is initially +uniformly accelerating with acceleration a at a proper +distance D = 1/a from the bifurcation surface of the +Rindler horizon. +The charge is then moved in the z- +direction to a different orbit of the same boost Killing +field, so that it has uniform acceleration a′ and lies at +proper distance D′ = 1/a′ from the Rindler horizon. After +the charge has reached its new location, the electric field +on H + +R is again given by eq. (3.10), but its value, E′ +U, +will be different from its value at early times. Maxwell’s +equations on H + +R imply that +DAEA = ∂V EU +(3.11) +where DA is the derivative operator on the R2 cross- +sections of the horizon and capital Latin indices from +the early alphabet are raised and lowered with the met- +ric, δAB, on the cross sections. Eq. (3.11) implies that +EA ̸= 0 whenever ∂V EU ̸= 0, so there will be radiation +through the horizon as the charge is being moved. Most +importantly, it implies that +DA +� +� +∞ +� +−∞ +dV EA +� +� = ∆EU +(3.12) +where ∆EU = E′ +U −EU is the change in the radial electric +field between the charge at positions D′ and D. Now, in +a gauge where Aana = 0 on the horizon, the transverse +(i.e., x-y) components of the electric field are related to +the corresponding components of the vector potential by +EA = −∂V AA. +(3.13) +Since the transverse components of the Coulomb field of a +static charge vanish, we may replace the vector potential +AA by the “Coulomb subtracted” vector potential AA +defined by eq.(2.9), so we have +EA = −∂V AA. +(3.14) +It then follows immediately from eq. (3.12) that the dif- +ference, ∆AA, between the final and initial values of AA +is given by +DA(∆AA) = −∆EU +(3.15) +independently of the manner in which the charge is moved +from D to D′. Equation (3.15) is an exact mathemati- +cal analog of the electromagnetic memory effect at null +infinity [31]. +12A uniformly accelerating charge has a nonvanishing inertial energy +current flux Tabta through both H + +R and I +, where ta denotes a +Minkowski time translation. However, the flux of “boost energy” +Tabba vanishes at both H + +R and I +. +For the explicit solution eq. (3.10), we have +∆EU ≈ qda3(1 − a2ρ2) +(1 + a2ρ2)3 +. +(3.16) +where d = D′ − D and we have assumed that +d ≪ D = 1 +a . +(3.17) +From eq. (3.15), we find that ∆AA points in the ˆρ- +direction and has magnitude +|∆AA| = ∆Aρ ≈ +qda4ρ2 +(1 + a2ρ2)2 . +(3.18) +The key point is that even though EA = 0 at both late +and early times, AA does return to its original value at +late times, and the change, ∆AA, in the vector potential +between late and early times is determined only by the +initial and final positions of the charge. +We now consider the quantized radiation through the +horizon resulting from the displacement of the charge, +assuming that, after the displacement, the charge is held +at its new position, D′, forever. +For the Fock space +associated with the Minkowski vacuum state, the map K : +S → Hin that associates one-particle states to classical +solutions is given by taking the positive frequency part of +the classical solution with respect to inertial time, with the +inner product on Hin given by the Klein-Gordon product. +For the electromagnetic field on H + +R in a gauge where +Aana on H + +R , the “free data” on H + +R is the pull-back, +AA, of the vector potential. For two classical solutions +with data A1,A and A2,A on H + +R , the inner product of +their corresponding one-particle states is given by [15, 32] +⟨KA1| KA2⟩H + +R = 2 +� +R2 +dxdy +∞ +� +0 +ωdω +2π δAB ˆ +A1,A ˆ +A2,B +(3.19) +where ˆ +AA(ω, xB) is the Fourier transform of AA(V, xB) +with respect to the affine parameter V . By the same +reasoning as led to eq. (2.15), the expected number of +photons on H + +R in the coherent state associated to any +classical solution AA is simply +⟨N⟩ = ∥KA∥2 +H + +R +(3.20) +where the norm is defined by the inner product eq. (3.19). +However, since ∆AA +̸= +0, the Fourier transform, +ˆ +AA(ω, xB), of AA diverges as 1/ω as ω → 0. +It fol- +lows that the integrand of the expression for the norm +given by the right side of eq. (3.19) also diverges as 1/ω as +ω → 0, so the integral is logarithmically divergent. Thus, +||KA||2 +H + +R = ∞. Therefore, if Alice displaces a charged +particle to a different orbit of the boost Killing field and +the particle remains on this new uniformly accelerated +trajectory forever, an infinite number of “soft horizon + +8 +photons” will be radiated through the Rindler horizon +regardless of how quickly or slowly this process is done. +This is an exact mathematical analog of the infrared di- +vergences that occur at null infinity in QED for processes +with nonzero memory (see e.g., [33–35]). +Now suppose that Alice displaces the particle a z- +distance d ≪ D = 1/a from D to D′ = D+d as above, but +instead of leaving the particle at D′ forever, she leaves it +there for proper time13 T and then returns it to D. In this +case, the transverse components of the vector potential, +AA, return to their initial values at late times, so there +is no “memory effect” at the horizon. Correspondingly, +there are no infrared divergences in the expected number +of photons that propagate through H + +R . Nevertheless, if +T is very large then the expected number of photons ⟨N⟩ +will be correspondingly large. To see this, we note that +if, for convenience, we work in a gauge where AA = 0 +initially, then during the era at which the particle is at D′, +AA will be given by the right side of eq. (3.18). If we keep +the manner in which the particle is moved from D to D′ +as well as from D′ to D fixed but take T to be very large, +the asymptotic behavior of the norm eq. (3.19) will be +dominated by the low-frequency contribution from the era +of time T that the particle is displaced. The logarithmic +divergence at ω = 0 that would occur if the particle re- +mained at D′ forever is now effectively cut off at frequency +ω ∼ 1/V , where V denotes the affine time duration on +the horizon H + +R over which the particle remains at D′. +We obtain +⟨N⟩ = ||KA||2 +HR ∼ q2d2a2 ln +� +V +min[V1, V2] +� +(3.21) +where V1, V2 ≪ V are the durations of affine time over +which the particle is displaced from D to D′ and from +D′ back to D, so that 1/min[V1, V2] provides an effective +high-frequency cutoff. However, the affine time V on the +horizon is related to boost Killing time on the horizon by +V = V0 exp(av) +(3.22) +and the boost Killing time v corresponds to the proper +time T in Alice’s lab. Thus, we obtain +⟨N⟩ ∼ q2d2a3T +(Rindler, EM) . +(3.23) +Therefore, no matter how slowly the particle is displaced, +it is forced to radiate a number of “soft Rindler horizon +photons” through the Rindler horizon that is proportional +to the time T that the particle remains on the displaced +trajectory. +We are now in a position to fully analyze Alice’s exper- +iment. Alice’s lab is uniformly accelerating with acceler- +13We have normalized the boost Killing field ba so that Killing time +equals proper time on the orbit at D with acceleration a. Since we +assume d = D′ − D ≪ D, Killing time and proper time are also +(nearly) equal on the orbit at D′. Thus, T is also the elapsed Killing +time that Alice keeps the particle at D′. +ation a in Minkowski spacetime. She puts her particle +of charge q into a superposition of states separated by +z-distance d ≪ 1/a and keeps these components sta- +tionary in her lab for a proper time T. +She then re- +combines the components and determines their coher- +ence14. By the analysis of sec. 2, the decoherence is given +by eq. (2.18). However, for large T, the calculation of +||K [Gret(j1 − j2)] ||2 corresponds precisely to the calcu- +lation we have given above of the number of photons +radiated through the Rindler horizon when a charge is +displaced for a time T. Thus, we obtain +||K +� +Gret(j1 − j2) +� +||2 ∼ q2d2a3T. +(3.24) +In other words, for large T, Alice’s superposition will de- +cohere due to radiation of “soft Rindler horizon photons,” +as +D = 1 − exp(−ΓradT) +(3.25) +where the “decoherence rate” Γrad, is given by, +Γrad = q2d2a3. +(3.26) +Thus, restoring the constants c, ℏ, and ϵ0, Alice’s par- +ticle will decohere within a time +TD ∼ ϵ0ℏc6 +a3q2d2 +(Rindler, EM) +(3.27) +∼ 1033 years +�g +a +�3 +· +�e +q +�2 +· +�m +d +�2 +. +(3.28) +Thus, if Alice’s lab uniformly accelerates at one g in +flat spacetime and she separates an electron into two +components one meter apart, she would not be able to +maintain coherence of the electron for more than 1033 +years. +A similar analysis holds in the gravitational case15 +where Alice separates a massive body with mass m across +a distance d and maintains this superposition for a time +T. In the gravitational case, the “electric part” of the +perturbed Weyl tensor Eab = Cacbdncnd plays an analo- +gous role to the electric field Ea in the electromagnetic +version of the gedankenexperiment. For a uniformly ac- +celerating point mass, the only non-vanishing compo- +nent of the electric part of the Weyl tensor on H + +R is +EUU = Cacbdℓancℓbnd. +Gravitational radiation on the horizon is described +by the pullback, EAB, of Eab, which vanishes for the +static point mass. However, the process of quasistatically +moving the static point mass involves a change in EUU +on H + +R . The (once-contracted) Bianchi identity on the +14The coherence can be determined as described in footnote 6. +15In the gravitational case, additional stress-energy will be needed +to keep Alice’s particle in uniform acceleration. We will ignore the +gravitational effects of this additional stress-energy. + +9 +horizon yields +DAEAB = ∂V EUB, +DAEUA = ∂V EUU +(3.29) +which implies that +DADBEAB = ∂2 +V EUU +(3.30) +which is closely analogous to eq. (3.11). As in the elec- +tromagnetic case, if a uniformly accelerating point mass +is quasistatically moved there is necessarily gravitational +radiation through H + +R . +To determine the number of “Rindler horizon gravitons” +emitted we quantize the linearized gravitational field. For +a metric perturbation hab in a gauge where habna = 0 +and δABhAB = 0, the “free data” on H + +R +is hAB. A +“particle” in the standard Fock space associated to the +Poincaré invariant vacuum is then a positive frequency +solution with respect to affine parameter V and the inner +product on the one-particle Hilbert space is given by a +direct analog of eq. (3.19) with the vector potential AA +replaced with the metric perturbation hAB, namely +⟨Kh1| Kh2⟩H + +R = 1 +8 +� +R2 +dxdy +∞ +� +0 +ωdω +2π δABδCDˆh1,ACˆh2,BD. +(3.31) +Finally, EAB is related to the metric perturbation hAB +by +EAB = −1 +2∂2 +V hAB . +(3.32) +Equations (3.30) and (3.32) directly imply that a per- +manent change, ∆EUU ̸= 0, in the U-U component of +the electric part of the Weyl tensor on H + +R +implies a +permanent change, ∆hAB ̸= 0, in the perturbed metric +on H + +R between early and late times. In the quantum +theory, as in the electromagnetic case, this implies a log- +arithmic infrared divergence in the number of gravitons +emitted through H + +R in the process where a uniformly +accelerating charge is moved to a new orbit of the same +boost Killing field and then remains at the new position +forever. +The analysis of Alice’s experiment proceeds in a similar +manner to the electromagnetic case. Alice does not main- +tain the relative separation of her wavefunction forever +but closes her superposition after a proper time T. As +before, the number of entangling gravitons emitted to +the Rindler horizon is logarithmically growing in affine +time and therefore linearly growing in the proper time +duration T of Alice’s experiment. We obtain +⟨N⟩ ∼ m2d4a5T +(Rindler, GR) . +(3.33) +Thus, restoring constants, we find that the Rindler hori- +zon decoheres the quantum superposition of a uniformly +accelerating massive body in a time +T GR +D +∼ +ℏc10 +Gm2d4a5 +(Rindler, GR) +(3.34) +∼ 2 fs +�MMoon +m +�2 +· +�RMoon +d +�4 +· +�g +a +�5 +. +(3.35) +Therefore, if the Moon were accelerating at one g and +occupied a quantum state with its center of mass super- +posed by a spatial separation of the order of its own radius +then it would decohere within about 2 femtoseconds. Of +course, it would not be easy to put the moon in such a +coherent quantum superposition. +Note the acceleration of a stationary observer outside +of a black hole who is reasonably far16 (D ≳ M) from the +event horizon is a ∼ M/D2. If we substitute a = M/D2 +in eqs. (3.27) and (3.34), we obtain eqs. (2.21) and (2.22), +respectively. Therefore, it might be tempting to believe +that what is important in all cases is the acceleration of +Alice’s lab. However, this is not the case. In particular, +if we replace the black hole by an ordinary star (and if +there are no dissipative effects in the star), then there +will not be any analogous decoherence effect, even though +the acceleration of Alice’s lab is the same as in the case +of a black hole. Furthermore, as we shall see in sec. 4, +decoherence effects associated with the cosmological hori- +zon occur in de Sitter spacetime even for nonaccelerating +observers. It is the presence of a Killing horizon that +is the essential ingredient for the fundamental rate of +decoherence of quantum superpositions as described in +this paper. +We now consider another potential cause of decoherence, +namely Unruh radiation. +2. +Decoherence Due to Scattering of Unruh +Radiation +The Minkowski vacuum state restricted to a Rindler +wedge is a thermal state at the Unruh temperature +T = a +2π +(3.36) +relative to the notion of time translations defined by +the Lorentz boost Killing field ba, eq. (3.2). Thus, the +superposition state of Alice’s particle will be buffeted by +this thermal bath of Unruh radiation. Scattering of this +radiation will cause some decoherence of Alice’s particle. +Indeed, since this decoherence should occur at a steady +rate while the superposition is kept stationary (and thus +the decoherence will be proportional to T), one might even +16It should be emphasized that the estimates made in [14] that yielded +eqs.(2.21) and (2.22) assumed that Alice’s lab is reasonably far from +the black hole. If Alice’s lab is extremely close to the black hole +(i.e., at a distance D ≪ M from the horizon), then the black hole +analysis would reduce to the Rindler case analyzed here. + +10 +suspect that scattering of Unruh radiation could be the +same effect as found in the previous section but expressed +in a different language. The purpose of this subsection +is to show that this is not the case, i.e., decoherence +due to scattering of Unruh radiation and decoherence +due to radiation of “soft” photons/gravitons through the +horizon are distinct effects. Furthermore, we shall show +that, for reasonable parameter choices, the decoherence +rate due to the scattering of Unruh radiation is smaller +than the decoherence rate due to emitted radiation as +obtained in the previous section. We will consider only +the electromagnetic case in this subsection. +The decoherence rate of a spatial superposition due +to collisions with particles in an environment has been +analyzed in [36–39], and we will adapt this analysis to +obtain a rough estimate of the decoherence caused by the +scattering of Unruh radiation. As in eq. (2.1), Alice has +a particle of charge q in a state |ψ⟩ = (|ψ1⟩ + |ψ2⟩)/ +√ +2, +where |ψ1⟩ and |ψ2⟩ are spatially separated by a distance +d. Since we require d ≪ 1/a (see eq. (3.17)) and since +the typical wavelength of Unruh photons at temperature +eq. (3.36) is λ ∼ 1/a, we are in the scattering regime +where λ ≫ d. +In an elastic scattering event between +Alice’s particle and a photon in the Unruh radiation, the +final outgoing state of the photon will depend upon which +branch of the superposition the photon scattered off of. +Let |χ1⟩ denote the outgoing state of the Unruh photon +for scattering off of |ψ1⟩ and let |χ2⟩ denote the outgoing +state for scattering off of |ψ2⟩. Decoherence will occur to +the extent to which these outgoing states of the scattered +Unruh photon are distinguishable, i.e., D = 1−| ⟨χ1|χ2⟩ |. +In order to obtain a rough estimate of the decoherence +resulting from a single scattering event, we consider the +corresponding Minkowski process of the scattering of a +photon of momentum p off of an inertial superposition +separated by d, with d ≪ 1/p. Assuming that the charged +particle states |ψ1⟩ and |ψ2⟩ are identical except for their +location, the scattered photon states |χ1⟩ and |χ2⟩ should +differ only by the action of the translation operator e−i ⃗P·⃗d, +i.e., +|χ2⟩ ≈ e−i ⃗P·⃗d |χ1⟩ +(3.37) +where ⃗P denotes the photon momentum operator. Ex- +panding the exponential, we obtain the following rough +estimate of the decoherence resulting from a single scat- +tering event involving a photon of momentum p +1 − | ⟨χ1|χ2⟩ | ∼ p2d2 +(3.38) +where we have ignored any dependence on the angle be- +tween the incoming momentum ⃗p and the separation ⃗d. +We will take eq. (3.38) as our estimate of the decoherence +of Alice’s particle resulting from the scattering of a single +Unruh photon of “Rindler momentum” p (i.e., of energy +ϵ = p with respect to the boost Killing field ba). +The total decoherence rate due to scattering of Unruh +radiation is then given by +Γscatt ∼ d2 +∞ +� +0 +dp p2ϱ(p)σ(p) +(3.39) +where ϱ(p) is the number density of photons at momentum +p (so ϱ(p) is also the incoming flux of photons) and σ(p) +is the scattering cross-section. For a thermal distribution +of photons17 we have +ϱ(p) ∼ +p2 +ep/T − 1. +(3.40) +We take σ to be given by the Thomson cross-section +σ = 8π +3 +q4 +(4πm)2 , +(3.41) +where m is the mass of Alice’s particle. Putting this all +together, our estimate of the decoherence rate due to +scattering of Unruh photons is +Γscatt ∼ q4d2a5 +m2 +(Rindler, EM) . +(3.42) +Comparing eq. (3.42) to the rate of decoherence, Γrad +due to the emission of soft photons given by eq. (3.26), +one can immediately see that the effects are distinct. +In particular, Γrad has no dependence on the mass, m, +of Alice’s particle, whereas Γscatt does depend on m on +account of the mass dependence of the scattering cross- +section. The ratio of these decoherence rates is given +by +Γscatt +Γrad +∼ q2a2 +m2 = +�q/m +D +�2 +(3.43) +Now, q/m is the “charge radius” of Alice’s particle and, +as argued in [3], it represents a fundamental lower bound +to the spread of a charged particle due to vacuum fluc- +tuations of the electromagnetic field. Therefore, in order +that |ψ1⟩ and |ψ2⟩ not overlap, we must have d > q/m. +Since d ≪ D, we conclude that +Γscatt +Γrad +≪ 1 +(3.44) +i.e., the contribution to decoherence from the scattering +of Unruh radiation is negligible compared with the de- +coherence due to emission of soft photons through the +Rindler horizon. +A similar analysis holds for a charged particle superpo- +sition outside of a black hole. It is worth noting, that the +17The factor of p2 in the numerator of eq. (3.40) arises from the density +of states in Minkowski spacetime. We ignore here any differences +between the Minkowski and Rindler densities of states. + +11 +decoherence effects due to scattering of Hawking radiation +will decrease with distance, D, from the black hole only +as 1/D2 for large D, giving, +Γscatt ∼ +q4d2 +m2M 3 +1 +D2 +(black hole, EM). +(3.45) +On the other hand, by eq. (2.21) the decoherence effects +of radiation of soft photons through the horizon decreases +with D as 1/D6. Thus at sufficiently large D, the deco- +herence effects due to scattering of Hawking radiation +will dominate. However, in this regime, both effects are +extremely small. +3. +Decoherence From the Inertial Perspective +In our analysis of the decoherence of a spatial superpo- +sition in the presence of a black hole [14] as well as in our +analysis of the decoherence of a spatial superposition in +Rindler spacetime given above in sec. 3.1, it may appear +that we have introduced a radical new mechanism for de- +coherence, namely radiation of soft photons and gravitons +through a horizon. The main purpose of this subsection +is to show that, in fact, the decoherence we derived in the +Rindler case can also be obtained by entirely conventional +means. In the Rindler case, we are simply considering a +uniformly accelerating superposition in Minkowski space- +time. The radiation of entangling photons to infinity from +such a superposition can be calculated in the inertial view- +point by standard methods, without introducing concepts +such as a Rindler horizon. It is instructive to calculate +the decoherence from the inertial viewpoint both in order +to validate the results of sec. 3.1 as well as to gain insight +into how the emitted “soft photons” would be interpreted +by an inertial observer. As we shall see, the entangling +photons as seen by inertial observer at large distances +near θ = 0 will be “hard” even though, from her point of +view, Alice has performed the experiment adiabatically. +We will restrict our analysis in this subsection to the +electromagnetic case. +The Liénard-Wiechert solution for the potential of a +point charge in Minkowski spacetime following an arbi- +trary worldline Xµ(τ) is, in Lorenz gauge, +Aµ(x) = 1 +4π +1 +α +q +|⃗x − ⃗X(tret)| +dXµ +dt (tret) +(3.46) +where +α ≡ 1 − ˆn · d ⃗X +dt (tret) +and ˆn = ⃗x − ⃗X(tret) +|⃗x − ⃗X(tret)| +. +(3.47) +For a uniformly accelerated trajectory with acceleration +a, we have +Xµ(τ) = +�1 +a sinh(aτ), 0, 0, 1 +a cosh(aτ) +� +. +(3.48) +In Bondi coordinates (u, r, θ, φ) with +u ≡ t − r +(3.49) +the future light cone of an event at proper time τ on the +worldline eq. (3.48) reaches null infinity at +au = sinh(aτ) − cos θ cosh(aτ). +(3.50) +Electromagnetic radiation is described by the pullback +of the electromagnetic field, eq. (3.46), to null infinity. +Taking the limit as r → ∞ at fixed u, we obtain18 +AA(u, θ, φ) = −q +4π +sinh(aτ) sin θ +cosh(aτ) − cos θ sinh(aτ)(dθ)A +(3.51) +where, in this subsection, capital indices from the early +alphabet denote angular components on the 2-sphere cross- +sections of I +. We will be concerned with the difference, +at fixed (u, θ, φ), between the electromagnetic radiation +of a particle following the trajectory eq. (3.48) and a +particle following a similar trajectory that is displaced in +the z-direction by a proper distance d ≪ 1/a and thus +has +δa = a2d. +(3.52) +We denote this difference by +Ad +A(u, θ, φ) ≡ AA(a + δa) − AA(a) ≈ δa +�∂AA +∂a +� +u,θ +(3.53) +From eq. (3.51), we obtain +Ad +A = −a2qd +4π +u sin θ +(cosh(aτ) − cos θ sinh(aτ))3 (dθ)A +(3.54) +where eq. (3.50) was used to compute (∂τ/∂a)(u,θ). +In her experiment, Alice starts with her particle in a +uniformly accelerating state. Over a proper time T1, she +separates it into two uniformly accelerating components +separated by a distance d as above. +She keeps these +components separated for a proper time T, and she then +recombines them over a proper time T2. The difference +between the radiation fields of these components is given +by +AA ≡ A1,A − A2,A = F(τ)Ad +A +(3.55) +where the smooth function F is such that F(τ) = 0 for +τ < −T1 and τ > T +T2, whereas F(τ) = 1 for 0 < τ < T. +18The vector potential is not smooth at I + in Lorenz gauge but +one can do an asymptotic gauge transformation such that Aa is +smooth at I +. Such a gauge transformation does not affect the +angular components AA at I + [35], so we can calculate AA using +our Lorenz gauge expression. + +12 +The entangling photon content is then given by +⟨N⟩ = ||KA||2 = 2 +� +S2 +dΩ +∞ +� +0 +ωdω +2π +ˆ +AA ˆ +AA +(3.56) +where +ˆ +AA(ω, θ, φ) denotes the Fourier transform of +AA(u, θ, φ) with respect to u, i.e., +ˆ +AA(ω, θ, φ) = +∞ +� +−∞ +du eiωuAA(u, θ, φ). +(3.57) +We are interested in estimating ⟨N⟩ for large T. +In order to evaluate the Fourier transform integral, it +is useful to note that, at fixed a, we have +du +dτ = cosh(aτ) − cos θ sinh(aτ) +(3.58) +and +d2u +dτ 2 = a2u. +(3.59) +It follows that +d +du +� +1 +du/dτ +� += +1 +du/dτ +d +dτ +� +1 +du/dτ +� += +−a2u +(cosh(aτ) − cos θ sinh(aτ))3 +(3.60) +Thus, we have +Ad +A = qd sin θ +4π +(dθ)A +d +du +� +1 +du/dτ +� +(3.61) +and +ˆ +AA = qd sin θ +4π +(dθ)A +∞ +� +−∞ +du eiωuF(τ) d +du +� +1 +du/dτ +� +. +(3.62) +Integrating by parts, we obtain +ˆ +AA(ω, xA) = − qd sin θ +4π +(dθ)A +� +iω +∞ +� +−∞ +du eiωu F(τ) +du/dτ ++ +∞ +� +−∞ +du eiωu +F ′(τ) +(du/dτ)2 +� +. +(3.63) +The second term in this equation contributes only during +the time intervals (−T1, 0) and (T, T + T2) when Alice +opens and closes the superposition. For large T, its con- +tribution can be shown to be negligible compared with +the first term. Therefore, we have +ˆ +AA(ω, xA) ≈ −(dθ)A +iωqd sin θ +4π +I +(3.64) +where +I ≡ +∞ +� +−∞ +du eiωu F(τ) +du/dτ . +(3.65) +To evaluate I, we approximate F by a step function in +the τ-interval [0, T]. The corresponding interval, [u0, uT ], +in u is +u0 = −1 +a cos θ +uT = 1 +2a +� +eaT (1 − cos θ) − e−aT (1 + cos θ) +� +. +(3.66) +Noting that +du +dτ = +� +a2u2 + sin2 θ +(3.67) +we obtain +I ≈ +uT +� +u0 +du +eiωu +� +a2u2 + sin2 θ +. +(3.68) +It can be seen that for large T, the dominant contribution +to I will come from small angles, θ ≪ 1. For aT ≫ 1, the +upper limit of the integral may then be approximated as +uT ≈ 1 +4aeaT θ2 − 1 +ae−aT +for θ ≪ 1 +∼ +� +0 +for θ2/4 < e−aT +1 +4aθ2eaT +for θ2/4 ≥ e−aT . +(3.69) +For aT ≫ 1, the contribution to I from θ2/4 < e−aT +can be shown to make a negligible contribution to ⟨N⟩, +eq. (3.56). Therefore, we may approximate I as +I ∼ Θ(θ2 − 4e−aT ) +exp(aT )θ2/(4a) +� +−1/a +du +eiωu +� +a2u2 + sin2 θ +(3.70) +where +Θ(x) ≡ +� +0 +for x < 0 +1 +for x ≥ 0. +(3.71) +For 0 < ω < 4ae−aT /θ2, we may bound I by replacing +eiωu by 1. The integral can then be evaluated explic- +itly, and it can be shown that for aT ≫ 1, the con- +tribution to ⟨N⟩ from this frequency range is negligi- +ble. For ω > 4ae−aT /θ2, the integrand is oscillatory for +u > exp(aT)θ2/(4a), and, for aT ≫ 1, we will make neg- +ligible error in our estimate of ⟨N⟩ if we replace the upper +limit of eq. (3.70) by ∞. We will also make a negligible +error by replacing the lower limit by 0. Thus, for aT ≫ 1, + +13 +we may approximate I as +I ∼ Θ(θ2−4e−aT )Θ(ω−4ae−aT /θ2) +∞ +� +0 +du +eiωu +� +a2u2 + sin2 θ +. +(3.72) +Evaluating the integral we obtain +I ∼ 1 +aΘ(θ2 − 4e−aT )Θ(ω − 4ae−aT /θ2) +�1 +2iπI0(sin θω/a) ++K0(sin θω/a) − 1 +2iπLLL0(sin θω/a) +� +(3.73) +where I0, K0 are Bessel functions and LLL0 is a Struve +function. This expression is highly suppressed for ω > a/θ, +so we can expand in θω/a and truncate the function above +ω = a/θ to obtain, +I ∼ −1 +aΘ(1−θω/a)Θ(θ2−4e−aT )Θ(ω−4ae−aT /θ2) ln (θω/a) . +(3.74) +Note that the restrictions ω < a/θ, and θ > 2e−aT/2 im- +ply a frequency cutoff at ω ∼ aeaT/2/2. By eqs.(3.74) and +(3.64), the frequency spectrum of ˆ +AA goes as ω ln(ω/a) +up to this cutoff, i.e., the spectrum is “hard” and becomes +increasingly so for large T. This contrasts with the in- +creasingly “soft” spectrum on the Rindler horizon, which +goes as 1/ω down to a low frequency cutoff ∼ 1/V ∝ e−aT . +Thus, the “soft horizon photons” from the Rindler per- +spective are “hard” photons from the inertial perspective. +From eq. (3.56) for ⟨N⟩ together with our expression +eq. (3.64) for ˆ +AA and the expression eq. (3.74) that we +have just derived for I, we obtain +⟨N⟩ ∼ +�qd +a +�2 � +dωdθ θ3ω3 +� +ln ωθ +a +�2 +(3.75) +where the region of ω-θ integration is determined by the Θ- +functions appearing in eq. (3.74) as well as the geometrical +restriction θ ≲ 1. We can break up this region into the +portion with ω ≤ a and the portion with ω > a. Since +the region with ω ≤ a and θ ≲ 1 is bounded and the +integrand of eq. (3.75) is bounded in this region, the +contribution to ⟨N⟩ from ω ≲ a is bounded by a constant +that is independent of T. We may therefore discard this +contribution. In the region ω > a, the third Θ-function +in eq. (3.74) is redundant, and the integration region is +a ≤ω≤ aeaT/2/2 +(3.76) +2e−aT/2 ≤θ≤ a +ω . +(3.77) +For aT ≫ 1, we will make negligible error by replacing +the lower limit of θ by 0. We thereby obtain +⟨N⟩ ∼ +�qd +a +�2 a exp(aT/2)/2 +� +a +dω +a/ω +� +0 +dθ θ3ω3 +� +ln ωθ +a +�2 +. +(3.78) +Making the change of variables from θ to +x = ω +a θ +(3.79) +we find that the θ-integral becomes +a/ω +� +0 +dθ θ3ω3 +� +ln ωθ +a +�2 += a +ω a3 +1 +� +0 +dx x3(ln x)2 ∼ a4 +ω . +(3.80) +Thus, we obtain +⟨N⟩ ∼ +�qd +a +�2 +a4 +a exp(aT/2)/2 +� +a +dω +ω +∼ a2q2d2 ln[exp(aT/2)] +∼ a3q2d2T. +(3.81) +This estimate agrees with eq. (3.23). +Thus, we have succeeded—with considerable effort!—in +our goal of deriving the decoherence of Alice’s superpo- +sition by entirely conventional means. It is notable how +much simpler the calculation of sec. 3.1 was compared to +the calculation that we have just completed. +4. +COSMOLOGICAL HORIZONS DECOHERE +QUANTUM SUPERPOSITIONS +In this section, we apply our analysis to de Sitter space- +time. +The de Sitter metric in a static patch is given +by +ds2 = −f(r)dt2 + f(r)−1dr2 + r2qABdxAdxB +(4.1) +where, in this section, xA are angular coordinates on the +2-sphere, qAB is the unit round metric on the 2-sphere, +and +f(r) = 1 − r2/R2 +H +(4.2) +where RH (the “Hubble radius”) is a constant. +The +coordinate singularity at r = RH corresponds to the +“cosmological horizon,” which is a Killing horizon of the +static Killing field (∂/∂t)a. The relation between “affine +time,” V , and “Killing time,” v, on the future cosmological +horizon is +V = ev/RH. +(4.3) +The general analysis of sec. 2 applies to the decoherence +of a static superposition in de Sitter spacetime. The esti- +mates of the decoherence due to emission of soft photons +and gravitons through the cosmological horizon when Al- +ice keeps the superposition present for a time T can be +made in exact parallel with the analysis of sec. 3 in the +Rindler case and [14] in the black hole case. The only +noteworthy new ingredient in de Sitter spacetime is that + +14 +the worldline r = 0 is an orbit of the static Killing field +that is inertial, i.e., non-accelerating. We now estimate +the decoherence of a spatial superposition created in Al- +ice’s lab at r = 0 and thereby show that decoherence will +occur even though Alice’s lab is not accelerating. +By Gauss’ law, a point charge placed at r = 0 will give +rise to a radial electric field EU on the future cosmological +horizon given by +EU ∼ +q +R2 +H +(4.4) +where EU = Fabℓanb on the horizon with na = (∂/∂V )a +tangent to the affinely parametrized null generators of +the horizon and ℓa = (∂/∂U)a a radial null vector with +naℓa = −1. The change in the electric field on the horizon +resulting from a displacement of the charge to r = d ≪ +RH is +∆EU ∼ qd +R3 +H +. +(4.5) +By paralleling the steps that led to eq. (3.18) above, we +find that the change in the tangential components of the +vector potential at the horizon is +|∆AA| ≡ +� +R−2 +H qAB∆AA∆AB +�1/2 ∼ qd +R2 +H +. +(4.6) +By paralleling the steps that led to eq. (3.23)—assuming +that the electromagnetic field is initially in the de Sitter +invariant vacuum (see footnote 7)—we obtain the estimate +⟨N⟩ ∼ q2d2 +R3 +H +T +(de Sitter, EM) . +(4.7) +Thus, restoring constants, the decoherence time due to +the presence of the cosmological horizon is +TD ∼ ℏϵ0R3 +H +q2d2 +(de Sitter, EM) . +(4.8) +Since d ≪ RH, the decoherence time will be much larger +than the Hubble time RH/c unless q is extremely large +relative to the Planck charge qP ≡ √ϵ0ℏc. Nevertheless, +we see that decoherence does occur despite the fact that +Alice’s lab is inertial. +A similar analysis applies in the gravitational case for +a spatial superposition of a massive particle in Alice’s lab +at r = 0. In parallel with the derivation given in sec. 3.1 +above, we find +⟨N⟩ ∼ m2d4 +R5 +H +T +(de Sitter, GR) +(4.9) +which leads to a decoherence time +T GR +D +∼ +ℏR5 +H +Gm2d4 +(de Sitter, GR) . +(4.10) +ACKNOWLEDGMENTS +D.L.D. acknowledges support as a Fannie and John +Hertz Foundation Fellow holding the Barbara Ann Cana- +van Fellowship and as an Eckhardt Graduate Scholar +in the Physical Sciences Division at the University of +Chicago. This research was supported in part by NSF +Grant No. 21-05878 to the University of Chicago. +[1] S. Bose, A. Mazumdar, G. W. Morley, H. Ulbricht, +M. Toroš, M. Paternostro, A. Geraci, P. Barker, M. S. +Kim, and G. Milburn, Spin Entanglement Witness for +Quantum Gravity, Phys. Rev. Lett. 119, 240401 (2017), +arXiv:1707.06050 [quant-ph]. +[2] C. Marletto and V. Vedral, Gravitationally-induced en- +tanglement between two massive particles is sufficient +evidence of quantum effects in gravity, Phys. Rev. Lett. +119, 240402 (2017), arXiv:1707.06036 [quant-ph]. +[3] A. Belenchia, R. M. Wald, F. Giacomini, E. Castro-Ruiz, +v. Brukner, and M. Aspelmeyer, Quantum Superposition +of Massive Objects and the Quantization of Gravity, Phys. +Rev. D 98, 126009 (2018), arXiv:1807.07015 [quant-ph]. +[4] M. Christodoulou and C. Rovelli, On the possibility of lab- +oratory evidence for quantum superposition of geometries, +Physics Letters B 792, 64 (2019). +[5] F. Giacomini, E. Castro-Ruiz, and v. Brukner, Quantum +mechanics and the covariance of physical laws in quan- +tum reference frames, Nature Commun. 10, 494 (2019), +arXiv:1712.07207 [quant-ph]. +[6] C. Gonzalez-Ballestero, M. Aspelmeyer, L. Novotny, +R. Quidant, and O. Romero-Isart, Levitodynamics: Lev- +itation and control of microscopic objects in vacuum, +Science 374, 3027 (2021), arXiv:2111.05215 [quant-ph]. +[7] D. L. Danielson, G. Satishchandran, and R. M. Wald, +Gravitationally mediated entanglement: Newtonian field +versus gravitons, Phys. Rev. D 105, 086001 (2022), +arXiv:2112.10798 [quant-ph]. +[8] D. Carney, Newton, entanglement, and the graviton, Phys. +Rev. D 105, 024029 (2022), arXiv:2108.06320 [quant-ph]. +[9] M. +Christodoulou, +A. +Di +Biagio, +M. +Aspelmeyer, +v. Brukner, C. Rovelli, and R. Howl, Locally mediated en- +tanglement through gravity from first principles, (2022), +arXiv:2202.03368 [quant-ph]. +[10] D. Carney, Y. Chen, A. Geraci, H. Müller, C. D. Panda, +P. C. E. Stamp, and J. M. Taylor, Snowmass 2021 +White Paper: Tabletop experiments for infrared quan- +tum gravity, in 2022 Snowmass Summer Study (2022) +arXiv:2203.11846 [gr-qc]. +[11] T. Feng and V. Vedral, Amplification of gravitationally +induced entanglement, Phys. Rev. D 106, 066013 (2022), +arXiv:2202.09737 [quant-ph]. +[12] R. Zhou, R. J. Marshman, S. Bose, and A. Mazum- +dar, Catapulting towards massive and large spatial quan- +tum superposition, Phys. Rev. Res. 4, 043157 (2022), +arXiv:2206.04088 [quant-ph]. + +15 +[13] C. Overstreet, J. Curti, M. Kim, P. Asenbaum, M. A. +Kasevich, and F. Giacomini, Inference of gravitational +field superposition from quantum measurements, (2022), +arXiv:2209.02214 [quant-ph]. +[14] D. L. Danielson, G. Satishchandran, and R. M. Wald, +Black holes decohere quantum superpositions, Int. J. Mod. +Phys. D 31, 2241003 (2022), arXiv:2205.06279 [hep-th]. +[15] B. S. Kay and R. M. Wald, Theorems on the Unique- +ness and Thermal Properties of Stationary, Nonsingular, +Quasifree States on Space-Times with a Bifurcate Killing +Horizon, Phys. Rept. 207, 49 (1991). +[16] S. W. Hawking and G. F. R. Ellis, The Large Scale Struc- +ture of Space-Time, Cambridge Monographs on Mathe- +matical Physics (Cambridge University Press, 2011). +[17] S. W. Hawking, Black holes in general relativity, Commun. +Math. Phys. 25, 152 (1972). +[18] S. Alexakis, A. D. Ionescu, and S. Klainerman, Hawking’s +local rigidity theorem without analyticity, Geometric and +Functional Analysis 20, 845 (2010), arXiv:0902.1173 [gr- +qc]. +[19] B. Allen, Vacuum States in de Sitter Space, Phys. Rev. +D 32, 3136 (1985). +[20] B. Allen and T. Jacobson, Vector Two Point Functions +in Maximally Symmetric Spaces, Commun. Math. Phys. +103, 669 (1986). +[21] B. Allen, The Graviton Propagator in De Sitter Space, +Phys. Rev. D 34, 3670 (1986). +[22] R. M. Wald, Quantum Field Theory in Curved Space- +Time and Black Hole Thermodynamics, Chicago Lectures +in Physics (University of Chicago Press, Chicago, IL, +1995). +[23] C. Yang and D. Feldman, The S Matrix in the Heisenberg +Representation, Phys. Rev. 79, 972 (1950). +[24] W. G. Unruh and R. M. Wald, What happens when an +accelerating observer detects a rindler particle, Phys. Rev. +D 29, 1047 (1984). +[25] E. T. Whittaker, On electric phenomena in gravitational +fields, Proc. Roy. Soc. Lond. A, 116, 720 (1927). +[26] H. Bondi and T. Gold, The field of a uniformly accel- +erated charge, with special reference to the problem of +gravitational acceleration, Proc. Roy. Soc. Lond. A 229, +416 (1955). +[27] F. Rohrlich, The equations of motion of classical charges, +Annals of Physics 13, 93 (1961). +[28] D. G. Boulware, Radiation from a uniformly accelerated +charge, Annals of Physics 124, 169 (1980). +[29] H. Padmanabhan and T. Padmanabhan, Aspects of elec- +trostatics in a weak gravitational field, Gen. Rel. Grav. +42, 1153 (2010), arXiv:0910.0926 [gr-qc]. +[30] E. Eriksen and Ø. Grøn, Electrodynamics of hyperboli- +cally accelerated charges v. the field of a charge in the +rindler space and the milne space, Annals of Physics 313, +147 (2004). +[31] L. Bieri and D. Garfinkle, An electromagnetic analogue +of gravitational wave memory, Class. Quant. Grav. 30, +195009 (2013), arXiv:1307.5098 [gr-qc]. +[32] C. Dappiaggi, V. Moretti, and N. Pinamonti, Hadamard +States From Light-like Hypersurfaces (Springer, Cham, +2017) arXiv:1706.09666 [math-ph]. +[33] A. Ashtekar, Asymptotic Quantization: Based On 1984 +Naples Lectures, Monographs and Textbooks in Physical +Science (Bibliopolis, Naples, Italy, 1987). +[34] A. Strominger, Lectures on the Infrared Structure of Grav- +ity and Gauge Theory (Princeton University Press, 2018) +arXiv:1703.05448 [hep-th]. +[35] G. Satishchandran and R. M. Wald, Asymptotic behavior +of massless fields and the memory effect, Phys. Rev. D +99, 084007 (2019), arXiv:1901.05942 [gr-qc]. +[36] E. Joos and H. D. Zeh, The emergence of classical proper- +ties through interaction with the environment, Zeitschrift +für Physik B Condensed Matter 59, 223 (1985). +[37] M. R. Gallis and G. N. Fleming, Environmental and +spontaneous localization, Phys. Rev. A 42, 38 (1990). +[38] L. Diósi, Quantum master equation of a particle in a gas +environment, Europhysics Letters 30, 63 (1995). +[39] K. Hornberger and J. E. Sipe, Collisional decoherence +reexamined, Phys. Rev. A 68, 012105 (2003). + diff --git a/1dAyT4oBgHgl3EQfPfaV/content/tmp_files/load_file.txt b/1dAyT4oBgHgl3EQfPfaV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..58690b998da26c823756aee1c5f35bb295aff094 --- /dev/null +++ b/1dAyT4oBgHgl3EQfPfaV/content/tmp_files/load_file.txt @@ -0,0 +1,947 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf,len=946 +page_content='Killing Horizons Decohere Quantum Superpositions Daine L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Danielson,1, ∗ Gautam Satishchandran,2, 1, † and Robert M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Wald1, ‡ 1Enrico Fermi Institute and Department of Physics, The University of Chicago, 933 East 56th Street, Chicago, Illinois 60637, USA 2Princeton Gravity Initiative, Princeton University, Jadwin Hall, Washington Road, Princeton NJ 08544, USA (Dated: January 3, 2023) We recently showed that if a massive (or charged) body is put in a quantum spatial superposition, the mere presence of a black hole in its vicinity will eventually decohere the superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In this paper we show that, more generally, decoherence of stationary superpositions will occur in any spacetime with a Killing horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' This occurs because, in effect, the long-range field of the body is registered on the Killing horizon which, we show, necessitates a flux of “soft horizon gravitons/photons” through the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The Killing horizon thereby harvests “which path” information of quantum superpositions and will decohere any quantum superposition in a finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' It is particularly instructive to analyze the case of a uniformly accelerating body in a quantum superposition in flat spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' As we show, from the Rindler perspective the superposition is decohered by “soft gravitons/photons” that propagate through the Rindler horizon with negligible (Rindler) energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We show that this decoherence effect is distinct from—and larger than—the decoherence resulting from the presence of Unruh radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We further show that from the inertial perspective, the decoherence is due to the radiation of high frequency (inertial) gravitons/photons to null infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (The notion of gravitons/photons that propagate through the Rindler horizon is the same notion as that of gravitons/photons that propagate to null infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=') We also analyze the decoherence of a spatial superposition due to the presence of a cosmological horizon in de Sitter spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We provide estimates of the decoherence time for such quantum superpositions in both the Rindler and cosmological cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' INTRODUCTION Consider a stationary spacetime in which an experimen- talist, Alice, is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Alice’s lab is stationary, and she has control of a charged or massive body (hereinafter re- ferred to as a “particle”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' She sends her particle through a Stern-Gerlach apparatus or other device that puts her par- ticle in a quantum superposition of two spatially separated states1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' She keeps these spatially separated components stationary for a time T and then recombines them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Will Alice be able to maintain the coherence of these compo- nents, so that, when recombined, the final state of her particle will be pure—or will decoherence have occurred, so that the final state of her particle will be mixed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Ordinarily, any decoherence effects will be dominated by “environmental influences,” i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', additional degrees of freedom present in Alice’s lab that interact with her particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We assume that Alice has perfect control of her laboratory and its environment so that there is no deco- herence from ordinary environmental effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' However, for a charged or massive particle, Alice cannot perfectly control the electromagnetic or gravitational field, since her particle acts as a source for these fields and some ∗ daine@uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='edu † gautam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='satish@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='edu ‡ rmwa@uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='edu 1 Quantum spatial superpositions of massive bodies have been of recent interest in both theoretical as well as proposed experimental probes of fundamental properties of quantum gravity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', [1–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' radiation will be emitted during the portions of her ex- periment where she separates and recombines her particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Nevertheless, in Minkowski spacetime, if her lab is sta- tionary in the ordinary, inertial sense, she can perform her experiment in a sufficiently adiabatic manner that negligible decohering radiation is emitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In principle, she can keep the particle separated for an arbitrarily long time T and still maintain coherence when the components are recombined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In a recent paper [14], we showed that the above situ- ation changes dramatically if a black hole is present in the spacetime—even though the experiment is carried out entirely in the black hole’s exterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In effect, a black hole horizon harvests “which path” information about any quantum superposition in its exterior, via the long-range fields sourced by the superposed matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We showed that this results in the unavoidable radiation of entangling “soft photons or gravitons” through the horizon that carry the “which path” information into the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Con- sequently, the mere presence of the black hole implies a fundamental rate of decoherence on the quantum super- position2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Although the rate of decoherence will be small if the black hole is far away, the coherence decays expo- nentially in the time, T, that the spatial superposition is maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Thus, in any spacetime with a black hole, there will be essentially complete decoherence within a 2 In QED, this effect applies only to superpositions of charged particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' However, since all matter sources gravity, the quantum gravitational decoherence applies to all superpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='00026v1 [hep-th] 30 Dec 2022 2 finite time3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The purpose of this paper is to generalize the results of [14] to spacetimes with Killing horizons, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', spacetimes with a Killing vector field such that there is a null surface to which the Killing field is normal (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', [15] for a discussion of properties of Killing horizons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The event horizon of a stationary black hole is a Killing horizon [16–18], so spacetimes with Killing horizons encompass the case of stationary spacetimes that contain black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' However, there are many cases of interest where Killing horizons are present without the presence of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' One such case is that of Minkowski spacetime, where the Rindler horizon is a Killing horizon with respect to the Lorentz boost Killing field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Another such case is de Sitter spacetime, where the cosmological horizon is a Killing horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We will show that in these cases, a spatial superposition that is kept stationary (with respect to the symmetry generating the Killing horizon) will decohere in a manner similar to the black hole case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We will also provide an estimate of the maximum amount of time during which coherence can be maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The case of the Rindler horizon is particularly instruc- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The relevant symmetry here is that of Lorentz boosts, so Alice’s lab will be “stationary” if it is uniformly accelerating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Our analysis based upon radiation through the Rindler horizon shows that decoherence of a uniformly accelerating spatially separated superposition occurs be- cause of the emission of “soft” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', very low frequency) gravitons or photons, where the frequency is defined rel- ative to an affine parameter on the Rindler horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' As we shall show, the decoherence effect of this radiation of soft gravitons or photons is distinct from the (smaller) decoherence effect resulting from the presence of Unruh radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' To gain further insight, we also analyze the decohering radiation in the electromagnetic case from the inertial point of view, using the Liénard-Wiechert solution to determine the radiation at future null infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' As we shall show, the decohering photons are of high frequency at null infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 2 we provide a general discussion of the deco- herence of a quantum superposition due to radiation in a stationary spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 3 we consider the decoher- ence of a uniformly accelerating superposition, analyzing it from both the Rindler and Minkowski viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We also show that this decoherence is distinct from (and larger than) the decoherence effects due to the presence of Un- ruh radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 4 we analyze the decoherence in de Sitter spacetime associated with the cosmological horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We will work in Planck units where G = c = ℏ = kB = 1 and, in electromagnetic formulas, we also put ϵ0 = 1, but we will restore these constants in our formulas that give estimates for decoherence times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Lower case Latin indices represent abstract spacetime indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Upper case Latin in- dices from the early alphabet correspond to spatial indices 3 This maximal coherence time for superpositions in the exterior can be much smaller than the evaporation time of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' on horizons or null infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' DECOHERENCE DUE TO RADIATION IN A STATIONARY SPACETIME In this section, we will give a general analysis of the decoherence of a spatial superposition in a stationary spacetime due to emission of radiation by the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Our analysis applies both to the decoherence of a charged body due to emission of electromagnetic radiation and to the decoherence of a gravitating body due to emission of linearized gravitational radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The analyses of these two cases are very closely parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In order to avoid repetition, we will analyze only the electromagnetic case in detail, but near the end of this section, we will state the corresponding results in the linearized gravitational case, which can be obtained straightforwardly by replacing the vector potential Aa with the perturbed metric hab, the charge-current ja with the stress-energy Tab, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Consider a charged particle4 in a stationary spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We assume that the particle is initially in a stationary state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The particle is then put through a Stern-Gerlach (or other) apparatus, resulting in it being in a superposition state5 |ψ⟩ = 1 √ 2 (|ψ1⟩ + |ψ2⟩) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='1) where |ψ1⟩ and |ψ2⟩ are normalized states that are spa- tially separated after passing through the apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The particle is then recombined via a reversing Stern-Gerlach (or other) apparatus and returns to a stationary state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We are particularly interested in the case where, between separation and recombination, |ψ1⟩ and |ψ2⟩ are kept stationary for a long period of time, T, but we do not make any such assumption in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We wish to estimate how much decoherence due to emission of elec- tromagnetic radiation will have occurred by the time of recombination6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 4 As already indicated above, the “particle” need not be an elementary particle but could be a “nanoparticle” or any other body whose only relevant degree of freedom for our analysis is its center of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 5 For simplicity, we have assumed that we have a 50-50 superposition of |ψ1⟩ and |ψ2⟩, but this assumption is not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 6 The decoherence of Alice’s particle can be experimentally deter- mined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We assume that Alice’s particle initially has spin in the positive x-direction and thus is in a 50-50 superposition of z-spin after passing through the initial Stern-Gerlach apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' After recombination, Alice measures the x-spin of her particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' If coherence of the superposition eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='1) has been maintained, then (assuming that Alice has made appropriate corrections if there are any phase differences between the paths) the spin will always be found to be in the positive x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' On the other hand, if any coherence has been lost, the particle will not be in a state of definite spin, and the spin will sometimes be found to be in the negative x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' By repeating the experiment many times, Alice can, in principle, determine the decoherence to any desired accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 3 A key assumption that we shall make is that the fluctu- ations in the charge-current operator ja in the states |ψ1⟩ and |ψ2⟩ are negligibly small over the scales of interest so that we can treat the charge current in each of these states as c-number sources in Maxwell’s equations, given by ja 1 = ⟨ψ1|ja|ψ1⟩ and ja 2 = ⟨ψ2|ja|ψ2⟩, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In the initial and final stationary eras, |ψ1⟩ and |ψ2⟩ are assumed to coincide spatially (though they may differ in other characteristics, such as spin) so that ja 1 = ja 2 at very early and very late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In order to proceed further, we must specify the initial state of the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Since, prior to going through the Stern-Gerlach apparatus, the charge is as- sumed to be stationary, at early times we may subtract the “Coulomb field” Cin a of the charge, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', at early times we may consider the electromagnetic field observable Ain a = Aa − Cin a 1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='2) where Cin a is the (assumed to be unique) stationary clas- sical solution to Maxwell’s equations with the early time stationary charged particle source ja 1 = ja 2 and Aa is the vector potential operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We need not assume any specific choice of gauge for Ain a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Then Ain a satisfies the source-free Maxwell’s equations at early times, and we may extend its definition to all times by requiring it to satisfy the source-free Maxwell equations everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The initial state of the electromagnetic field may be specified by giving the “radiation state” of Ain a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The choice of this state depends on the physical situation being considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' If the spacetime were globally stationary—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', if the stationary Killing field were everywhere timelike, so, in particular, there are no Killing horizons—it would be natural to assume that the initial state of the radiation is the stationary vacuum state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', the ground state relative to the time translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' For the case of a black hole spacetime, it would be correspondingly natural to assume that the initial state of the radiation is that of the Unruh vacuum, since for a black hole formed by gravitational collapse, the state of a quantum field is expected to approach the Unruh vacuum after the black hole has “settled down” to a stationary state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' For the case of Minkowski spacetime, we take the initial state of the radiation to be the ordinary (inertial) Minkowski vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' For de Sitter spacetime, we take the initial state of the radiation to be the de Sitter invariant vacuum7 for the electromagnetic field [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We denote the initial state of the radiation in all of the above cases by |Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In each of the above cases, |Ψ0⟩ is a pure, quasi-free (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', Gaussian) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' It follows (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', [22] or appendix A of [15]) that we can construct a one-particle Hilbert space Hin and corresponding Fock space F(Hin) wherein |Ψ0⟩ plays the role of the vacuum state and the field operator 7 A de Sitter invariant vacuum state does not exist for the massless scalar field [19] but such a state does exist for the electromagnetic field [20] and linearized gravitational field [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Ain a is represented on F(Hin) by Ain a (f a) = ia(Kσf) − ia†(Kσf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='3) Here f a a divergence-free8 test function, σf denotes the advanced minus retarded solution to Maxwell’s equations with source f a, and K : S → Hin denotes the map taking the space S of classical solutions to their representatives in the one-particle Hilbert space Hin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The commutator of the creation and annihilation operators in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='3) is given by [a(Kσf), a†(Kσg)] = ⟨Kσf|Kσg⟩ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='4) where ⟨Kσf|Kσg⟩ is the inner product on Hin, which is given by a natural generalization of the Klein-Gordon inner product to electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' For the case of a globally stationary spacetime in the stationary vacuum state, Kσf corresponds to taking the positive frequency part of σf with respect to the time translations generating the stationary symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' For the case of a stationary black hole in the Unruh vacuum state, Kσf corresponds to taking the positive frequency part of σf with respect to affine time on the past horizon and with respect to Killing time at past null infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' For Minkowski spacetime in the inertial Minkowski vacuum, Kσf corresponds to taking the positive frequency part of σf with respect to inertial time translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Equiv- alently, Kσf, in this case, corresponds to the solution obtained by taking the positive frequency part of the re- striction of σf to any null hyperplane N (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', any Rindler horizon) with respect to an affine parametrization of the null geodesics generating N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' For de Sitter spacetime in the de Sitter invariant vacuum, Kσf corresponds to the solution obtained by taking the positive frequency part of the restriction of σf to any cosmological horizon with respect to an affine parametrization of the null geodesics generating that horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Under the above assumption that the charge-current of |ψ1⟩ and |ψ2⟩ can be treated as c-number sources, the electromagnetic field Ai,a in the presence of the charge in state |ψi⟩ for i = 1, 2 is given in terms of the source free field Ain a by [23] Ai,a = Ain a + Gret a (jb i )1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='5) where Gret a (jb i ) denotes the classical retarded solution for source jb i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In particular, since the field Ain a is in state |Ψ0⟩, the correlation functions of the electromagnetic field 8 Restriction of the smearing to divergence-free test functions is necessary and sufficient to eliminate the gauge dependence of Ain a (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='101 of [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 4 Ai,a for |ψi⟩ are given by9 ⟨Ai,a1(x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Ai,an(xn)⟩ = ⟨Ψ0| � Ain a1(x1) + Gret a1 (jb i )(x1)1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' � Ain an(xn) + Gret an (jb i )(xn)1) � |Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='6) Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='6) is valid at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' However, at late times—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', to the future of any Cauchy surface Σ corresponding to the time at which recombination has occurred—we can again subtract off the common sta- tionary Coulomb field, Cout a , of ja 1 = ja 2 to obtain the source-free field10 Aout i,a that describes the radiation at late times for the states |ψi⟩, Aout i,a = Ai,a − Cout a 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='7) By eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='6), at late times, the correlation functions of Aout a are given by ⟨Aout i,a1(x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Aout i,an(xn)⟩ = ⟨Ψ0| � Ain a1(x1) + Ai,a1(x1)1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' � Ain an(xn) + Ai,an(xn)1) � |Ψ0⟩ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='8) where Ai,a = Gret a (jb i ) − Cout a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='9) Note that Ai,a is a classical solution of the source-free Maxwell equations in the late-time region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The correlation functions eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='8) on any late-time Cauchy surface are precisely those of the coherent state |Ψi⟩ = e− 1 2 ∥KAi∥2 exp � a†(KAi) � |Ψ0⟩ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='10) where the norm is that of the one-particle inner product of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Thus, the coherent state |Ψ1⟩ describes the “out” radiation state corresponding to charged particle state |ψ1⟩ and the coherent state |Ψ2⟩ describes the “out” radiation state corresponding to charged particle state |ψ2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The joint “out” state, |Υ⟩, of the particle-radiation system is given by |Υ⟩ = 1 √ 2 (|ψ1⟩ ⊗ |Ψ1⟩ + |ψ2⟩ ⊗ |Ψ2⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='11) Therefore, the decoherence of |ψ1⟩ and |ψ2⟩ due to emis- sion of electromagnetic radiation is given by D = 1 − | ⟨Ψ1|Ψ2⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='12) 9 It is understood that each of the xk variables should be smeared with a divergence-free test vector field fa k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 10Note that Ain a did not have a subscript “i” whereas Ai,a and Aout i,a do carry such subscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' This is a consequence of the fact that we are working in the “in” representation—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', the Heisenberg representation on the Hilbert space F(Hin)—so Ain a does not depend on the sources, but the other fields do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We wish to evaluate D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' By the general formula for the inner product of coherent states, we have | ⟨Ψ1|Ψ2⟩ | = exp � −1 2||K(A1 − A2)||2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='13) Now, in the late-time era, A1,a−A2,a is just the difference between the classical retarded solutions with sources ja 1 and ja 2, A1,a −A2,a = Gret a (jb 1)−Gret a (jb 2) = Gret a (jb 1 −jb 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='14) Consider the coherent state associated with Gret a (jb 1 − jb 2) in the late-time era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We refer to photons in this state as entangling photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' By the general properties of coherent states, the expected number, ⟨N⟩, of entangling photons is given by ⟨N⟩ ≡ ||K � Gret(j1 − j2) � ||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='15) Thus, we have | ⟨Ψ1|Ψ2⟩ | = exp � −1 2⟨N⟩ � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='16) so D = 1 − | ⟨Ψ1|Ψ2⟩ | = 1 − exp � −1 2⟨N⟩ � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='17) and we see that the necessary and sufficient condition for significant decoherence (D ∼ 1) is ⟨N⟩ ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We summarize the results that we have obtained above as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Under the assumptions we have made above, in order to calculate the decoherence, D, of the particle due to radiation, we carry out the following steps: (1) We obtain the expected charge current, ja 1 and ja 2, for the particle in states |ψ1⟩ and |ψ2⟩ of the super- position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2) We calculate the classical retarded solution, Gret a (jb 1 − jb 2) for the difference of these charge cur- rents, which is a source-free solution at late times, since ja 1 = ja 2 at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3) We calculate the one-particle state KGret(j1 − j2) corresponding to Gret a (jb 1 − jb 2) at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In the various cases, this corresponds to the follow- ing: (i) For a globally stationary spacetime initially in the stationary vacuum state, this one-particle state is the positive frequency part of the solution with respect to the time translations generating the stationary symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (ii) For the case of a station- ary black hole initially in the Unruh vacuum, the one-particle state is the positive frequency part of the solution with respect to affine time on the past horizon and with respect to Killing time at past null infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (iii) For Minkowski spacetime initially in the Minkowski vacuum, the one-particle state is the positive frequency part of the solution with 5 respect to inertial time or, equivalently, the posi- tive frequency part with respect to affine time on any Rindler horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (iv) For de Sitter spacetime initially in the de Sitter invariant vacuum, the one- particle state is the positive frequency part of the solution with respect to affine time on any cosmo- logical horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (4) We compute the squared norm, ∥K[Gret(j1 −j2)]∥2, of this one-particle state at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' This quan- tity is equal to the expected number of entangling photons, ⟨N⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The decoherence due to radiation is then given by D = 1 − exp � −1 2∥K � Gret(j1 − j2) � ∥2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='18) As previously stated, the above analysis extends straightforwardly to the linearized gravitational case, where the perturbed metric, hab, is treated as a linear quantum field propagating in the background classical stationary spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' To compute the decoherence due to gravitational radiation in this case, we carry out the above steps, replacing Aa by hab and the charge-current ja by the stress-energy tensor Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The retarded solu- tion Gret a (jb) for Maxwell’s equations is replaced by the retarded solution Gret ab (Tcd) for the linearized Einstein equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The map K : S → Hin is again obtained as in item (3) above and the inner product on Hin is again given by a natural generalization of the Klein-Gordon inner product to linearized gravitational fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The de- coherence due to gravitational radiation is then given by the analog of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The above analysis applies for any motion of the compo- nents of Alice’s superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We are primarily interested in the case where, during a time interval T1, Alice puts a particle of charge q (or mass m) into a spatial super- position, where the distance between the components of the particle wavefunction is d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' She then keeps this super- position stationary in her lab for a time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Finally, she recombines her particle over a time interval T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In Minkowski spacetime in the case where Alice’s lab is inertial, Gret a (jb 1 − jb 2) will be nonzero at null infinity only at the retarded times corresponding to the time intervals T1 and T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' A rough estimate of the number of entangling photons was obtained in [3] using the Larmor formula for radiation in these eras, which, in natural units, yields ⟨N⟩ ∼ q2d2 [min(T1, T2)]2 (Minkowski, EM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='19) The corresponding result in the linearized gravitational case is [3] ⟨N⟩ ∼ m2d4 [min(T1, T2)]4 (Minkowski, GR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='20) Therefore, if Alice recombines her particle sufficiently slowly that T1, T2 ≫ qd in the electromagnetic case or T1, T2 ≫ md2 in the gravitational case, then she can main- tain the quantum coherence of her particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In particular, Alice can keep the components of her particle separated for as long a time T as she likes without destruction of the coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' As shown in [14], the situation is quite different if a black hole is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In the electromagnetic case, even if T1, T2 ≫ qd so that a negligible number of entangling photons is emitted to infinity, there will be entangling radiation emitted into the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' For large T, the number of entangling photons increases with T as11 ⟨N⟩ ∼ M 3q2d2 D6 T (black hole, EM) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='21) where M is the mass of the black hole, D is the proper distance of Alice’s lab from the horizon of the black hole, and we assume that D ≳ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The corresponding result in the linearized gravitational case is ⟨N⟩ ∼ M 5m2d4 D10 T (black hole, GR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='22) Thus, the coherence of Alice’s particle will always be destroyed within a finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In the next two sections, we will apply the above anal- ysis to the cases of Rindler spacetime and de Sitter space- time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Although we will explicitly analyze only the Rindler and de Sitter cases, it will be clear from our analysis of the next two sections—as well as our analysis in [14]—that it can be applied to any Killing horizon, provided only that the initial “vacuum state” |Ψ0⟩ of the electromagnetic and/or linearized gravitational field corresponds to one- particle states that are positive frequency with respect to affine time on the future Killing horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' RINDLER HORIZONS DECOHERE QUANTUM SUPERPOSITIONS We now consider the case of Minkowski spacetime with Alice’s lab uniformly accelerating with acceleration a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Specifically, we take Alice’s lab to follow the orbit t = 1 a sinh(aτ), z = 1 a cosh(aτ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='1) of the boost Killing field ba = a � z � ∂ ∂t �a + t � ∂ ∂z �a� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='2) Here we have normalized ba such that baba = −1 on the worldline of Alice’s laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Thus, ba is the four- velocity of Alice’s laboratory and τ is the proper time in 11In the analysis of [14], we used the fact that the Unruh vacuum is well approximated by the Hartle-Hawking vacuum at low frequencies near the horizon of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 6 her lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We introduce the null coordinates U ≡ t − z, V ≡ t + z (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='3) and the corresponding vector fields na ≡ (∂/∂V )a, ℓa ≡ (∂/∂U)a, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='4) which are globally defined, future-directed null vector fields that satisfy ℓana = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In terms of these coordi- nates, the Minkowski spacetime metric is η = −dUdV + dx2 + dy2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='5) and the boost vector field is given by ba = a � − Uℓa + V na� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='6) The boost Killing field is null on the two “Rindler hori- zons,” i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', the two null planes U = 0 and V = 0, which divide Minkowski spacetime into four wedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The orbits of the boost Killing field are future-directed and time- like within the “right Rindler wedge” WR which is the region U < 0 and V > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Thus, the “right Rindler wedge” WR—where Alice performs her experiment—is a static, globally hyperbolic spacetime where the notion of “time translations” is defined by Lorentz boosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We refer to the null surface U = 0 as the future Rindler horizon and denote it as H + R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' On the region V > 0 of H + R , it is useful to introduce the coordinate v by V = V0eav (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='7) where V0 is an arbitrary constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Then, for V > 0 on H + R , we have ba�� HR+ = aV � ∂ ∂V �a���� HR+ = � ∂ ∂v �a���� HR+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='8) Since (∂/∂V )a on the horizon is tangent to the affinely parameterized null geodesic generators of H + R , we refer to V as the “affine time” on H + R , whereas we refer to v as the “boost Killing time” on H + R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Decoherence Due to Radiation of Soft Photons/Gravitons Through the Rindler Horizon We are now in position to apply the results of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 2 to the Rindler case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We will first analyze the electromag- netic case and then give the corresponding results in the gravitational case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We assume that the electromagnetic field is initially in the Minkowski vacuum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We assume that Alice possesses a charged particle that is initially stationary (with respect to the boost Killing field) in her (uniformly accelerating) lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' She then creates a quantum spatial superposition which is held stationary (with respect to the boost Killing field) for a proper time T and is then recombined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We wish to know the degree of decoherence of Alice’s particle due to emission of radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We may directly apply the analysis of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 2 to answer this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The future Rindler horizon H + R (U = 0) does not meet the technical requirements of being a Cauchy surface for Minkowski spacetime, since there are inextendible time- like curves that remain in the past of H + R as well as inextendible timelike curves that lie in the future of H + R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' However, as argued in [24], it is effectively a Cauchy sur- face for determining evolution of solutions to the wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' This is most easily seen in the conformally completed spacetime, where H + R is the past light cone of a point p ∈ I + except for the single generator that lies on I + and it also is the future light cone of a point on p′ ∈ I − except for the single generator that lies on I −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Data on the full past light cone of p would determine a solution to the past of H + R and data on the full future light cone of p′ would determine a solution to the future of H + R , thereby determining a solution everywhere in Minkowski spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' However, for solutions with ap- propriate decay, the data on the missing null geodesic generators of I + and I − can be determined by conti- nuity from the data on H + R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Consequently, data on H + R suffices to uniquely characterize solutions with appropri- ate decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Consequently, the “out” states |Ψ1⟩ and |Ψ2⟩ of the radiation are completely determined by data on H + R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Note that this contrasts sharply with the black hole case, where one would need data on both the future event horizon and future null infinity to characterize the “out” state of radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The decoherence of Alice’s particle due to radiation is given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In order to evaluate this, we first consider a classical point charge of charge q in the “right Rindler wedge” WR that is stationary with respect to the boost Killing field and lies at proper distance D from the bifurcation surface of the Rindler horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Such a charge will be uniformly accelerating with acceleration a given by a = 1 D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='9) The explicit solution for such a stationary charge in the Rindler wedge has long been known [25–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The only nonvanishing component of the electromagnetic field in the region V > 0 of H + R is EU ≡ Fabℓanb = 2a2q π(1 + a2ρ2)2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='10) where ρ2 ≡ x2 + y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Electromagnetic radiation through the Rindler horizon is described by the pullback, EA, of the electric field Ea = Fabnb to H + R , where the capital Latin indices from the early alphabet denote spatial com- ponents in the x and y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Since EA = 0 on the horizon for a uniformly accelerated charge, one may say that a charge held stationary in Alice’s lab does not pro- duce any radiation as determined on H + R —even though a uniformly accelerated charge radiates (inertial) energy 7 to future null infinity12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Now consider the case where the point charge is initially uniformly accelerating with acceleration a at a proper distance D = 1/a from the bifurcation surface of the Rindler horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The charge is then moved in the z- direction to a different orbit of the same boost Killing field, so that it has uniform acceleration a′ and lies at proper distance D′ = 1/a′ from the Rindler horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' After the charge has reached its new location, the electric field on H + R is again given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='10), but its value, E′ U, will be different from its value at early times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Maxwell’s equations on H + R imply that DAEA = ∂V EU (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='11) where DA is the derivative operator on the R2 cross- sections of the horizon and capital Latin indices from the early alphabet are raised and lowered with the met- ric, δAB, on the cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='11) implies that EA ̸= 0 whenever ∂V EU ̸= 0, so there will be radiation through the horizon as the charge is being moved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Most importantly, it implies that DA � � ∞ � −∞ dV EA � � = ∆EU (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='12) where ∆EU = E′ U −EU is the change in the radial electric field between the charge at positions D′ and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Now, in a gauge where Aana = 0 on the horizon, the transverse (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', x-y) components of the electric field are related to the corresponding components of the vector potential by EA = −∂V AA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='13) Since the transverse components of the Coulomb field of a static charge vanish, we may replace the vector potential AA by the “Coulomb subtracted” vector potential AA defined by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='9), so we have EA = −∂V AA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='14) It then follows immediately from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='12) that the dif- ference, ∆AA, between the final and initial values of AA is given by DA(∆AA) = −∆EU (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='15) independently of the manner in which the charge is moved from D to D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='15) is an exact mathemati- cal analog of the electromagnetic memory effect at null infinity [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 12A uniformly accelerating charge has a nonvanishing inertial energy current flux Tabta through both H + R and I +, where ta denotes a Minkowski time translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' However, the flux of “boost energy” Tabba vanishes at both H + R and I +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' For the explicit solution eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='10), we have ∆EU ≈ qda3(1 − a2ρ2) (1 + a2ρ2)3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='16) where d = D′ − D and we have assumed that d ≪ D = 1 a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='17) From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='15), we find that ∆AA points in the ˆρ- direction and has magnitude |∆AA| = ∆Aρ ≈ qda4ρ2 (1 + a2ρ2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='18) The key point is that even though EA = 0 at both late and early times, AA does return to its original value at late times, and the change, ∆AA, in the vector potential between late and early times is determined only by the initial and final positions of the charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We now consider the quantized radiation through the horizon resulting from the displacement of the charge, assuming that, after the displacement, the charge is held at its new position, D′, forever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' For the Fock space associated with the Minkowski vacuum state, the map K : S → Hin that associates one-particle states to classical solutions is given by taking the positive frequency part of the classical solution with respect to inertial time, with the inner product on Hin given by the Klein-Gordon product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' For the electromagnetic field on H + R in a gauge where Aana on H + R , the “free data” on H + R is the pull-back, AA, of the vector potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' For two classical solutions with data A1,A and A2,A on H + R , the inner product of their corresponding one-particle states is given by [15, 32] ⟨KA1| KA2⟩H + R = 2 � R2 dxdy ∞ � 0 ωdω 2π δAB ˆ A1,A ˆ A2,B (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='19) where ˆ AA(ω, xB) is the Fourier transform of AA(V, xB) with respect to the affine parameter V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' By the same reasoning as led to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='15), the expected number of photons on H + R in the coherent state associated to any classical solution AA is simply ⟨N⟩ = ∥KA∥2 H + R (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='20) where the norm is defined by the inner product eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' However, since ∆AA ̸= 0, the Fourier transform, ˆ AA(ω, xB), of AA diverges as 1/ω as ω → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' It fol- lows that the integrand of the expression for the norm given by the right side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='19) also diverges as 1/ω as ω → 0, so the integral is logarithmically divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Thus, ||KA||2 H + R = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Therefore, if Alice displaces a charged particle to a different orbit of the boost Killing field and the particle remains on this new uniformly accelerated trajectory forever, an infinite number of “soft horizon 8 photons” will be radiated through the Rindler horizon regardless of how quickly or slowly this process is done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' This is an exact mathematical analog of the infrared di- vergences that occur at null infinity in QED for processes with nonzero memory (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', [33–35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Now suppose that Alice displaces the particle a z- distance d ≪ D = 1/a from D to D′ = D+d as above, but instead of leaving the particle at D′ forever, she leaves it there for proper time13 T and then returns it to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In this case, the transverse components of the vector potential, AA, return to their initial values at late times, so there is no “memory effect” at the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Correspondingly, there are no infrared divergences in the expected number of photons that propagate through H + R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Nevertheless, if T is very large then the expected number of photons ⟨N⟩ will be correspondingly large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' To see this, we note that if, for convenience, we work in a gauge where AA = 0 initially, then during the era at which the particle is at D′, AA will be given by the right side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' If we keep the manner in which the particle is moved from D to D′ as well as from D′ to D fixed but take T to be very large, the asymptotic behavior of the norm eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='19) will be dominated by the low-frequency contribution from the era of time T that the particle is displaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The logarithmic divergence at ω = 0 that would occur if the particle re- mained at D′ forever is now effectively cut off at frequency ω ∼ 1/V , where V denotes the affine time duration on the horizon H + R over which the particle remains at D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We obtain ⟨N⟩ = ||KA||2 HR ∼ q2d2a2 ln � V min[V1, V2] � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='21) where V1, V2 ≪ V are the durations of affine time over which the particle is displaced from D to D′ and from D′ back to D, so that 1/min[V1, V2] provides an effective high-frequency cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' However, the affine time V on the horizon is related to boost Killing time on the horizon by V = V0 exp(av) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='22) and the boost Killing time v corresponds to the proper time T in Alice’s lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Thus, we obtain ⟨N⟩ ∼ q2d2a3T (Rindler, EM) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='23) Therefore, no matter how slowly the particle is displaced, it is forced to radiate a number of “soft Rindler horizon photons” through the Rindler horizon that is proportional to the time T that the particle remains on the displaced trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We are now in a position to fully analyze Alice’s exper- iment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Alice’s lab is uniformly accelerating with acceler- 13We have normalized the boost Killing field ba so that Killing time equals proper time on the orbit at D with acceleration a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Since we assume d = D′ − D ≪ D, Killing time and proper time are also (nearly) equal on the orbit at D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Thus, T is also the elapsed Killing time that Alice keeps the particle at D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' ation a in Minkowski spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' She puts her particle of charge q into a superposition of states separated by z-distance d ≪ 1/a and keeps these components sta- tionary in her lab for a proper time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' She then re- combines the components and determines their coher- ence14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' By the analysis of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 2, the decoherence is given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' However, for large T, the calculation of ||K [Gret(j1 − j2)] ||2 corresponds precisely to the calcu- lation we have given above of the number of photons radiated through the Rindler horizon when a charge is displaced for a time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Thus, we obtain ||K � Gret(j1 − j2) � ||2 ∼ q2d2a3T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='24) In other words, for large T, Alice’s superposition will de- cohere due to radiation of “soft Rindler horizon photons,” as D = 1 − exp(−ΓradT) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='25) where the “decoherence rate” Γrad, is given by, Γrad = q2d2a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='26) Thus, restoring the constants c, ℏ, and ϵ0, Alice’s par- ticle will decohere within a time TD ∼ ϵ0ℏc6 a3q2d2 (Rindler, EM) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='27) ∼ 1033 years �g a �3 �e q �2 �m d �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='28) Thus, if Alice’s lab uniformly accelerates at one g in flat spacetime and she separates an electron into two components one meter apart, she would not be able to maintain coherence of the electron for more than 1033 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' A similar analysis holds in the gravitational case15 where Alice separates a massive body with mass m across a distance d and maintains this superposition for a time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In the gravitational case, the “electric part” of the perturbed Weyl tensor Eab = Cacbdncnd plays an analo- gous role to the electric field Ea in the electromagnetic version of the gedankenexperiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' For a uniformly ac- celerating point mass, the only non-vanishing compo- nent of the electric part of the Weyl tensor on H + R is EUU = Cacbdℓancℓbnd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Gravitational radiation on the horizon is described by the pullback, EAB, of Eab, which vanishes for the static point mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' However, the process of quasistatically moving the static point mass involves a change in EUU on H + R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The (once-contracted) Bianchi identity on the 14The coherence can be determined as described in footnote 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 15In the gravitational case, additional stress-energy will be needed to keep Alice’s particle in uniform acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We will ignore the gravitational effects of this additional stress-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 9 horizon yields DAEAB = ∂V EUB, DAEUA = ∂V EUU (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='29) which implies that DADBEAB = ∂2 V EUU (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='30) which is closely analogous to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' As in the elec- tromagnetic case, if a uniformly accelerating point mass is quasistatically moved there is necessarily gravitational radiation through H + R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' To determine the number of “Rindler horizon gravitons” emitted we quantize the linearized gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' For a metric perturbation hab in a gauge where habna = 0 and δABhAB = 0, the “free data” on H + R is hAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' A “particle” in the standard Fock space associated to the Poincaré invariant vacuum is then a positive frequency solution with respect to affine parameter V and the inner product on the one-particle Hilbert space is given by a direct analog of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='19) with the vector potential AA replaced with the metric perturbation hAB, namely ⟨Kh1| Kh2⟩H + R = 1 8 � R2 dxdy ∞ � 0 ωdω 2π δABδCDˆh1,ACˆh2,BD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='31) Finally, EAB is related to the metric perturbation hAB by EAB = −1 2∂2 V hAB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='32) Equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='30) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='32) directly imply that a per- manent change, ∆EUU ̸= 0, in the U-U component of the electric part of the Weyl tensor on H + R implies a permanent change, ∆hAB ̸= 0, in the perturbed metric on H + R between early and late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In the quantum theory, as in the electromagnetic case, this implies a log- arithmic infrared divergence in the number of gravitons emitted through H + R in the process where a uniformly accelerating charge is moved to a new orbit of the same boost Killing field and then remains at the new position forever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The analysis of Alice’s experiment proceeds in a similar manner to the electromagnetic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Alice does not main- tain the relative separation of her wavefunction forever but closes her superposition after a proper time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' As before, the number of entangling gravitons emitted to the Rindler horizon is logarithmically growing in affine time and therefore linearly growing in the proper time duration T of Alice’s experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We obtain ⟨N⟩ ∼ m2d4a5T (Rindler, GR) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='33) Thus, restoring constants, we find that the Rindler hori- zon decoheres the quantum superposition of a uniformly accelerating massive body in a time T GR D ∼ ℏc10 Gm2d4a5 (Rindler, GR) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='34) ∼ 2 fs �MMoon m �2 �RMoon d �4 �g a �5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='35) Therefore, if the Moon were accelerating at one g and occupied a quantum state with its center of mass super- posed by a spatial separation of the order of its own radius then it would decohere within about 2 femtoseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Of course, it would not be easy to put the moon in such a coherent quantum superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Note the acceleration of a stationary observer outside of a black hole who is reasonably far16 (D ≳ M) from the event horizon is a ∼ M/D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' If we substitute a = M/D2 in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='27) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='34), we obtain eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='21) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='22), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Therefore, it might be tempting to believe that what is important in all cases is the acceleration of Alice’s lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' However, this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In particular, if we replace the black hole by an ordinary star (and if there are no dissipative effects in the star), then there will not be any analogous decoherence effect, even though the acceleration of Alice’s lab is the same as in the case of a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Furthermore, as we shall see in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 4, decoherence effects associated with the cosmological hori- zon occur in de Sitter spacetime even for nonaccelerating observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' It is the presence of a Killing horizon that is the essential ingredient for the fundamental rate of decoherence of quantum superpositions as described in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We now consider another potential cause of decoherence, namely Unruh radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Decoherence Due to Scattering of Unruh Radiation The Minkowski vacuum state restricted to a Rindler wedge is a thermal state at the Unruh temperature T = a 2π (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='36) relative to the notion of time translations defined by the Lorentz boost Killing field ba, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Thus, the superposition state of Alice’s particle will be buffeted by this thermal bath of Unruh radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Scattering of this radiation will cause some decoherence of Alice’s particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Indeed, since this decoherence should occur at a steady rate while the superposition is kept stationary (and thus the decoherence will be proportional to T), one might even 16It should be emphasized that the estimates made in [14] that yielded eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='21) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='22) assumed that Alice’s lab is reasonably far from the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' If Alice’s lab is extremely close to the black hole (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', at a distance D ≪ M from the horizon), then the black hole analysis would reduce to the Rindler case analyzed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 10 suspect that scattering of Unruh radiation could be the same effect as found in the previous section but expressed in a different language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The purpose of this subsection is to show that this is not the case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', decoherence due to scattering of Unruh radiation and decoherence due to radiation of “soft” photons/gravitons through the horizon are distinct effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Furthermore, we shall show that, for reasonable parameter choices, the decoherence rate due to the scattering of Unruh radiation is smaller than the decoherence rate due to emitted radiation as obtained in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We will consider only the electromagnetic case in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The decoherence rate of a spatial superposition due to collisions with particles in an environment has been analyzed in [36–39], and we will adapt this analysis to obtain a rough estimate of the decoherence caused by the scattering of Unruh radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' As in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='1), Alice has a particle of charge q in a state |ψ⟩ = (|ψ1⟩ + |ψ2⟩)/ √ 2, where |ψ1⟩ and |ψ2⟩ are spatially separated by a distance d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Since we require d ≪ 1/a (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='17)) and since the typical wavelength of Unruh photons at temperature eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='36) is λ ∼ 1/a, we are in the scattering regime where λ ≫ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In an elastic scattering event between Alice’s particle and a photon in the Unruh radiation, the final outgoing state of the photon will depend upon which branch of the superposition the photon scattered off of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Let |χ1⟩ denote the outgoing state of the Unruh photon for scattering off of |ψ1⟩ and let |χ2⟩ denote the outgoing state for scattering off of |ψ2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Decoherence will occur to the extent to which these outgoing states of the scattered Unruh photon are distinguishable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', D = 1−| ⟨χ1|χ2⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In order to obtain a rough estimate of the decoherence resulting from a single scattering event, we consider the corresponding Minkowski process of the scattering of a photon of momentum p off of an inertial superposition separated by d, with d ≪ 1/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Assuming that the charged particle states |ψ1⟩ and |ψ2⟩ are identical except for their location, the scattered photon states |χ1⟩ and |χ2⟩ should differ only by the action of the translation operator e−i ⃗P·⃗d, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', |χ2⟩ ≈ e−i ⃗P·⃗d |χ1⟩ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='37) where ⃗P denotes the photon momentum operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Ex- panding the exponential, we obtain the following rough estimate of the decoherence resulting from a single scat- tering event involving a photon of momentum p 1 − | ⟨χ1|χ2⟩ | ∼ p2d2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='38) where we have ignored any dependence on the angle be- tween the incoming momentum ⃗p and the separation ⃗d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We will take eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='38) as our estimate of the decoherence of Alice’s particle resulting from the scattering of a single Unruh photon of “Rindler momentum” p (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', of energy ϵ = p with respect to the boost Killing field ba).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The total decoherence rate due to scattering of Unruh radiation is then given by Γscatt ∼ d2 ∞ � 0 dp p2ϱ(p)σ(p) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='39) where ϱ(p) is the number density of photons at momentum p (so ϱ(p) is also the incoming flux of photons) and σ(p) is the scattering cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' For a thermal distribution of photons17 we have ϱ(p) ∼ p2 ep/T − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='40) We take σ to be given by the Thomson cross-section σ = 8π 3 q4 (4πm)2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='41) where m is the mass of Alice’s particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Putting this all together, our estimate of the decoherence rate due to scattering of Unruh photons is Γscatt ∼ q4d2a5 m2 (Rindler, EM) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='42) Comparing eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='42) to the rate of decoherence, Γrad due to the emission of soft photons given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='26), one can immediately see that the effects are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In particular, Γrad has no dependence on the mass, m, of Alice’s particle, whereas Γscatt does depend on m on account of the mass dependence of the scattering cross- section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The ratio of these decoherence rates is given by Γscatt Γrad ∼ q2a2 m2 = �q/m D �2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='43) Now, q/m is the “charge radius” of Alice’s particle and, as argued in [3], it represents a fundamental lower bound to the spread of a charged particle due to vacuum fluc- tuations of the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Therefore, in order that |ψ1⟩ and |ψ2⟩ not overlap, we must have d > q/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Since d ≪ D, we conclude that Γscatt Γrad ≪ 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='44) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', the contribution to decoherence from the scattering of Unruh radiation is negligible compared with the de- coherence due to emission of soft photons through the Rindler horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' A similar analysis holds for a charged particle superpo- sition outside of a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' It is worth noting, that the 17The factor of p2 in the numerator of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='40) arises from the density of states in Minkowski spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We ignore here any differences between the Minkowski and Rindler densities of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 11 decoherence effects due to scattering of Hawking radiation will decrease with distance, D, from the black hole only as 1/D2 for large D, giving, Γscatt ∼ q4d2 m2M 3 1 D2 (black hole, EM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='45) On the other hand, by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='21) the decoherence effects of radiation of soft photons through the horizon decreases with D as 1/D6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Thus at sufficiently large D, the deco- herence effects due to scattering of Hawking radiation will dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' However, in this regime, both effects are extremely small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Decoherence From the Inertial Perspective In our analysis of the decoherence of a spatial superpo- sition in the presence of a black hole [14] as well as in our analysis of the decoherence of a spatial superposition in Rindler spacetime given above in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='1, it may appear that we have introduced a radical new mechanism for de- coherence, namely radiation of soft photons and gravitons through a horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The main purpose of this subsection is to show that, in fact, the decoherence we derived in the Rindler case can also be obtained by entirely conventional means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In the Rindler case, we are simply considering a uniformly accelerating superposition in Minkowski space- time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The radiation of entangling photons to infinity from such a superposition can be calculated in the inertial view- point by standard methods, without introducing concepts such as a Rindler horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' It is instructive to calculate the decoherence from the inertial viewpoint both in order to validate the results of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='1 as well as to gain insight into how the emitted “soft photons” would be interpreted by an inertial observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' As we shall see, the entangling photons as seen by inertial observer at large distances near θ = 0 will be “hard” even though, from her point of view, Alice has performed the experiment adiabatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We will restrict our analysis in this subsection to the electromagnetic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The Liénard-Wiechert solution for the potential of a point charge in Minkowski spacetime following an arbi- trary worldline Xµ(τ) is, in Lorenz gauge, Aµ(x) = 1 4π 1 α q |⃗x − ⃗X(tret)| dXµ dt (tret) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='46) where α ≡ 1 − ˆn · d ⃗X dt (tret) and ˆn = ⃗x − ⃗X(tret) |⃗x − ⃗X(tret)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='47) For a uniformly accelerated trajectory with acceleration a, we have Xµ(τ) = �1 a sinh(aτ), 0, 0, 1 a cosh(aτ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='48) In Bondi coordinates (u, r, θ, φ) with u ≡ t − r (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='49) the future light cone of an event at proper time τ on the worldline eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='48) reaches null infinity at au = sinh(aτ) − cos θ cosh(aτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='50) Electromagnetic radiation is described by the pullback of the electromagnetic field, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='46), to null infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Taking the limit as r → ∞ at fixed u, we obtain18 AA(u, θ, φ) = −q 4π sinh(aτ) sin θ cosh(aτ) − cos θ sinh(aτ)(dθ)A (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='51) where, in this subsection, capital indices from the early alphabet denote angular components on the 2-sphere cross- sections of I +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We will be concerned with the difference, at fixed (u, θ, φ), between the electromagnetic radiation of a particle following the trajectory eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='48) and a particle following a similar trajectory that is displaced in the z-direction by a proper distance d ≪ 1/a and thus has δa = a2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='52) We denote this difference by Ad A(u, θ, φ) ≡ AA(a + δa) − AA(a) ≈ δa �∂AA ∂a � u,θ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='53) From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='51), we obtain Ad A = −a2qd 4π u sin θ (cosh(aτ) − cos θ sinh(aτ))3 (dθ)A (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='54) where eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='50) was used to compute (∂τ/∂a)(u,θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In her experiment, Alice starts with her particle in a uniformly accelerating state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Over a proper time T1, she separates it into two uniformly accelerating components separated by a distance d as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' She keeps these components separated for a proper time T, and she then recombines them over a proper time T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The difference between the radiation fields of these components is given by AA ≡ A1,A − A2,A = F(τ)Ad A (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='55) where the smooth function F is such that F(τ) = 0 for τ < −T1 and τ > T +T2, whereas F(τ) = 1 for 0 < τ < T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 18The vector potential is not smooth at I + in Lorenz gauge but one can do an asymptotic gauge transformation such that Aa is smooth at I +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Such a gauge transformation does not affect the angular components AA at I + [35], so we can calculate AA using our Lorenz gauge expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 12 The entangling photon content is then given by ⟨N⟩ = ||KA||2 = 2 � S2 dΩ ∞ � 0 ωdω 2π ˆ AA ˆ AA (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='56) where ˆ AA(ω, θ, φ) denotes the Fourier transform of AA(u, θ, φ) with respect to u, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', ˆ AA(ω, θ, φ) = ∞ � −∞ du eiωuAA(u, θ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='57) We are interested in estimating ⟨N⟩ for large T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In order to evaluate the Fourier transform integral, it is useful to note that, at fixed a, we have du dτ = cosh(aτ) − cos θ sinh(aτ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='58) and d2u dτ 2 = a2u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='59) It follows that d du � 1 du/dτ � = 1 du/dτ d dτ � 1 du/dτ � = −a2u (cosh(aτ) − cos θ sinh(aτ))3 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='60) Thus, we have Ad A = qd sin θ 4π (dθ)A d du � 1 du/dτ � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='61) and ˆ AA = qd sin θ 4π (dθ)A ∞ � −∞ du eiωuF(τ) d du � 1 du/dτ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='62) Integrating by parts, we obtain ˆ AA(ω, xA) = − qd sin θ 4π (dθ)A � iω ∞ � −∞ du eiωu F(τ) du/dτ + ∞ � −∞ du eiωu F ′(τ) (du/dτ)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='63) The second term in this equation contributes only during the time intervals (−T1, 0) and (T, T + T2) when Alice opens and closes the superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' For large T, its con- tribution can be shown to be negligible compared with the first term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Therefore, we have ˆ AA(ω, xA) ≈ −(dθ)A iωqd sin θ 4π I (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='64) where I ≡ ∞ � −∞ du eiωu F(τ) du/dτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='65) To evaluate I, we approximate F by a step function in the τ-interval [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The corresponding interval, [u0, uT ], in u is u0 = −1 a cos θ uT = 1 2a � eaT (1 − cos θ) − e−aT (1 + cos θ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='66) Noting that du dτ = � a2u2 + sin2 θ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='67) we obtain I ≈ uT � u0 du eiωu � a2u2 + sin2 θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='68) It can be seen that for large T, the dominant contribution to I will come from small angles, θ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' For aT ≫ 1, the upper limit of the integral may then be approximated as uT ≈ 1 4aeaT θ2 − 1 ae−aT for θ ≪ 1 ∼ � 0 for θ2/4 < e−aT 1 4aθ2eaT for θ2/4 ≥ e−aT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='69) For aT ≫ 1, the contribution to I from θ2/4 < e−aT can be shown to make a negligible contribution to ⟨N⟩, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Therefore, we may approximate I as I ∼ Θ(θ2 − 4e−aT ) exp(aT )θ2/(4a) � −1/a du eiωu � a2u2 + sin2 θ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='70) where Θ(x) ≡ � 0 for x < 0 1 for x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='71) For 0 < ω < 4ae−aT /θ2, we may bound I by replacing eiωu by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The integral can then be evaluated explic- itly, and it can be shown that for aT ≫ 1, the con- tribution to ⟨N⟩ from this frequency range is negligi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' For ω > 4ae−aT /θ2, the integrand is oscillatory for u > exp(aT)θ2/(4a), and, for aT ≫ 1, we will make neg- ligible error in our estimate of ⟨N⟩ if we replace the upper limit of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='70) by ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We will also make a negligible error by replacing the lower limit by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Thus, for aT ≫ 1, 13 we may approximate I as I ∼ Θ(θ2−4e−aT )Θ(ω−4ae−aT /θ2) ∞ � 0 du eiωu � a2u2 + sin2 θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='72) Evaluating the integral we obtain I ∼ 1 aΘ(θ2 − 4e−aT )Θ(ω − 4ae−aT /θ2) �1 2iπI0(sin θω/a) +K0(sin θω/a) − 1 2iπLLL0(sin θω/a) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='73) where I0, K0 are Bessel functions and LLL0 is a Struve function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' This expression is highly suppressed for ω > a/θ, so we can expand in θω/a and truncate the function above ω = a/θ to obtain, I ∼ −1 aΘ(1−θω/a)Θ(θ2−4e−aT )Θ(ω−4ae−aT /θ2) ln (θω/a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='74) Note that the restrictions ω < a/θ, and θ > 2e−aT/2 im- ply a frequency cutoff at ω ∼ aeaT/2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' By eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='74) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='64), the frequency spectrum of ˆ AA goes as ω ln(ω/a) up to this cutoff, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', the spectrum is “hard” and becomes increasingly so for large T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' This contrasts with the in- creasingly “soft” spectrum on the Rindler horizon, which goes as 1/ω down to a low frequency cutoff ∼ 1/V ∝ e−aT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Thus, the “soft horizon photons” from the Rindler per- spective are “hard” photons from the inertial perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='56) for ⟨N⟩ together with our expression eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='64) for ˆ AA and the expression eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='74) that we have just derived for I, we obtain ⟨N⟩ ∼ �qd a �2 � dωdθ θ3ω3 � ln ωθ a �2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='75) where the region of ω-θ integration is determined by the Θ- functions appearing in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='74) as well as the geometrical restriction θ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We can break up this region into the portion with ω ≤ a and the portion with ω > a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Since the region with ω ≤ a and θ ≲ 1 is bounded and the integrand of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='75) is bounded in this region, the contribution to ⟨N⟩ from ω ≲ a is bounded by a constant that is independent of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We may therefore discard this contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In the region ω > a, the third Θ-function in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='74) is redundant, and the integration region is a ≤ω≤ aeaT/2/2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='76) 2e−aT/2 ≤θ≤ a ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='77) For aT ≫ 1, we will make negligible error by replacing the lower limit of θ by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We thereby obtain ⟨N⟩ ∼ �qd a �2 a exp(aT/2)/2 � a dω a/ω � 0 dθ θ3ω3 � ln ωθ a �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='78) Making the change of variables from θ to x = ω a θ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='79) we find that the θ-integral becomes a/ω � 0 dθ θ3ω3 � ln ωθ a �2 = a ω a3 1 � 0 dx x3(ln x)2 ∼ a4 ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='80) Thus, we obtain ⟨N⟩ ∼ �qd a �2 a4 a exp(aT/2)/2 � a dω ω ∼ a2q2d2 ln[exp(aT/2)] ∼ a3q2d2T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='81) This estimate agrees with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Thus, we have succeeded—with considerable effort!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='—in our goal of deriving the decoherence of Alice’s superpo- sition by entirely conventional means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' It is notable how much simpler the calculation of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='1 was compared to the calculation that we have just completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' COSMOLOGICAL HORIZONS DECOHERE QUANTUM SUPERPOSITIONS In this section, we apply our analysis to de Sitter space- time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The de Sitter metric in a static patch is given by ds2 = −f(r)dt2 + f(r)−1dr2 + r2qABdxAdxB (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='1) where, in this section, xA are angular coordinates on the 2-sphere, qAB is the unit round metric on the 2-sphere, and f(r) = 1 − r2/R2 H (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='2) where RH (the “Hubble radius”) is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The coordinate singularity at r = RH corresponds to the “cosmological horizon,” which is a Killing horizon of the static Killing field (∂/∂t)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The relation between “affine time,” V , and “Killing time,” v, on the future cosmological horizon is V = ev/RH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='3) The general analysis of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 2 applies to the decoherence of a static superposition in de Sitter spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The esti- mates of the decoherence due to emission of soft photons and gravitons through the cosmological horizon when Al- ice keeps the superposition present for a time T can be made in exact parallel with the analysis of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 3 in the Rindler case and [14] in the black hole case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The only noteworthy new ingredient in de Sitter spacetime is that 14 the worldline r = 0 is an orbit of the static Killing field that is inertial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=', non-accelerating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' We now estimate the decoherence of a spatial superposition created in Al- ice’s lab at r = 0 and thereby show that decoherence will occur even though Alice’s lab is not accelerating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' By Gauss’ law, a point charge placed at r = 0 will give rise to a radial electric field EU on the future cosmological horizon given by EU ∼ q R2 H (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='4) where EU = Fabℓanb on the horizon with na = (∂/∂V )a tangent to the affinely parametrized null generators of the horizon and ℓa = (∂/∂U)a a radial null vector with naℓa = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' The change in the electric field on the horizon resulting from a displacement of the charge to r = d ≪ RH is ∆EU ∼ qd R3 H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='5) By paralleling the steps that led to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='18) above, we find that the change in the tangential components of the vector potential at the horizon is |∆AA| ≡ � R−2 H qAB∆AA∆AB �1/2 ∼ qd R2 H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='6) By paralleling the steps that led to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='23)—assuming that the electromagnetic field is initially in the de Sitter invariant vacuum (see footnote 7)—we obtain the estimate ⟨N⟩ ∼ q2d2 R3 H T (de Sitter, EM) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='7) Thus, restoring constants, the decoherence time due to the presence of the cosmological horizon is TD ∼ ℏϵ0R3 H q2d2 (de Sitter, EM) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='8) Since d ≪ RH, the decoherence time will be much larger than the Hubble time RH/c unless q is extremely large relative to the Planck charge qP ≡ √ϵ0ℏc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Nevertheless, we see that decoherence does occur despite the fact that Alice’s lab is inertial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' A similar analysis applies in the gravitational case for a spatial superposition of a massive particle in Alice’s lab at r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' In parallel with the derivation given in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='1 above, we find ⟨N⟩ ∼ m2d4 R5 H T (de Sitter, GR) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='9) which leads to a decoherence time T GR D ∼ ℏR5 H Gm2d4 (de Sitter, GR) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='10) ACKNOWLEDGMENTS D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' acknowledges support as a Fannie and John Hertz Foundation Fellow holding the Barbara Ann Cana- van Fellowship and as an Eckhardt Graduate Scholar in the Physical Sciences Division at the University of Chicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' This research was supported in part by NSF Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 21-05878 to the University of Chicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Bose, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Mazumdar, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Morley, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Ulbricht, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Toroš, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Paternostro, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Geraci, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Barker, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Kim, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Milburn, Spin Entanglement Witness for Quantum Gravity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 119, 240401 (2017), arXiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='06050 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Marletto and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Vedral, Gravitationally-induced en- tanglement between two massive particles is sufficient evidence of quantum effects in gravity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 119, 240402 (2017), arXiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='06036 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Belenchia, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Wald, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Giacomini, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Castro-Ruiz, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Brukner, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Aspelmeyer, Quantum Superposition of Massive Objects and the Quantization of Gravity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' D 98, 126009 (2018), arXiv:1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='07015 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Christodoulou and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Rovelli, On the possibility of lab- oratory evidence for quantum superposition of geometries, Physics Letters B 792, 64 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [5] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Giacomini, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Castro-Ruiz, and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Brukner, Quantum mechanics and the covariance of physical laws in quan- tum reference frames, Nature Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 10, 494 (2019), arXiv:1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='07207 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [6] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Gonzalez-Ballestero, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Aspelmeyer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Novotny, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Quidant, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Romero-Isart, Levitodynamics: Lev- itation and control of microscopic objects in vacuum, Science 374, 3027 (2021), arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='05215 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [7] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Danielson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Satishchandran, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Wald, Gravitationally mediated entanglement: Newtonian field versus gravitons, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' D 105, 086001 (2022), arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='10798 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Carney, Newton, entanglement, and the graviton, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' D 105, 024029 (2022), arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='06320 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Christodoulou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Di Biagio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Aspelmeyer, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Brukner, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Rovelli, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Howl, Locally mediated en- tanglement through gravity from first principles, (2022), arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='03368 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [10] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Carney, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Chen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Geraci, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Müller, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Panda, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Stamp, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Taylor, Snowmass 2021 White Paper: Tabletop experiments for infrared quan- tum gravity, in 2022 Snowmass Summer Study (2022) arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='11846 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [11] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Feng and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Vedral, Amplification of gravitationally induced entanglement, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' D 106, 066013 (2022), arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='09737 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [12] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Zhou, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Marshman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Bose, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Mazum- dar, Catapulting towards massive and large spatial quan- tum superposition, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 4, 043157 (2022), arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='04088 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 15 [13] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Overstreet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Curti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Kim, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Asenbaum, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Kasevich, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Giacomini, Inference of gravitational field superposition from quantum measurements, (2022), arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='02214 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [14] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Danielson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Satishchandran, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Wald, Black holes decohere quantum superpositions, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' D 31, 2241003 (2022), arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='06279 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [15] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Kay and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Wald, Theorems on the Unique- ness and Thermal Properties of Stationary, Nonsingular, Quasifree States on Space-Times with a Bifurcate Killing Horizon, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 207, 49 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Hawking and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Ellis, The Large Scale Struc- ture of Space-Time, Cambridge Monographs on Mathe- matical Physics (Cambridge University Press, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [17] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Hawking, Black holes in general relativity, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 25, 152 (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Alexakis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Ionescu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Klainerman, Hawking’s local rigidity theorem without analyticity, Geometric and Functional Analysis 20, 845 (2010), arXiv:0902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='1173 [gr- qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [19] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Allen, Vacuum States in de Sitter Space, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' D 32, 3136 (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [20] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Allen and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Jacobson, Vector Two Point Functions in Maximally Symmetric Spaces, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 103, 669 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [21] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Allen, The Graviton Propagator in De Sitter Space, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' D 34, 3670 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [22] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Wald, Quantum Field Theory in Curved Space- Time and Black Hole Thermodynamics, Chicago Lectures in Physics (University of Chicago Press, Chicago, IL, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [23] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Yang and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Feldman, The S Matrix in the Heisenberg Representation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 79, 972 (1950).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [24] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Unruh and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Wald, What happens when an accelerating observer detects a rindler particle, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' D 29, 1047 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [25] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Whittaker, On electric phenomena in gravitational fields, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' A, 116, 720 (1927).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [26] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Bondi and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Gold, The field of a uniformly accel- erated charge, with special reference to the problem of gravitational acceleration, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' A 229, 416 (1955).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [27] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Rohrlich, The equations of motion of classical charges, Annals of Physics 13, 93 (1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [28] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Boulware, Radiation from a uniformly accelerated charge, Annals of Physics 124, 169 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [29] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Padmanabhan and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Padmanabhan, Aspects of elec- trostatics in a weak gravitational field, Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 42, 1153 (2010), arXiv:0910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='0926 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [30] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Eriksen and Ø.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Grøn, Electrodynamics of hyperboli- cally accelerated charges v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' the field of a charge in the rindler space and the milne space, Annals of Physics 313, 147 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [31] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Bieri and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Garfinkle, An electromagnetic analogue of gravitational wave memory, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' 30, 195009 (2013), arXiv:1307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='5098 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [32] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Dappiaggi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Moretti, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Pinamonti, Hadamard States From Light-like Hypersurfaces (Springer, Cham, 2017) arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='09666 [math-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [33] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Ashtekar, Asymptotic Quantization: Based On 1984 Naples Lectures, Monographs and Textbooks in Physical Science (Bibliopolis, Naples, Italy, 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [34] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Strominger, Lectures on the Infrared Structure of Grav- ity and Gauge Theory (Princeton University Press, 2018) arXiv:1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='05448 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [35] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Satishchandran and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Wald, Asymptotic behavior of massless fields and the memory effect, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' D 99, 084007 (2019), arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content='05942 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [36] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Joos and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Zeh, The emergence of classical proper- ties through interaction with the environment, Zeitschrift für Physik B Condensed Matter 59, 223 (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [37] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Gallis and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Fleming, Environmental and spontaneous localization, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' A 42, 38 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [38] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Diósi, Quantum master equation of a particle in a gas environment, Europhysics Letters 30, 63 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' [39] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Hornberger and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Sipe, Collisional decoherence reexamined, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAyT4oBgHgl3EQfPfaV/content/2301.00026v1.pdf'} +page_content=' A 68, 012105 (2003).' metadata={'source': 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file mode 100644 index 0000000000000000000000000000000000000000..fe3fe9b7019d37236c1ded0cc65dfc208ffc475f --- /dev/null +++ b/7NFAT4oBgHgl3EQfoB2V/content/tmp_files/2301.08632v1.pdf.txt @@ -0,0 +1,1161 @@ +Generative Slate Recommendation with Reinforcement +Learning +Romain Deffayet +Naver Labs Europe +Meylan, France +University of Amsterdam +Amsterdam, The Netherlands +romain.deffayet@naverlabs.com +Thibaut Thonet +Naver Labs Europe +Meylan, France +thibaut.thonet@naverlabs.com +Jean-Michel Renders +Naver Labs Europe +Meylan, France +jean-michel.renders@naverlabs.com +Maarten de Rijke +University of Amsterdam +Amsterdam, The Netherlands +m.derijke@uva.nl +ABSTRACT +Recent research has employed reinforcement learning (RL) algo- +rithms to optimize long-term user engagement in recommender +systems, thereby avoiding common pitfalls such as user boredom +and filter bubbles. They capture the sequential and interactive na- +ture of recommendations, and thus offer a principled way to deal +with long-term rewards and avoid myopic behaviors. However, RL +approaches are intractable in the slate recommendation scenario +– where a list of items is recommended at each interaction turn – +due to the combinatorial action space. In that setting, an action +corresponds to a slate that may contain any combination of items. +While previous work has proposed well-chosen decompositions +of actions so as to ensure tractability, these rely on restrictive and +sometimes unrealistic assumptions. Instead, in this work we pro- +pose to encode slates in a continuous, low-dimensional latent space +learned by a variational auto-encoder. Then, the RL agent selects +continuous actions in this latent space, which are ultimately de- +coded into the corresponding slates. By doing so, we are able to +(i) relax assumptions required by previous work, and (ii) improve +the quality of the action selection by modeling full slates instead +of independent items, in particular by enabling diversity. Our ex- +periments performed on a wide array of simulated environments +confirm the effectiveness of our generative modeling of slates over +baselines in practical scenarios where the restrictive assumptions +underlying the baselines are lifted. Our findings suggest that repre- +sentation learning using generative models is a promising direction +towards generalizable RL-based slate recommendation. +CCS CONCEPTS +• Information systems → Recommender systems. +Permission to make digital or hard copies of all or part of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for components of this work owned by others than the +author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or +republish, to post on servers or to redistribute to lists, requires prior specific permission +and/or a fee. Request permissions from permissions@acm.org. +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +© 2023 Copyright held by the owner/author(s). Publication rights licensed to ACM. +ACM ISBN 978-1-4503-9407-9/23/02...$15.00 +https://doi.org/10.1145/3539597.3570412 +KEYWORDS +Slate recommendation, Reinforcement learning, Variational auto- +encoder +ACM Reference Format: +Romain Deffayet, Thibaut Thonet, Jean-Michel Renders, and Maarten de +Rijke. 2023. Generative Slate Recommendation with Reinforcement Learn- +ing. In Proceedings of the Sixteenth ACM International Conference on Web +Search and Data Mining (WSDM ’23), February 27-March 3, 2023, Singa- +pore, Singapore. ACM, New York, NY, USA, 9 pages. https://doi.org/10.1145/ +3539597.3570412 +1 +INTRODUCTION +Ubiquitous in online services, recommender systems (RSs) play a +key role personalization by catering to users’ identified tastes. Ide- +ally, they also diversify their offerings and help users discover new +interests [19]. In the latter case, RSs take on an active role, which +means that recommendations influence future user behavior, and +therefore their effects on users must be explicitly controlled. Such +effects can be detrimental: users may get bored if too many simi- +lar recommendations are made, and it has been well-documented +that users can end up in so-called filter bubbles or echo chambers +[4, 13, 28]. From the perspective of the online platform or the con- +tent provider, user boredom leads to poor retention and conversion +rates [17], while filter bubbles raise fairness and ethical issues for +which providers can be held accountable [26]. Conversely, RSs can +also positively impact users, for example, when users get interested +in new, unexpected topics or when the RS offers a fair represen- +tation of available options [1]. It is natural, therefore, to balance +exploitation (i.e., sticking to the known interests of the user) and +exploration (i.e., further probing the user’s interests) so as to avoid +always recommending similar items, and encourage recommenda- +tions that boost future engagement. The reinforcement learning +(RL) literature has proposed models and algorithms that aim to +optimize long-term metrics by acknowledging the causal effect of +recommendations on users [8, 36]. +In this work we consider the common scenario of slate recom- +mendation [8, 18, 31], which comes with specific challenges. At each +interaction turn, a slate recommender system recommends a list of +items from the collection, and the user interacts with zero, one or +several of those items. As a consequence, users may not examine +arXiv:2301.08632v1 [cs.IR] 20 Jan 2023 + +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +Romain Deffayet, Thibaut Thonet, Jean-Michel Renders, & Maarten de Rijke +all the recommended items, which leads to biases in the observed +interactions along with a complex interplay between items in the +same slate [27]. More importantly, the size of the action space, i.e., +the number of possible slates, prohibits the use of off-the-shelf RL +approaches [12]. Indeed, as slate recommendation is a combinato- +rial problem, the evaluation of all actions by the RL agent through +trial and error is simply intractable: even with as few as 1, 000 +items in the collection, the number of possible slates of size 10 is +approximately 9.6 × 1029. We propose to tackle this problem in +the context of a practical scenario, (S), which fits the second-stage +ranking phase [11] of many content recommendation platforms: +(S) The collection contains around a thousand items, and at each +turn of interaction the proposed model must select and rank +10 items to be presented to the user. +All our tractability and feasibility statements in this paper must +therefore be understood through the lens of this scenario (S). +To reduce the prohibitively large size of the combinatorial action +space, previous studies have proposed to decompose slates in a +tractable manner [8, 18, 31] – but at the cost of restrictive assump- +tions, e.g., concerning mutual independence of items in the slate, +knowledge of the user click model, availability of high-quality item +embeddings, or that at most one item per slate is clicked. +In contrast, in this work we propose to first learn a continuous, +low-dimensional latent representation of actions (i.e., slates), and +then let the agent take actions within this latent space during its +training phase. In practice, we obtain the latent representations +by introducing a generative modeling of slates (GeMS) based on a +variational auto-encoder (VAE) pre-trained on a dataset of observed +slates and clicks, collected from a previous version of the recom- +mender system. Such a dataset is usually available in industrial +recommendation settings. Therefore, we do not rely on restrictive +assumptions, and the fact that we represent full slates enables the +agent to improve the quality of its recommendations, instead of +using individual item representations. +Our contributions can be summarized as follows: +• We propose GeMS, a novel way to represent actions in RL for slate +recommendation, by pre-training a VAE on slates and associated +clicks. Unlike previous methods, GeMS is free of overly restrictive +assumptions and only requires logged interaction data. +• We provide a unified terminology to classify existing slate recom- +mendation approaches based on their underlying assumptions. +• We show on a wide array of simulated environments that previ- +ous methods underperform when their underlying assumptions +are lifted (i.e., in practical settings), while GeMS allows us to re- +cover highly rewarding policies without restrictive assumptions. +• To support the reproducibility of this work, we publicly release +the code for our approach, baselines and simulator.1 +2 +RELATED WORK +Long-term user engagement. Several studies have documented +the misalignment between short-term benefits and long-term user +engagement [1, 17], as well as the tendency of traditional recom- +mender systems to be detrimental to long-term outcomes [29]. Such +myopic behavior is known to cause boredom and decrease user re- +tention [1], which is prejudicial for both users and content providers. +1https://github.com/naver/gems. +This behavior also raises concerns such as the rich-get-richer issue +[8] and feeding close-mindedness [29]. Some previous studies tried +to counter this effect by explicitly maximizing diversity [33] or +by finding metrics correlated with long-term outcomes [2, 7]. In +contrast, in our work we directly optimize long-term metrics by +using reinforcement learning algorithms [8, 16, 36]. +Reinforcement learning for slate recommendation. The prob- +lem of slate recommendation with reinforcement learning (RL) has +been tackled in several previous studies, although the settings in +which solutions were tested vary and are sometimes not applicable +to our scenario (S). Chen et al. [8] and Bai et al. [3] assume a simple +user click model and independence of items within a slate in order +to reduce the problem to choosing individual items, which they +solve with the REINFORCE algorithm on a SoftMax policy. Ie et al. +[18] assume knowledge of the user’s click model and item relevance, +which allows them to perform combinatorial optimization for the +computation of Q-values. Sunehag et al. [31] take a continuous +action in the product space of item embeddings, i.e., one embed- +ding per slot in the slate, and pre-select nearest-neighbor items +for full-slate Q-function evaluation. Chen et al. [9] use properties +of the optimal Q-function to propose an elegant decomposition +of it and generate optimal slates autoregressively. We detail the +assumptions made by each of these approaches in Section 4, but +we had to discard [9] due to its prohibitively heavy computation: it +requires a number of neural network forward passes proportional +to the slate size times the number of items in the collection (i.e., +10,000 passes in scenario (S)), for each training or inference step. +Our proposed approach differs from previous work because we +do not manually decompose the slates using tractable heuristics +based on restrictive assumptions, but instead approximate the slate +generation process with a deep generative model. Our proposed +framework only has a single requirement, viz. the availability of +logged data with slates and associated clicks, as we will detail in +Section 4. The latter assumption is by no means restrictive as such +logged data is readily available in common industrial recommenda- +tion settings. +Latent action representations. While learning a latent repre- +sentation of states is very common in the RL literature [14, 30], +few studies have tackled the problem of latent action representa- +tion. Chandak et al. [6] train an action generation function in a +supervised manner, by learning to predict the action taken from +a pair of successive states. This is not directly applicable in our +case, because the true user state is not observable and successive +observations are simply clicks that appear to be too weak of a signal +to infer the slates leading to these clicks. Botteghi et al. [5] learn a +state-action world model and jointly train latent state and action +representations in a model-based fashion. +Learning a world model in our setting essentially amounts to the +latent modeling of slates and clicks (similar to our approach), while +also conditioning on an internal hidden state.2 The work by Zhou +et al. [35] is perhaps the closest work to ours, as it uses a variational +auto-encoder (VAE) to embed actions into a controllable latent space +before training an RL agent. However, it does not consider slates +but only simple, atomic actions. In contrast, Jiang et al. [20], Liu +2We tried a similar method in pilot experiments, but the additional conditioning only +deteriorated the results, so we only present the condition-free method in this paper. + +Generative Slate Recommendation with Reinforcement Learning +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +Figure 1: Our proposed framework for slate recommendation with reinforcement learning. We first pretrain our GeMS model on previously +collected logged data composed of slates and associated clicks (left), then we use the frozen decoder of GeMS to decode the RL agent’s low- +dimensional proto-action vector into a slate (right). +et al. [25] train VAEs to represent slates and their associated clicks, +but they do not investigate training an RL agent from the learned +latent representation. +To the best of our knowledge, we are the first to learn a latent +representation of slates for RL-based recommendation. +3 +METHOD +3.1 +Notations and problem definition +We consider a slate recommendation scenario in which a user inter- +acts with a recommender system (RS) throughout an episode of 𝑇 +turns. At every turn 𝑡 ∈ {1, . . . ,𝑇 }, the system recommends a slate +𝑎𝑡 = (𝑖1 +𝑡 , . . . ,𝑖𝑘 +𝑡 ) where (𝑖 𝑗 +𝑡 )1⩽𝑗⩽𝑘 are items from the collection I +and 𝑘 is the size of the slate set by the RS designer. The user can +click on zero, one or several items in the slate and the resulting +click vector 𝑐𝑡 = (𝑐1 +𝑡 , . . . ,𝑐𝑘 +𝑡 ),𝑐 𝑗 +𝑡 ∈ {0, 1} is returned to the RS. +The problem of maximizing the cumulative number of clicks +over an episode can be modeled as a partially observable Markov +decision process (POMDP) M𝑃 = (S, O, A, 𝑅,𝑇, Ω) defined by: +• A set of states S, which represent the unobservable state of the +user’s mind; +• A set of observations O accessible to the system. Here, obser- +vations are clicks from the previous interaction (𝑜𝑡 = 𝑐𝑡−1) and +therefore lie in the space of binary vectors of size 𝑘: O = {0, 1}𝑘; +• A set of actions A, which is the set of all possible slates composed +of items from the collection, i.e., |A| = +|I |! +( |I |−𝑘)!; +• A reward function 𝑅 : S × A → R, which we set to 𝑅(𝑠𝑡,𝑎𝑡) = +𝑟𝑡 = �𝑘 +𝑗=1 𝑐 𝑗 +𝑡 in order to reflect our long-term objective of maxi- +mizing the cumulative number of clicks; and +• A set of unknown transition and observation probabilities, re- +spectively 𝑇 : S × A × S → [0, 1] and Ω : S × A × O → [0, 1], +as well as a distribution over initial states 𝑆1 : S → [0, 1]. +Due to the unobserved nature of the true user state in the POMDP, it +is common to train agents by relying on a proxy of the state inferred +from available observations. The function that provides such proxy +is traditionally referred to as the belief encoder [21]. We also define +the concepts of a policy 𝜋 : S × A → [0, 1] and trajectory 𝜏 = +(𝑜𝑡,𝑎𝑡,𝑟𝑡)1⩽𝑡⩽𝑇 . In the remainder, we write 𝜏 ∼ 𝜋 to signify that +we obtain a trajectory by first sampling an initial state 𝑠1 from 𝑆1 +and then recursively sampling actions𝑇 −1 times from the policy 𝜋. +The goal can now be formulated as finding an optimal policy, i.e., a +policy maximizing the expected return 𝜋∗ ∈ arg max𝜋 E𝜏∼𝜋 [R(𝜏)] +with R(𝜏) = �𝑇 +𝑡=1 𝑟𝑡. Finally, given a state 𝑠 and action 𝑎, we define +the Q-function 𝑄𝜋 (𝑠,𝑎) = E𝜏∼𝜋,𝑠1=𝑠,𝑎1=𝑎 [R(𝜏)] and V-function +𝑉 𝜋 (𝑠) = E𝑎∼𝜋 (𝑠) [𝑄𝜋 (𝑠,𝑎)]. +3.2 +Overview of the framework +In our proposed framework, the interactions with the environment, +i.e., the user, can be described by the following repeated steps: +(1) The belief encoder summarizes the history of interactions with +the user into a state vector; +(2) The agent selects a proto-action based on this state; and +(3) The ranker (here resulting from a VAE model) decodes this +proto-action into a slate that is served to the user. +In the remainder of this section, we first detail our proposed gener- +ative modeling of slates (GeMS). GeMS is a deep generative model +that learns a low-dimensional latent space for slates and associated +clicks – thus constituting a convenient proto-action space for the RL +agent and allowing for tractable RL without resorting to restrictive +assumptions as in prior work [3, 8, 18, 31]. Then we describe how +GeMS is integrated as a ranker in our RL framework and we briefly +discuss the remaining RL components. This two-step process is +depicted in Figure 1. +3.3 +Generative Modeling of Slates (GeMS) +In order to instantiate our GeMS model, we propose to train a vari- +ational auto-encoder (VAE) on a precollected dataset D of logged +interactions, as illustrated in Figure 1 (left). A VAE aims to learn +a joint distribution over data samples (i.e., slates and clicks de- +noted as 𝑎 and 𝑐, respectively) and latent encodings (i.e., proto- +actions denoted as 𝑧) [22]. To do so, a parameterized distribution +𝑝𝜃 (𝑎,𝑐,𝑧) is trained to maximize the marginal likelihood of the data +𝑝𝜃 (𝑎,𝑐) = +∫ +𝑧 𝑝𝜃 (𝑎,𝑐,𝑧)𝑑𝑧. In practice, due to the intractability of +this integral, a parameterized distribution 𝑞𝜙 (𝑧|𝑎,𝑐) is introduced +as a variational approximation of the true posterior 𝑝𝜃 (𝑧|𝑎,𝑐) and +the VAE is trained by maximizing the evidence lower bound (ELBO): +LELBO +𝜃,𝜙 +=E𝑎,𝑐∼D +� +E𝑧∼𝑞𝜙 (·|𝑎,𝑐) [log 𝑝𝜃 (𝑎,𝑐|𝑧)]−KL +� +𝑞𝜙 (𝑧|𝑎,𝑐)∥𝑝(𝑧) +�� +, +where 𝑝(𝑧) is the prior distribution over the latent space, KL is the +Kullback-Leibler divergence [24], and 𝑧 is a sample from a Gaussian + +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +Romain Deffayet, Thibaut Thonet, Jean-Michel Renders, & Maarten de Rijke +distribution obtained using the reparameterization trick [22]. The +distributions 𝑞𝜙 (𝑧|𝑎,𝑐) and 𝑝𝜃 (𝑎,𝑐|𝑧) are usually referred to as the +encoder and the decoder, respectively. +The downstream performance of the RL agent we wish to ulti- +mately learn clearly depends on the upstream ability of the VAE +to properly reconstruct slates. However, as Liu et al. [25] observe, +an accurate reconstruction of slates may limit the agent’s capacity +to satisfy the user’s interests. Indeed, finding high-performance +continuous control policies requires smoothness and structure in +the latent space, which may be lacking if too much emphasis is +given to the reconstruction objective in comparison to the prior +matching objective enforced by the KL-divergence. Therefore, it +is necessary to balance reconstruction and controllability, which +is done by introducing an hyperparameter 𝛽 as weight for the KL +term in Eq. ??. Moreover, in order to promote additional structure +in the latent space, we add a click reconstruction term in the loss: +slates with similar short-term outcomes (i.e., clicks) are grouped +together during pre-training. Yet, we may want to avoid biasing +the learned representations towards click reconstruction too much, +as it may come at the cost of quality of the slate reconstruction. +Therefore, we introduce a hyperparameter 𝜆 to adjust this second +trade-off. We show the empirical impact of 𝛽 and 𝜆 in Section 6.3. +In our implementation, the prior 𝑝(𝑧) is set as a standard Gauss- +ian distribution N (0, I). The encoder 𝑞𝜙 (𝑧|𝑎,𝑐) is a Gaussian dis- +tribution with diagonal covariance N (𝜇𝜙 (𝑎,𝑐), diag(𝜎2 +𝜙 (𝑎,𝑐))), pa- +rameterized by a multi-layer perceptron (MLP). This MLP inputs +the concatenation of learnable item embeddings and associated +clicks over the whole slate, and outputs (𝜇𝜙 (𝑎,𝑐), log 𝜎𝜙 (𝑎,𝑐)). For +the decoder 𝑝𝜃 (𝑎,𝑐|𝑧), another MLP takes as input the latent sam- +ple 𝑧, and outputs the concatenation of reconstructed embeddings +e𝑗 +𝜃 (𝑧) and click probabilities 𝑝 𝑗,𝑐 +𝜃 (𝑐𝑗 |𝑧) for each slot 𝑗 in the slate. +We then derive logits for the item probabilities 𝑝 𝑗,𝑎 +𝜃 (𝑎𝑗 |𝑧) by taking +the dot-product of the reconstructed embedding e𝑗 +𝜃 (𝑧) with the +embeddings of all items in the collection. For collection items, we +use the current version of embeddings learned within the encoder, +but we prevent the gradient from back-propagating to them using +the stop-gradient operator to avoid potential degenerate solutions. +In summary, the VAE is pre-trained by maximizing the ELBO on +the task of reconstructing slates and corresponding clicks, i.e., by +minimizing LGeMS +𝜃,𝜙 += E𝑎,𝑐∼D [LGeMS +𝜃,𝜙 +(𝑎,𝑐)] with: +LGeMS +𝜃,𝜙 +(𝑎,𝑐) = +slate reconstruction +������������������������������������������������������ +𝑘 +∑︁ +𝑗=1 +log 𝑝 𝑗,𝑎 +𝜃 (𝑎𝑗 |𝑧𝜙 (𝑎,𝑐)) + +𝜆 +click reconstruction +������������������������������������������������������ +𝑘 +∑︁ +𝑗=1 +log 𝑝 𝑗,𝑐 +𝜃 (𝑐𝑗 |𝑧𝜙 (𝑎,𝑐)) + +𝛽 +KL-divergence +������������������������������������������������������������������������ +𝑑 +∑︁ +𝑖=1 +� +𝜎2 +𝜙,𝑖 + 𝜇2 +𝜙,𝑖 − log 𝜎𝜙,𝑖 − 1 +� +(1) +where 𝑧𝜙 (𝑎,𝑐) = 𝜇𝜙 (𝑎,𝑐) + diag(𝜎2 +𝜙 (𝑎,𝑐)) · 𝜖, for 𝜖 ∼ N (0, I). Here, +𝑑 is the dimension of the latent space, and 𝛽 and 𝜆 are hyperparam- +eters controlling the respective weight of the KL term and the click +reconstruction term. Note that the KL term takes this simple form +due to the Gaussian assumption on 𝑞𝜙 (𝑧|𝑎,𝑐) and the N (0, I) prior. +3.4 +RL agent and belief encoder +After the pre-training step described in Section 3.3, the parameters +of GeMS are frozen and we use its decoder as the ranker in our +RL framework. The RL agent can then be trained to maximize the +discounted return by taking proto-actions within the VAE’s latent +space. To generate a slate (𝑖1, . . . ,𝑖𝑘) from the agent’s proto-action +𝑧, we take for each slot 𝑗 ∈ {1, . . . ,𝑘} the most likely item according +to the decoder: 𝑖 𝑗 = arg max𝑖 ∈I 𝑝 𝑗,𝑎 +𝜙 (𝑖|𝑧). +Since our focus within the RL framework is on the choice of the +ranker, we adopt a standard implementation of the belief encoder +and the agent: the former is modeled by a gated recurrent unit +(GRU) [10] taking as input the concatenation of item embeddings +and respective clicks from each slate, and the latter is a soft actor- +critic (SAC) [15] algorithm. We chose SAC because it is a well- +established RL algorithm, known for its strong performance and +data-efficiency in continuous control. Additionally, SAC adds an +entropy term incentivizing exploration which we have noticed +during our experiments to be important to attain high performance +in highly stochastic recommendation environments. +4 +BASELINES AND THEIR ASSUMPTIONS +We evaluate our proposed method against four main baselines +derived from prior work. In this section, we describe these baselines +as well the assumptions on user behavior that they formulate in +order to make the combinatorial problem of slate recommendation +tractable. By doing so, we are able to compare the assumptions +made by these baselines and highlight the generality of our method +in Table 1. Note that we only report from previous studies the +mechanism used for slate generation, which is the topic of this +study, and ignore other design choices. +SoftMax. In [3, 8], the authors reduce the combinatorial problem +of slate optimization to the simpler problem of item optimization: +the policy network output is a softmax layer over all items in the +collection, and items are sampled with replacement to form slates. +Doing so requires the mild assumption that the Q-value of the slate +can be linearly decomposed into item-specific Q-values (DQ). But +more importantly, it also requires two strong assumptions, namely +users can click on at most one item per slate (1CL) and the returns +of items in the same slate are mutually independent (MI). Together, +these assumptions are restrictive, because their conjunction means +that the click probability of an item in the slate does not depend +on the item itself. Indeed, having dependent click probabilities +(to enforce the single click) and independent items in the slate is +compatible only if click probabilities do not depend on items. +SlateQ. Ie et al. [18] propose a model-based approach in which +the click behavior of the user is given, and Q-learning [34] is used +to plan and approximate users’ dynamic preferences. On top of +the earlier DQ and 1CL, it requires access to the true relevance and +click model (CM), which is an unfair advantage compared to other +methods. For computational efficiency reasons, we adopt the faster +variant referred to as QL-TT-TS in the original paper. +TopK. Even though, to the best of our knowledge, no work has +proposed this approach, we include it in our set of baselines as + +Generative Slate Recommendation with Reinforcement Learning +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +Table 1: Comparison of assumptions made by prior work. Our +method only requires access to logged interaction data. +1CL +DQ +MI +CM +SP +EIB +LD +SoftMax [3, 8] +✓ +✓ +✓ +✗ +✗ +✗ +✗ +SlateQ [18] +✓ +✓ +✗ +✓ +✗ +✗ +✗ +WkNN [31] +✗ +✓ +✗ +✗ +✓ +✓ +✓ +TopK +✗ +✗ +✗ +✗ +✓ +✗ +✓ +GeMS (Ours) +✗ +✗ +✗ +✗ +✗ +✗ +✓ +it is a natural way to deal with slate recommendation. The agent +takes continuous actions in the space of item embeddings, and we +generate slates by taking the 𝑘 items from the collection with the +closest embeddings to the action, according to a similarity metric +(the dot-product in practice). This method therefore assumes the +availability of logged data of past interactions (LD), in order to +pre-train item embeddings. In our experiments, we evaluate two +variants of this baseline: TopK (MF), where item embeddings are +learned by matrix factorization [23], and TopK (ideal), which uses +ideal item embeddings, i.e., the embeddings used internally by the +simulator (see Section 5.1). The latter version clearly has an unfair +advantage. Also, because ranking items this way assumes that the +most rewarding items should appear on top, it makes the sequential +presentation (SP) assumption from [31] that the true click model +is top-down and fading, i.e., if 𝑐(𝑖) indicates that item 𝑖 has been +clicked and 𝑙 ⩽ 𝑘 is the position of 𝑖 in slate 𝑎, then 𝑃(𝑐(𝑖)|𝑠,𝑎) = +𝑃(𝑐(𝑖)|𝑠,𝑎⩽𝑙) ⩽ 𝑃(𝑐(𝑖)|𝑠, ˜𝑎⩽𝑙−1), where 𝑎⩽𝑙 = (𝑖1, . . . ,𝑖𝑙−1,𝑖) and +˜𝑎⩽𝑙−1 = (𝑖1, . . . ,𝑖𝑙−2,𝑖). +WkNN. In [31], the authors propose a finer-grained and potentially +more capable variant of TopK referred to as Wolpertinger [12]: the +agent takes actions in the product-space of item embeddings over +slate slots, i.e., continuous actions of dimension 𝑘 ×𝑑, where 𝑑 is the +dimension of item embeddings. Then, for each slot in the slate, 𝑝 +candidate items are selected by Euclidean distance with embeddings +of items from the collection, and every candidate item’s contribution +to the Q-value is evaluated in a greedy fashion. Besides LD and DQ, +WkNN requires two strong assumptions to ensure submodularity +of the Q-function: sequential presentation SP and execution is best +(EIB), i.e., recommendations that are risky on the short term are +never worth it. Formally, this translates as: P(𝑅(𝑠, 𝜋1(𝑠)) = 0) ⩾ +P(𝑅(𝑠, 𝜋2(𝑠)) = 0) ⇒ 𝑉 𝜋1 (𝑠) ⩽ 𝑉 𝜋2 (𝑠) for any policies 𝜋1, 𝜋2. +Note that it partly defeats the purpose of long-term optimization. +In Table 1, we summarize the assumptions made by each baseline. +In comparison to prior work, our proposed framework has a single +assumption: the availability of logged data with slates and asso- +ciated clicks (LD), as Table 1 indicates. This assumption is by no +means restrictive as such logged data is readily available in common +industrial recommendation settings. +On top of these baselines, we also include a random policy and +a short-term oracle as reference points. The short-term oracle +has access to the true user and item embeddings, enabling it to +select the items with the highest relevance probability in each slate. +Therefore, at each turn of interaction, it gives an upper bound on +the immediate reward but it is unable to cope with boredom and +influence phenomena. +5 +EXPERIMENTAL SETUP +5.1 +Simulator +We design a simulator that allows us to observe the effect of lifting +the assumptions required by the baselines, and we experiment with +several simulator variants to ensure generalizability. We summarize +our main design choices below and refer the reader to our code +available online3 for a more detailed description. +Item and user embeddings. Following scenario (S), our simula- +tor includes 1, 000 items. We consider a cold-start situation where +users are generated on-the-fly for each new trajectory. Items and +users are randomly assigned embeddings of size 20, corresponding +to ten 2-dimensional topics: e = (e1, . . . , e10). Each 2-dimensional +vector e𝑡 is meant to capture the existence of subtopics within +topic 𝑡. The embedding of a user or item 𝑥 is generated using the +following process: (i) sample topic propensities 𝑤𝑡𝑥 ∼ U(0, 1) and +normalize such that � +𝑡 𝑤𝑡𝑥 = 1; (ii) sample topic-specific compo- +nents 𝜖𝑡𝑥 ∼ N (0, 0.4 · I2) and rescale as e𝑡𝑥 = 𝑤𝑡𝑥 · min(|𝜖𝑡𝑥 |, 1)); +and (iii) normalize the embedding e𝑥 = (e1𝑥, . . . , e10 +𝑥 ) such that +∥e𝑥 ∥ = 1. Each item is associated to a main topic, defined as +𝑡(𝑖) = arg max1⩽𝑡⩽10 ∥e𝑡 +𝑖 ∥. +To accomodate different types of content and platforms, we +derive two variants of item embeddings in the simulator: one with +embeddings obtained as described above, and one with embeddings +for which we square and re-normalize each component. In Section 6, +we highlight this difference in peakedness by referring to the former +as diffuse embeddings and the latter as focused embeddings. +Relevance computation. The relevance probability of item 𝑖 for +user 𝑢 is a monotonically increasing function of the dot-product +between their respective embeddings: rel(𝑖,𝑢) = 𝜎(e𝑖𝑇 e𝑢), where +𝜎 is a sigmoid function. +Boredom and influence effects. User embeddings can be af- +fected by two mechanisms: boredom and influence. Each item 𝑖 +clicked by user 𝑢 influences the user embedding in the next interac- +tion turn as: e𝑢 ← 𝜔e𝑢 +(1−𝜔)e𝑖, where we set 𝜔 = 0.9 in practice. +Additionally, if in the last 10 items clicked by user 𝑢 five have the +same main topic 𝑡𝑏, then 𝑢 gets bored with this topic, meaning we +put e𝑡𝑏 +𝑢 = 0 for 5 turns. These mechanisms have been defined to +penalize myopic behavior and encourage long-term strategies. +Click model. Users click on recommended items according to a +position-based model, i.e., the click probability is the product of +item-specific attractiveness and rank-specific examination probabil- +ities: P(𝑐|𝑖,𝑟) = 𝐴𝑖 × 𝐸𝑟. Specifically, we define for an item located +at rank 𝑟: 𝐸𝑟 = 𝜈𝜀𝑟 + (1 − 𝜈)𝜀𝑘+1−𝑟 with 𝜀 = 0.85. It is a mixture of +the terms 𝜀𝑟 and 𝜀𝑘+1−𝑟, which respectively capture the top-down +and bottom-up browsing behaviors. We use two variants of this +click model in our experiments: TopDown with 𝜈 = 1.0 and Mixed +with 𝜈 = 0.5. The attractiveness of an item is set to its relevance +in TopDown and Mixed. In addition, we consider a third variant +DivPen which also penalizes slates that lack diversity: 𝐴𝑖 is down- +weighted by a factor of 3 if more than 4 items from the slate have +the same main topic (as in Mixed, we also set 𝜈 = 0.5 for DivPen). +In summary, our experiments are performed on 6 simulator variants +defined by the choice of item embedding peakedness (diffuse item +3https://naver/github/gems + +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +Romain Deffayet, Thibaut Thonet, Jean-Michel Renders, & Maarten de Rijke +Table 2: Average cumulative number of clicks on the test set for our 6 simulated environments. Bold: best method; underlined: 2nd-best +method; †: statistically significantly better than all other methods. 95% confidence intervals are given in parentheses. Methods grouped under +“Disclosed env.” have access to privileged information about the environment and can therefore not be fairly compared with “Undisclosed +env.” methods. +Focused item embeddings +Diffuse item embeddings +Method +TopDown +Mixed +DivPen +TopDown +Mixed +DivPen +Disclosed +env. + + +Short-term oracle +SAC+TopK (ideal) +SlateQ +107.7 +101.6 +85.4 +96.7 +94.6 +78.8 +429.0 (±5.9) +384.1 (±13.5) +386.3 (±15.5) +373.9 (±25.0) +371.9 (±36.4) +341.3 (±55.3) +206.5 (±4.1) +202.7 (±3.4) +119.0 (±3.9) +209.5 (±5.4) +192.7 (±5.1) +117.8 (±5.8) +Undisclosed +env. + + +Random +REINFORCE+SoftMax +SAC+WkNN +SAC+TopK (MF) +SAC+GeMS (Ours) +33.8 (±0.2) +33.9 (±0.2) +33.6 (±0.2) +33.3 (±0.2) +33.2 (±0.2) +32.9 (±0.2) +248.1 (±19.3) +233.5 (±18.5) +249.1 (±11.6) +249.5 (±15.3) +214.7 (±25.0) +213.8 (±27.1) +98.5 (±8.9) +97.7 (±10.8) +95.5 (±9.9) +107.2 (±8.9) +89.8 (±7.4) +92.5 (±5.0) +254.4 (±17.1) +232.7 (±19.4) +242.2 (±15.4) +249.7 (±10.3) +184.1 (±1.3) +231.4 (±13.3) +305.3†(±21.9) +242.6 (±21.5) +254.1 (±27.7) +300.0†(±42.8) +260.6†(±27.2) +249.6 (±37.6) +embeddings or focused item embeddings) and the choice of click +model (TopDown, Mixed, or DivPen). +5.2 +Implementation and evaluation details +Our implementation aims to be as standard as possible, considering +the literature on RL, in order to ensure reproducibility. All base- +lines are paired with SAC [15], except SlateQ which is based on +Q-Learning [34], and SoftMax, which we pair with REINFORCE [32] +because it requires a discrete action space and a discretized variant +of SAC led to lower performance in our experiments. We implement +all agents using two-layer neural networks as function approxima- +tors, and use target networks for Q-functions in Slate-Q and SAC. +For hyperparameters common to baselines and our method, we +first performed a grid search over likely regions of the space on +baselines, and re-used the selected values for our method. For all +methods we use the Adam optimizer with learning rates of 0.001 +for Q-networks and 0.003 for policy networks when applicable, as +well as a discount factor 𝛾 = 0.8 and a polyak averaging parameter +𝜏 = 0.002. For the hyperparameters specific to our method (𝑑, 𝛽 +and 𝜆), we perform a grid search on the TopDown environment +with focused item embeddings and select the combination with +the highest validation return. This combination is then re-used +on all other environments. The searched ranges were defined as +𝑑 ∈ {16, 32}, 𝛽 ∈ {0.1, 0.2, 0.5, 1.0, 2.0} and 𝜆 ∈ {0.0, 0.2, 0.5, 1.0}. +For methods making the (LD) assumption, we generated a dataset +of 100K user trajectories (with 100 interactions turns each) from an +𝜖-greedy oracle policy with 𝜖 = 0.5, i.e., each recommended item is +selected either uniformly randomly or by an oracle, with equal prob- +abilities. The VAE in GeMS is trained on this dataset for 10 epochs +with a batch size of 256 and a learning rate of 0.001. For approaches +requiring pre-trained item embeddings (TopK and WkNN), we learn +a simple matrix factorization model on the generated dataset by +considering as positive samples the pairs composed of the user in +the trajectory and each clicked item in their recommended slates. +In all of our experiments, we compare average cumulative re- +wards over 10 seeded runs, corresponding to ten initializations of +the agent’s parameters. In the case of GeMS, the seed also controls +the initialization of the VAE model during pre-training. We train +agents for 100K steps. Each step corresponds to a user trajectory, +composed of 100 interaction turns (i.e., 100 slates successively pre- +sented to the user) for a unique user. Every 1, 000 training steps, we +also evaluate the agents on 200 validation user trajectories. Finally, +the agents are tested by selecting the checkpoint with the highest +validation return and applying it on 500 test user trajectories. Con- +fidence intervals use Student’s 𝑡-distribution, and statistical tests +are Welch’s 𝑡-test. Both are based on a 95% confidence level. +6 +RESULTS +In our experiments, we investigate the following research ques- +tions: (RQ1) How does our slate recommendation framework based +on GeMS compare to previous methods when the underlying as- +sumptions of the latter are lifted? (RQ2) Does the proposed GeMS +framework effectively balance immediate and future rewards to +avoid boredom? (RQ3) How do the balancing hyperparameters 𝛽 +and 𝜆 in GeMS impact the downstream RL performance? +6.1 +Comparison of our method against +baselines (RQ1) +In this section, we compare the performance of our method and +baselines on a wide array of simulated environments, corresponding +to the six environments described in Section 5.1. +Overview of the results. Table 2 shows the average test return +(i.e., cumulated reward or cumulated number of clicks) after train- +ing on 100K user trajectories. We group methods into two cate- +gories: Disclosed env., i.e., methods leveraging hidden environment +information, and Undisclosed env., i.e., methods that consider the +environment as a black-box and are therefore practically applicable. +A first observation we can draw, regardless of the specific environ- +ment used, is that the short-term oracle is easily beaten by most +approaches. Indeed, the simulator penalizes short-sighted recom- +mendations that lead to boredom: in these environments, diversity +is required to reach higher returns. We can also observe the superi- +ority of SAC+TopK (Ideal). This is not surprising, as this method +benefits from an unfair advantage – access to true item embed- +dings – but it suggests that practically applicable methods could be +augmented with domain knowledge to improve their performance. +However, despite having access to privileged information, SlateQ’s +performance is subpar, especially in DivPen environments. Its lower + +Generative Slate Recommendation with Reinforcement Learning +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +(a) Short-term oracle. +(b) SAC+GeMS with 𝛾 = 0. +(c) SAC+GeMS with 𝛾 = 0.8. +Figure 2: Distribution of the relevance scores of items recommended by (a) a short-term oracle, (b) SAC+GeMS with 𝛾 = 0 and (c) SAC+GeMS +with 𝛾 = 0.8. Boredom penalizes item scores and is visualized by orange areas. The myopic approaches (left, center) lead to more boredom +than the long-term approach (right), and therefore to lower average item scores (solid red lines). +performance might be explained by its approximate optimization +strategy and restrictive single-click assumption. +Overall comparison of methods. The proposed SAC+GeMS com- +pares favorably to baselines across the range of environments we sim- +ulate. Out of the 6 tested environments, SAC+GeMS obtained the +best average results on all of them, among which 3 show a statisti- +cally significant improvement over all other methods. SAC+WkNN +performs very poorly: we hypothesize that the approach suffers +from the curse of dimensionality due to the larger action space +(200 dimensions in our experiments) and the assumption made +by the approach that candidate items need to be close to target +item embeddings according to the Euclidean distance. SAC+TopK +(MF) is more competitive, but the large difference with SAC+TopK +(ideal) suggests that TopK is very sensitive to the quality of item +embeddings. Despite its very restrictive assumptions and lack of the- +oretical guarantees in our setup, REINFORCE+SoftMax was a very +competitive baseline overall. However, while its best checkpoint +had high return, its training was unstable and failed to converge in +our experiments, which suggests it may be unreliable. +Comparisons across environments. The TopDown environ- +ment is the easiest for most methods, regardless of the type of +item embeddings. This is not surprising as all methods besides +Random either assume a top-down click model, sample items in +a top-down fashion or rely on data from a top-down logging pol- +icy. However, it is worth noting that other factors can dominate +the performance, such as sub-optimality of item embeddings for +SAC+TopK (MF). Conversely, DivPen was harder for most methods, +because it requires a strong additional constraint to obtain high +returns: intra-slate diversity must be high. SAC+GeMS was also af- +fected by these dynamics, but remained able to beat other methods +by generating diverse slates. Finally, the use of diffused item embed- +dings does not appear to cause lower returns for GeMS, compared +with focused ones, but is associated with larger confidence intervals +for SAC+GeMS: indeed, pivot items spanning multiple topics are +more likely to be attractive, at the expense of more fine-grained +strategies, making the training process uncertain. +6.2 +GeMS overcomes boredom to improve its +return (RQ2) +In Section 1 we highlighted that long-term optimization with RL +can penalize myopic behavior such as recommending only highly +relevant but similar items, which may lead to boredom. In this sec- +tion, we verify that SAC+GeMS is able to adapt its slate selection +to cope with boredom. We recall that in our simulated environ- +ments (detailed in Section 5.1), users get bored of a particular topic +whenever 5 of their latest 10 clicks were on items from that topic. +When a topic is saturated, its corresponding dimensions in the user +embedding are set to 0, which has the effect of diminishing the +attractiveness of future items presented to the user. It is therefore +necessary to avoid boredom in order to reach higher returns, even +if it comes at the cost of lower immediate rewards. +In this section, we compare three approaches on the TopDown +environment with focused item embeddings: (i) the short-term ora- +cle (STO) always maximizing the immediate reward, (ii) SAC+GeMS +with 𝛾 = 0.8 (i.e., our proposed method) where 𝛾 is the discount +factor of the RL algorithm, and (iii) SAC+GeMS with 𝛾 = 0 which +does not explicitly include future rewards in its policy gradient. In +this environment, SAC+GeMS𝛾=0.8 achieves an average test return +of 305.3, while SAC+GeMS𝛾=0 reaches 194.3, and STO only ob- +tains 107.7. These results suggest that long-term optimization is +indeed required to reach higher returns. It may seem surprising +that SAC+GeMS𝛾=0 gets better returns than STO, but its training +objective incentivizes average immediate rewards, which implicitly +encourages it to avoid low future rewards. However, adopting an +explicit mechanism to account for its causal effect on the user (i.e., +setting 𝛾 = 0.8) allows SAC+GeMS to improve its decision-making. +In Figure 2, we plot the distribution of item scores (i.e., the dot- +product between internal user and item embeddings as defined in +Section 5.1) for the items recommended in slates by each of the +three methods, with the same seed for all three plots. The dashed +vertical line shows the score threshold of 0.28 needed to reach a +relevance probability of 0.5. Therefore, items on the left of this +line have a lower click probability while items on the right have a +higher click probability. The color indicates how many topics were +saturated when the agent recommended that particular item whose +score is plotted: one can see that when the user is bored of at least +one topic, items become less attractive as scores are reduced. +When no topic is saturated (i.e., yellow distribution), STO rec- + +average score +ithreshold +12 +1 +Number of +10 +saturated topics +0 +1 +8 +2 +1 +PDF +6 +4 +2 +00 +0.1 +0.2 +0.3 +0.4 +0.5 +Scoreaverage score +ithreshold +12 +Number of +saturated topics +10 +0 +1 +2 +8 +PDF +6 +4 +2 +00 +0.1 +0.2 +0.3 +0.4 +0.5 +Scorethreshold . + average score +18 +Number of +16 +saturated topics +14 +0 +1 +12 +2 +PDF +10 +8 +6 +4 +2 +00 +0.1 +0.2 +0.3 +0.4 +0.5 +ScoreWSDM ’23, February 27-March 3, 2023, Singapore, Singapore +Romain Deffayet, Thibaut Thonet, Jean-Michel Renders, & Maarten de Rijke +(a) Impact of 𝛽 for 𝜆 = 0.5. +(b) Impact of 𝜆 for 𝛽 = 1.0. +Figure 3: Average cumulative number of clicks on the validation set obtained by SAC+GeMS with its best validation checkpoint, for different +values of 𝛽 and 𝜆 (defined in Section 3.3). We also display 95% confidence intervals. +ommends items with excellent scores (above the threshold and +up to 0.45): as a consequence, STO gets high immediate rewards. +However, by doing so it incurs a lot of boredom (large orange +areas). Overall, it leads to lower expected scores (solid red line) +and therefore fewer clicks. Conversely, SAC+GeMS𝛾=0.8 sacrifices +some immediate reward (yellow distribution shifted to the left) but +causes very little boredom (small orange area). Overall, by trading +off relevance and diversity, SAC+GeMS𝛾=0.8 yields good immediate +rewards while limiting boredom. It therefore gets higher average +scores. SAC+GeMS𝛾=0 exhibits an intermediate behavior due to its +limited capabilities: it recommends items of varying relevance, yet +leads to substantial boredom (larger orange area than for 𝛾 = 0.8). +6.3 +Balancing hyperparameters 𝛽 and 𝜆 (RQ3) +In Section 3.3, we suggested that the choice of 𝛽 and 𝜆 leads to trade- +offs that may impact the downstream performance of SAC+GeMS. +As a reminder, 𝛽 adjusts the importance of accurate reconstruction +versus smoothness and structure in the latent space (i.e., controlla- +bility), while 𝜆 weights the click reconstruction with respect to the +slate reconstruction. Next, we verify our intuition on the importance +of these trade-offs by reporting (in Figure 3) the best validation +return obtained for different values of said hyperparameters, on +the TopDown environment with focused item embeddings. +Figure 3a suggests that, indeed, there exists a “sweet spot” in the +selection of 𝛽. It confirms the intuition described in Section 3.3 and +the observation of Liu et al. [25]: 𝛽 must be appropriately balanced +in order to ensure high performance on the downstream RL task. +Specifically, we found that choosing 𝛽 = 1.0 leads to the highest +return overall, regardless of whether a latent dimension of 16 or +32 is used. +The impact on the downstream performance of the trade-off +between slate and click reconstruction (Figure 3b) is less prominent +but can still be observed. It justifies our choice to add the click +reconstruction term in the loss (Eq. 1), even though clicks output by +GeMS’ decoder are not used during RL training. This also confirms +the importance of introducing and adjusting the hyperparameter 𝜆: +modeling clicks jointly with slates improves the final performance of +SAC+GeMS, but properly weighting the click reconstruction objective +with respect to the slate reconstruction objective is necessary. +7 +CONCLUSION +We have presented GeMS, a slate representation learning method +based on variational auto-encoders for slate recommendation with +reinforcement learning. This method has the notable advantage +of being flexible, allowing full-slate modeling and lightweight as- +sumptions, in contrast with existing approaches. +Findings and broader impact. Our experiments across a wide +array of environments demonstrate that GeMS compares favor- +ably against existing slate representation methods in practical set- +tings. Moreover, our empirical analysis highlights that it effectively +balances immediate and future rewards, and that the trade-offs +imposed by 𝛽 and 𝜆 significantly impact the RL downstream perfor- +mance, indicating that properly balancing these hyperparameters is +critical. Our work suggests that generative models are a promising +direction for representing rich actions such as slates. +Limitations. Our simulated experiments demonstrate the effec- +tiveness of GeMS for representing slates in an RL framework. How- +ever, it is well-known that online training of RL agents is too expen- +sive and risky, and that in practice agents must be trained offline, i.e., +directly from logged data [8]. We did not address here the specific +challenges of offline RL, as we wished to isolate the contribution of +the slate representation to downstream performance. +Future work. In future work, we will investigate how generative +models can be leveraged in the offline setting, in different scenarios, +or with even richer actions. We also plan to look into improvements +of the architectures used for structured action representations, for +example by using domain knowledge and user models. +ACKNOWLEDGMENTS +This research was (partially) funded by the Hybrid Intelligence +Center, a 10-year program funded by the Dutch Ministry of Educa- +tion, Culture and Science through the Netherlands Organisation for +Scientific Research, https://hybrid-intelligence-centre.nl. All con- +tent represents the opinion of the authors, which is not necessarily +shared or endorsed by their respective employers and/or sponsors. +REFERENCES +[1] Ashton Anderson, Lucas Maystre, Ian Anderson, Rishabh Mehrotra, and Mounia +Lalmas. 2020. Algorithmic Effects on the Diversity of Consumption on Spotify. +In WWW ’20. 2155–2165. +[2] Susan Athey, Raj Chetty, Guido W Imbens, and Hyunseung Kang. 2019. The + +325 +T↑ +300 +ks + click +275 + number of +250 +Latent dim +Cumulative ( +225 +16 +32 +200 +T +175 +150 - +0.0 +0.2 +0.5 +1.0 +2.0 +beta300 +1 +T + clicks +T +250 + number of +Latent dim +Cumulative +200 +16 +32 +150 +100 - +0.0 +0.2 +0.5 +1.0 +lambdaGenerative Slate Recommendation with Reinforcement Learning +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +Surrogate Index: Combining Short-Term Proxies to Estimate Long-Term Treatment +Effects More Rapidly and Precisely. Technical Report. National Bureau of Economic +Research. +[3] Xueying Bai, Jian Guan, and Hongning Wang. 2019. +A Model-Based Rein- +forcement Learning with Adversarial Training for Online Recommendation. +In NeurIPS ’19. 10734–10745. +[4] Eytan Bakshy, Solomon Messing, and Lada Adamic. 2015. Exposure to Ideo- +logically Diverse News and Opinion on Facebook. Science 348, 6239 (2015), +1130–1132. +[5] Nicolò Botteghi, Mannes Poel, Beril Sirmaçek, and Christoph Brune. 2021. Low- +Dimensional State and Action Representation Learning with MDP Homomor- +phism Metrics. arXiv:2107.01677 (2021). +[6] Yash Chandak, Georgios Theocharous, James Kostas, Scott Jordan, and Philip +Thomas. 2019. Learning Action Representations for Reinforcement Learning. In +ICML ’19. 941–950. +[7] Praveen Chandar, Brian St. Thomas, Lucas Maystre, Vijay Pappu, Roberto Sanchis- +Ojeda, Tiffany Wu, Ben Carterette, Mounia Lalmas, and Tony Jebara. 2022. Using +Survival Models to Estimate User Engagement in Online Experiments. In WWW +’22. 3186–3195. +[8] Minmin Chen, Alex Beutel, Paul Covington, Sagar Jain, Francois Belletti, and +Ed H. Chi. 2019. Top-K Off-Policy Correction for a REINFORCE Recommender +System. In WSDM ’19. 456–464. +[9] Xinshi Chen, Shuang Li, Hui Li, Shaohua Jiang, Yuan Qi, and Le Song. 2019. +Generative Adversarial User Model for Reinforcement Learning Based Recom- +mendation System. In ICML ’19. 1052–1061. +[10] Kyunghyun Cho, Bart van Merrienboer, Çaglar Gülçehre, Dzmitry Bahdanau, +Fethi Bougares, Holger Schwenk, and Yoshua Bengio. 2014. Learning Phrase +Representations using RNN Encoder-Decoder for Statistical Machine Translation. +In EMNLP ’14. 1724–1734. +[11] Van Dang, Michael Bendersky, and W. Bruce Croft. 2013. Two-Stage Learning to +Rank for Information Retrieval. In ECIR ’13. 423–434. +[12] Gabriel Dulac-Arnold, Richard Evans, Hado van Hasselt, Peter Sunehag, Timothy +Lillicrap, Jonathan Hunt, Timothy Mann, Theophane Weber, Thomas Degris, +and Ben Coppin. 2015. Deep Reinforcement Learning in Large Discrete Action +Spaces. arXiv:1512.07679 (2015). +[13] Seth R. Flaxman, Sharad Goel, and Justin M. Rao. 2016. Filter Bubbles, Echo +Chambers, and Online News Consumption. Public Opinion Quarterly 80, S1 +(2016), 298–320. +[14] David Ha and Jürgen Schmidhuber. 2018. Recurrent World Models Facilitate +Policy Evolution. In NeurIPS ’18. 2455–2467. +[15] Tuomas Haarnoja, Aurick Zhou, Pieter Abbeel, and Sergey Levine. 2018. Soft +Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a +Stochastic Actor. In ICML ’18. 1856–1865. +[16] Christian Hansen, Rishabh Mehrotra, Casper Hansen, Brian Brost, Lucas Maystre, +and Mounia Lalmas. 2021. Shifting Consumption towards Diverse Content on +Music Streaming Platforms. In WSDM ’21. 238–246. +[17] Henning Hohnhold, Deirdre O’Brien, and Diane Tang. 2015. Focusing on the +Long-Term: It’s Good for Users and Business. In KDD ’15. 1849–1858. +[18] Eugene Ie, Vihan Jain, Jing Wang, Sanmit Narvekar, Ritesh Agarwal, Rui Wu, +Heng-Tze Cheng, Tushar Chandra, and Craig Boutilier. 2019. SlateQ: A Tractable +Decomposition for Reinforcement Learning with Recommendation Sets. In IJ- +CAI ’19. 2592–2599. +[19] Dietmar Jannach, Pearl Pu, Francesco Ricci, and Markus Zanker. 2021. Recom- +mender Systems: Past, Present, Future. AI Mag. 42, 3 (2021), 3–6. +[20] Ray Jiang, Sven Gowal, Yuqiu Qian, Timothy A. Mann, and Danilo J. Rezende. +2019. Beyond Greedy Ranking: Slate Optimization via List-CVAE. In ICLR ’19. +[21] Leslie Pack Kaelbling, Michael L. Littman, and Anthony R. Cassandra. 1998. +Planning and Acting in Partially Observable Stochastic Domains. Artificial +Intelligence 101, 1 (1998), 99–134. +[22] Diederik Kingma and Max Welling. 2014. Auto-Encoding Variational Bayes. In +ICLR ’14. +[23] Yehuda Koren, Robert M. Bell, and Chris Volinsky. 2009. Matrix Factorization +Techniques for Recommender Systems. Computer 42, 8 (2009), 30–37. +[24] Solomon Kullback and Richard A. Leibler. 1951. On Information and Sufficiency. +The Annals of Mathematical Statistics 22, 1 (1951), 79–86. +[25] Shuchang Liu, Fei Sun, Yingqiang Ge, Changhua Pei, and Yongfeng Zhang. 2021. +Variation Control and Evaluation for Generative Slate Recommendations. In +WWW ’21. 436–448. +[26] Farzan Masrour, Tyler Wilson, Heng Yan, Pang-Ning Tan, and Abdol-Hossein +Esfahanian. 2020. Bursting the Filter Bubble: Fairness-Aware Network Link +Prediction. In AAAI ’20. 841–848. +[27] James McInerney, Brian Brost, Praveen Chandar, Rishabh Mehrotra, and Benjamin +Carterette. 2020. Counterfactual Evaluation of Slate Recommendations with +Sequential Reward Interactions. In KDD ’20. 1779–1788. +[28] Eli Pariser. 2011. The Filter Bubble: What the Internet Is Hiding from You. The +Penguin Press. +[29] Wilbert Samuel Rossi, Jan Willem Polderman, and Paolo Frasca. 2021. The Closed +Loop between Opinion Formation and Personalised Recommendations. IEEE +Transactions on Control of Network Systems (2021). +[30] Adam Stooke, Kimin Lee, Pieter Abbeel, and Michael Laskin. 2021. Decoupling +Representation Learning from Reinforcement Learning. In ICML ’21. 9870–9879. +[31] Peter Sunehag, Richard Evans, Gabriel Dulac-Arnold, Yori Zwols, Daniel Visentin, +and Ben Coppin. 2015. +Deep Reinforcement Learning with Attention for +Slate Markov Decision Processes with High-Dimensional States and Actions. +arXiv:1512.01124 (2015). +[32] Richard Sutton and Andrew Barto. 2018. Reinforcement Learning: An Introduction. +MIT Press, 326–329. +[33] Isaac Waller and Ashton Anderson. 2019. Generalists and Specialists: Using +Community Embeddings to Quantify Activity Diversity in Online Platforms. In +WWW ’19. 1954–1964. +[34] Christopher Watkins and Peter Dayan. 1992. Q-learning. Machine Learning 8 +(1992), 279–292. +[35] Wenxuan Zhou, Sujay Bajracharya, and David Held. 2020. PLAS: Latent Action +Space for Offline Reinforcement Learning. In CoRL ’20. 1719–1735. +[36] Lixin Zou, Long Xia, Zhuoye Ding, Jiaxing Song, Weidong Liu, and Dawei Yin. +2019. Reinforcement Learning to Optimize Long-Term User Engagement in +Recommender Systems. In KDD ’19. 2810–2818. + diff --git a/7NFAT4oBgHgl3EQfoB2V/content/tmp_files/load_file.txt b/7NFAT4oBgHgl3EQfoB2V/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..23a3113f1bec5c7e4b8195f239fa4b76d63c846f --- /dev/null +++ b/7NFAT4oBgHgl3EQfoB2V/content/tmp_files/load_file.txt @@ -0,0 +1,823 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf,len=822 +page_content='Generative Slate Recommendation with Reinforcement Learning Romain Deffayet Naver Labs Europe Meylan, France University of Amsterdam Amsterdam, The Netherlands romain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='deffayet@naverlabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='com Thibaut Thonet Naver Labs Europe Meylan, France thibaut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='thonet@naverlabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='com Jean-Michel Renders Naver Labs Europe Meylan, France jean-michel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='renders@naverlabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='com Maarten de Rijke University of Amsterdam Amsterdam, The Netherlands m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='derijke@uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='nl ABSTRACT Recent research has employed reinforcement learning (RL) algo- rithms to optimize long-term user engagement in recommender systems, thereby avoiding common pitfalls such as user boredom and filter bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' They capture the sequential and interactive na- ture of recommendations, and thus offer a principled way to deal with long-term rewards and avoid myopic behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' However, RL approaches are intractable in the slate recommendation scenario – where a list of items is recommended at each interaction turn – due to the combinatorial action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In that setting, an action corresponds to a slate that may contain any combination of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' While previous work has proposed well-chosen decompositions of actions so as to ensure tractability, these rely on restrictive and sometimes unrealistic assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Instead, in this work we pro- pose to encode slates in a continuous, low-dimensional latent space learned by a variational auto-encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Then, the RL agent selects continuous actions in this latent space, which are ultimately de- coded into the corresponding slates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' By doing so, we are able to (i) relax assumptions required by previous work, and (ii) improve the quality of the action selection by modeling full slates instead of independent items, in particular by enabling diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Our ex- periments performed on a wide array of simulated environments confirm the effectiveness of our generative modeling of slates over baselines in practical scenarios where the restrictive assumptions underlying the baselines are lifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Our findings suggest that repre- sentation learning using generative models is a promising direction towards generalizable RL-based slate recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' CCS CONCEPTS Information systems → Recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Copyrights for components of this work owned by others than the author(s) must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Abstracting with credit is 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+page_content='3570412 KEYWORDS Slate recommendation, Reinforcement learning, Variational auto- encoder ACM Reference Format: Romain Deffayet, Thibaut Thonet, Jean-Michel Renders, and Maarten de Rijke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Generative Slate Recommendation with Reinforcement Learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In Proceedings of the Sixteenth ACM International Conference on Web Search and Data Mining (WSDM ’23), February 27-March 3, 2023, Singa- pore, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' ACM, New York, NY, USA, 9 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='1145/ 3539597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3570412 1 INTRODUCTION Ubiquitous in online services, recommender systems (RSs) play a key role personalization by catering to users’ identified tastes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Ide- ally, they also diversify their offerings and help users discover new interests [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In the latter case, RSs take on an active role, which means that recommendations influence future user behavior, and therefore their effects on users must be explicitly controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Such effects can be detrimental: users may get bored if too many simi- lar recommendations are made, and it has been well-documented that users can end up in so-called filter bubbles or echo chambers [4, 13, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' From the perspective of the online platform or the con- tent provider, user boredom leads to poor retention and conversion rates [17], while filter bubbles raise fairness and ethical issues for which providers can be held accountable [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Conversely, RSs can also positively impact users, for example, when users get interested in new, unexpected topics or when the RS offers a fair represen- tation of available options [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' It is natural, therefore, to balance exploitation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', sticking to the known interests of the user) and exploration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', further probing the user’s interests) so as to avoid always recommending similar items, and encourage recommenda- tions that boost future engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The reinforcement learning (RL) literature has proposed models and algorithms that aim to optimize long-term metrics by acknowledging the causal effect of recommendations on users [8, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In this work we consider the common scenario of slate recom- mendation [8, 18, 31], which comes with specific challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' At each interaction turn, a slate recommender system recommends a list of items from the collection, and the user interacts with zero, one or several of those items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' As a consequence, users may not examine arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='08632v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='IR] 20 Jan 2023 WSDM ’23, February 27-March 3, 2023, Singapore, Singapore Romain Deffayet, Thibaut Thonet, Jean-Michel Renders, & Maarten de Rijke all the recommended items, which leads to biases in the observed interactions along with a complex interplay between items in the same slate [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' More importantly, the size of the action space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', the number of possible slates, prohibits the use of off-the-shelf RL approaches [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Indeed, as slate recommendation is a combinato- rial problem, the evaluation of all actions by the RL agent through trial and error is simply intractable: even with as few as 1, 000 items in the collection, the number of possible slates of size 10 is approximately 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='6 × 1029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' We propose to tackle this problem in the context of a practical scenario, (S), which fits the second-stage ranking phase [11] of many content recommendation platforms: (S) The collection contains around a thousand items, and at each turn of interaction the proposed model must select and rank 10 items to be presented to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' All our tractability and feasibility statements in this paper must therefore be understood through the lens of this scenario (S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' To reduce the prohibitively large size of the combinatorial action space, previous studies have proposed to decompose slates in a tractable manner [8, 18, 31] – but at the cost of restrictive assump- tions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', concerning mutual independence of items in the slate, knowledge of the user click model, availability of high-quality item embeddings, or that at most one item per slate is clicked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In contrast, in this work we propose to first learn a continuous, low-dimensional latent representation of actions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', slates), and then let the agent take actions within this latent space during its training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In practice, we obtain the latent representations by introducing a generative modeling of slates (GeMS) based on a variational auto-encoder (VAE) pre-trained on a dataset of observed slates and clicks, collected from a previous version of the recom- mender system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Such a dataset is usually available in industrial recommendation settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Therefore, we do not rely on restrictive assumptions, and the fact that we represent full slates enables the agent to improve the quality of its recommendations, instead of using individual item representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Our contributions can be summarized as follows: We propose GeMS, a novel way to represent actions in RL for slate recommendation, by pre-training a VAE on slates and associated clicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Unlike previous methods, GeMS is free of overly restrictive assumptions and only requires logged interaction data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' We provide a unified terminology to classify existing slate recom- mendation approaches based on their underlying assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' We show on a wide array of simulated environments that previ- ous methods underperform when their underlying assumptions are lifted (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', in practical settings), while GeMS allows us to re- cover highly rewarding policies without restrictive assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' To support the reproducibility of this work, we publicly release the code for our approach, baselines and simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='1 2 RELATED WORK Long-term user engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Several studies have documented the misalignment between short-term benefits and long-term user engagement [1, 17], as well as the tendency of traditional recom- mender systems to be detrimental to long-term outcomes [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Such myopic behavior is known to cause boredom and decrease user re- tention [1], which is prejudicial for both users and content providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='com/naver/gems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' This behavior also raises concerns such as the rich-get-richer issue [8] and feeding close-mindedness [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Some previous studies tried to counter this effect by explicitly maximizing diversity [33] or by finding metrics correlated with long-term outcomes [2, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In contrast, in our work we directly optimize long-term metrics by using reinforcement learning algorithms [8, 16, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Reinforcement learning for slate recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The prob- lem of slate recommendation with reinforcement learning (RL) has been tackled in several previous studies, although the settings in which solutions were tested vary and are sometimes not applicable to our scenario (S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [8] and Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [3] assume a simple user click model and independence of items within a slate in order to reduce the problem to choosing individual items, which they solve with the REINFORCE algorithm on a SoftMax policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Ie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [18] assume knowledge of the user’s click model and item relevance, which allows them to perform combinatorial optimization for the computation of Q-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Sunehag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [31] take a continuous action in the product space of item embeddings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', one embed- ding per slot in the slate, and pre-select nearest-neighbor items for full-slate Q-function evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [9] use properties of the optimal Q-function to propose an elegant decomposition of it and generate optimal slates autoregressively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' We detail the assumptions made by each of these approaches in Section 4, but we had to discard [9] due to its prohibitively heavy computation: it requires a number of neural network forward passes proportional to the slate size times the number of items in the collection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', 10,000 passes in scenario (S)), for each training or inference step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Our proposed approach differs from previous work because we do not manually decompose the slates using tractable heuristics based on restrictive assumptions, but instead approximate the slate generation process with a deep generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Our proposed framework only has a single requirement, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' the availability of logged data with slates and associated clicks, as we will detail in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The latter assumption is by no means restrictive as such logged data is readily available in common industrial recommenda- tion settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Latent action representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' While learning a latent repre- sentation of states is very common in the RL literature [14, 30], few studies have tackled the problem of latent action representa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Chandak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [6] train an action generation function in a supervised manner, by learning to predict the action taken from a pair of successive states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' This is not directly applicable in our case, because the true user state is not observable and successive observations are simply clicks that appear to be too weak of a signal to infer the slates leading to these clicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Botteghi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [5] learn a state-action world model and jointly train latent state and action representations in a model-based fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Learning a world model in our setting essentially amounts to the latent modeling of slates and clicks (similar to our approach), while also conditioning on an internal hidden state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='2 The work by Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [35] is perhaps the closest work to ours, as it uses a variational auto-encoder (VAE) to embed actions into a controllable latent space before training an RL agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' However, it does not consider slates but only simple, atomic actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In contrast, Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [20], Liu 2We tried a similar method in pilot experiments, but the additional conditioning only deteriorated the results, so we only present the condition-free method in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Generative Slate Recommendation with Reinforcement Learning WSDM ’23, February 27-March 3, 2023, Singapore, Singapore Figure 1: Our proposed framework for slate recommendation with reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' We first pretrain our GeMS model on previously collected logged data composed of slates and associated clicks (left), then we use the frozen decoder of GeMS to decode the RL agent’s low- dimensional proto-action vector into a slate (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [25] train VAEs to represent slates and their associated clicks, but they do not investigate training an RL agent from the learned latent representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' To the best of our knowledge, we are the first to learn a latent representation of slates for RL-based recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 3 METHOD 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='1 Notations and problem definition We consider a slate recommendation scenario in which a user inter- acts with a recommender system (RS) throughout an episode of 𝑇 turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' At every turn 𝑡 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' ,𝑇 }, the system recommends a slate 𝑎𝑡 = (𝑖1 𝑡 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' ,𝑖𝑘 𝑡 ) where (𝑖 𝑗 𝑡 )1⩽𝑗⩽𝑘 are items from the collection I and 𝑘 is the size of the slate set by the RS designer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The user can click on zero, one or several items in the slate and the resulting click vector 𝑐𝑡 = (𝑐1 𝑡 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' ,𝑐𝑘 𝑡 ),𝑐 𝑗 𝑡 ∈ {0, 1} is returned to the RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The problem of maximizing the cumulative number of clicks over an episode can be modeled as a partially observable Markov decision process (POMDP) M𝑃 = (S, O, A, 𝑅,𝑇, Ω) defined by: A set of states S, which represent the unobservable state of the user’s mind;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' A set of observations O accessible to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Here, obser- vations are clicks from the previous interaction (𝑜𝑡 = 𝑐𝑡−1) and therefore lie in the space of binary vectors of size 𝑘: O = {0, 1}𝑘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' A set of actions A, which is the set of all possible slates composed of items from the collection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', |A| = |I |!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' ( |I |−𝑘)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' A reward function 𝑅 : S × A → R, which we set to 𝑅(𝑠𝑡,𝑎𝑡) = 𝑟𝑡 = �𝑘 𝑗=1 𝑐 𝑗 𝑡 in order to reflect our long-term objective of maxi- mizing the cumulative number of clicks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' and A set of unknown transition and observation probabilities, re- spectively 𝑇 : S × A × S → [0, 1] and Ω : S × A × O → [0, 1], as well as a distribution over initial states 𝑆1 : S → [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Due to the unobserved nature of the true user state in the POMDP, it is common to train agents by relying on a proxy of the state inferred from available observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The function that provides such proxy is traditionally referred to as the belief encoder [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' We also define the concepts of a policy 𝜋 : S × A → [0, 1] and trajectory 𝜏 = (𝑜𝑡,𝑎𝑡,𝑟𝑡)1⩽𝑡⩽𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In the remainder, we write 𝜏 ∼ 𝜋 to signify that we obtain a trajectory by first sampling an initial state 𝑠1 from 𝑆1 and then recursively sampling actions𝑇 −1 times from the policy 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The goal can now be formulated as finding an optimal policy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', a policy maximizing the expected return 𝜋∗ ∈ arg max𝜋 E𝜏∼𝜋 [R(𝜏)] with R(𝜏) = �𝑇 𝑡=1 𝑟𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Finally, given a state 𝑠 and action 𝑎, we define the Q-function 𝑄𝜋 (𝑠,𝑎) = E𝜏∼𝜋,𝑠1=𝑠,𝑎1=𝑎 [R(𝜏)] and V-function 𝑉 𝜋 (𝑠) = E𝑎∼𝜋 (𝑠) [𝑄𝜋 (𝑠,𝑎)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='2 Overview of the framework In our proposed framework, the interactions with the environment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', the user, can be described by the following repeated steps: (1) The belief encoder summarizes the history of interactions with the user into a state vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' (2) The agent selects a proto-action based on this state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' and (3) The ranker (here resulting from a VAE model) decodes this proto-action into a slate that is served to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In the remainder of this section, we first detail our proposed gener- ative modeling of slates (GeMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' GeMS is a deep generative model that learns a low-dimensional latent space for slates and associated clicks – thus constituting a convenient proto-action space for the RL agent and allowing for tractable RL without resorting to restrictive assumptions as in prior work [3, 8, 18, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Then we describe how GeMS is integrated as a ranker in our RL framework and we briefly discuss the remaining RL components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' This two-step process is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3 Generative Modeling of Slates (GeMS) In order to instantiate our GeMS model, we propose to train a vari- ational auto-encoder (VAE) on a precollected dataset D of logged interactions, as illustrated in Figure 1 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' A VAE aims to learn a joint distribution over data samples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', slates and clicks de- noted as 𝑎 and 𝑐, respectively) and latent encodings (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', proto- actions denoted as 𝑧) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' To do so, a parameterized distribution 𝑝𝜃 (𝑎,𝑐,𝑧) is trained to maximize the marginal likelihood of the data 𝑝𝜃 (𝑎,𝑐) = ∫ 𝑧 𝑝𝜃 (𝑎,𝑐,𝑧)𝑑𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In practice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' due to the intractability of this integral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' a parameterized distribution 𝑞𝜙 (𝑧|𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝑐) is introduced as a variational approximation of the true posterior 𝑝𝜃 (𝑧|𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝑐) and the VAE is trained by maximizing the evidence lower bound (ELBO): LELBO 𝜃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝜙 =E𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝑐∼D � E𝑧∼𝑞𝜙 (·|𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝑐) [log 𝑝𝜃 (𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝑐|𝑧)]−KL � 𝑞𝜙 (𝑧|𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝑐)∥𝑝(𝑧) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' where 𝑝(𝑧) is the prior distribution over the latent space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' KL is the Kullback-Leibler divergence [24],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' and 𝑧 is a sample from a Gaussian WSDM ’23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' February 27-March 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Singapore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Singapore Romain Deffayet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Thibaut Thonet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Jean-Michel Renders,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' & Maarten de Rijke distribution obtained using the reparameterization trick [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The distributions 𝑞𝜙 (𝑧|𝑎,𝑐) and 𝑝𝜃 (𝑎,𝑐|𝑧) are usually referred to as the encoder and the decoder, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The downstream performance of the RL agent we wish to ulti- mately learn clearly depends on the upstream ability of the VAE to properly reconstruct slates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' However, as Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [25] observe, an accurate reconstruction of slates may limit the agent’s capacity to satisfy the user’s interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Indeed, finding high-performance continuous control policies requires smoothness and structure in the latent space, which may be lacking if too much emphasis is given to the reconstruction objective in comparison to the prior matching objective enforced by the KL-divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Therefore, it is necessary to balance reconstruction and controllability, which is done by introducing an hyperparameter 𝛽 as weight for the KL term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='. Moreover, in order to promote additional structure in the latent space, we add a click reconstruction term in the loss: slates with similar short-term outcomes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', clicks) are grouped together during pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Yet, we may want to avoid biasing the learned representations towards click reconstruction too much, as it may come at the cost of quality of the slate reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Therefore, we introduce a hyperparameter 𝜆 to adjust this second trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' We show the empirical impact of 𝛽 and 𝜆 in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In our implementation, the prior 𝑝(𝑧) is set as a standard Gauss- ian distribution N (0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The encoder 𝑞𝜙 (𝑧|𝑎,𝑐) is a Gaussian dis- tribution with diagonal covariance N (𝜇𝜙 (𝑎,𝑐), diag(𝜎2 𝜙 (𝑎,𝑐))), pa- rameterized by a multi-layer perceptron (MLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' This MLP inputs the concatenation of learnable item embeddings and associated clicks over the whole slate, and outputs (𝜇𝜙 (𝑎,𝑐), log 𝜎𝜙 (𝑎,𝑐)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' For the decoder 𝑝𝜃 (𝑎,𝑐|𝑧), another MLP takes as input the latent sam- ple 𝑧, and outputs the concatenation of reconstructed embeddings e𝑗 𝜃 (𝑧) and click probabilities 𝑝 𝑗,𝑐 𝜃 (𝑐𝑗 |𝑧) for each slot 𝑗 in the slate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' We then derive logits for the item probabilities 𝑝 𝑗,𝑎 𝜃 (𝑎𝑗 |𝑧) by taking the dot-product of the reconstructed embedding e𝑗 𝜃 (𝑧) with the embeddings of all items in the collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' For collection items, we use the current version of embeddings learned within the encoder, but we prevent the gradient from back-propagating to them using the stop-gradient operator to avoid potential degenerate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In summary, the VAE is pre-trained by maximizing the ELBO on the task of reconstructing slates and corresponding clicks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' by minimizing LGeMS 𝜃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝜙 = E𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝑐∼D [LGeMS 𝜃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝜙 (𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝑐)] with: LGeMS 𝜃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝜙 (𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝑐) = slate reconstruction ������������������������������������������������������ 𝑘 ∑︁ 𝑗=1 log 𝑝 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝑎 𝜃 (𝑎𝑗 |𝑧𝜙 (𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝑐)) + 𝜆 click reconstruction ������������������������������������������������������ 𝑘 ∑︁ 𝑗=1 log 𝑝 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝑐 𝜃 (𝑐𝑗 |𝑧𝜙 (𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝑐)) + 𝛽 KL-divergence ������������������������������������������������������������������������ 𝑑 ∑︁ 𝑖=1 � 𝜎2 𝜙,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝑖 + 𝜇2 𝜙,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝑖 − log 𝜎𝜙,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝑖 − 1 � (1) where 𝑧𝜙 (𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝑐) = 𝜇𝜙 (𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝑐) + diag(𝜎2 𝜙 (𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='𝑐)) · 𝜖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' for 𝜖 ∼ N (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Here, 𝑑 is the dimension of the latent space, and 𝛽 and 𝜆 are hyperparam- eters controlling the respective weight of the KL term and the click reconstruction term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Note that the KL term takes this simple form due to the Gaussian assumption on 𝑞𝜙 (𝑧|𝑎,𝑐) and the N (0, I) prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='4 RL agent and belief encoder After the pre-training step described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3, the parameters of GeMS are frozen and we use its decoder as the ranker in our RL framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The RL agent can then be trained to maximize the discounted return by taking proto-actions within the VAE’s latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' To generate a slate (𝑖1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' ,𝑖𝑘) from the agent’s proto-action 𝑧, we take for each slot 𝑗 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' ,𝑘} the most likely item according to the decoder: 𝑖 𝑗 = arg max𝑖 ∈I 𝑝 𝑗,𝑎 𝜙 (𝑖|𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Since our focus within the RL framework is on the choice of the ranker, we adopt a standard implementation of the belief encoder and the agent: the former is modeled by a gated recurrent unit (GRU) [10] taking as input the concatenation of item embeddings and respective clicks from each slate, and the latter is a soft actor- critic (SAC) [15] algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' We chose SAC because it is a well- established RL algorithm, known for its strong performance and data-efficiency in continuous control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Additionally, SAC adds an entropy term incentivizing exploration which we have noticed during our experiments to be important to attain high performance in highly stochastic recommendation environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 4 BASELINES AND THEIR ASSUMPTIONS We evaluate our proposed method against four main baselines derived from prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In this section, we describe these baselines as well the assumptions on user behavior that they formulate in order to make the combinatorial problem of slate recommendation tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' By doing so, we are able to compare the assumptions made by these baselines and highlight the generality of our method in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Note that we only report from previous studies the mechanism used for slate generation, which is the topic of this study, and ignore other design choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' SoftMax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In [3, 8], the authors reduce the combinatorial problem of slate optimization to the simpler problem of item optimization: the policy network output is a softmax layer over all items in the collection, and items are sampled with replacement to form slates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Doing so requires the mild assumption that the Q-value of the slate can be linearly decomposed into item-specific Q-values (DQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' But more importantly, it also requires two strong assumptions, namely users can click on at most one item per slate (1CL) and the returns of items in the same slate are mutually independent (MI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Together, these assumptions are restrictive, because their conjunction means that the click probability of an item in the slate does not depend on the item itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Indeed, having dependent click probabilities (to enforce the single click) and independent items in the slate is compatible only if click probabilities do not depend on items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' SlateQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Ie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [18] propose a model-based approach in which the click behavior of the user is given, and Q-learning [34] is used to plan and approximate users’ dynamic preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' On top of the earlier DQ and 1CL, it requires access to the true relevance and click model (CM), which is an unfair advantage compared to other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' For computational efficiency reasons, we adopt the faster variant referred to as QL-TT-TS in the original paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' TopK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Even though, to the best of our knowledge, no work has proposed this approach, we include it in our set of baselines as Generative Slate Recommendation with Reinforcement Learning WSDM ’23, February 27-March 3, 2023, Singapore, Singapore Table 1: Comparison of assumptions made by prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Our method only requires access to logged interaction data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 1CL DQ MI CM SP EIB LD SoftMax [3, 8] ✓ ✓ ✓ ✗ ✗ ✗ ✗ SlateQ [18] ✓ ✓ ✗ ✓ ✗ ✗ ✗ WkNN [31] ✗ ✓ ✗ ✗ ✓ ✓ ✓ TopK ✗ ✗ ✗ ✗ ✓ ✗ ✓ GeMS (Ours) ✗ ✗ ✗ ✗ ✗ ✗ ✓ it is a natural way to deal with slate recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The agent takes continuous actions in the space of item embeddings, and we generate slates by taking the 𝑘 items from the collection with the closest embeddings to the action, according to a similarity metric (the dot-product in practice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' This method therefore assumes the availability of logged data of past interactions (LD), in order to pre-train item embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In our experiments, we evaluate two variants of this baseline: TopK (MF), where item embeddings are learned by matrix factorization [23], and TopK (ideal), which uses ideal item embeddings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', the embeddings used internally by the simulator (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The latter version clearly has an unfair advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Also, because ranking items this way assumes that the most rewarding items should appear on top, it makes the sequential presentation (SP) assumption from [31] that the true click model is top-down and fading, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', if 𝑐(𝑖) indicates that item 𝑖 has been clicked and 𝑙 ⩽ 𝑘 is the position of 𝑖 in slate 𝑎, then 𝑃(𝑐(𝑖)|𝑠,𝑎) = 𝑃(𝑐(𝑖)|𝑠,𝑎⩽𝑙) ⩽ 𝑃(𝑐(𝑖)|𝑠, ˜𝑎⩽𝑙−1), where 𝑎⩽𝑙 = (𝑖1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' ,𝑖𝑙−1,𝑖) and ˜𝑎⩽𝑙−1 = (𝑖1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' ,𝑖𝑙−2,𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' WkNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In [31], the authors propose a finer-grained and potentially more capable variant of TopK referred to as Wolpertinger [12]: the agent takes actions in the product-space of item embeddings over slate slots, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', continuous actions of dimension 𝑘 ×𝑑, where 𝑑 is the dimension of item embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Then, for each slot in the slate, 𝑝 candidate items are selected by Euclidean distance with embeddings of items from the collection, and every candidate item’s contribution to the Q-value is evaluated in a greedy fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Besides LD and DQ, WkNN requires two strong assumptions to ensure submodularity of the Q-function: sequential presentation SP and execution is best (EIB), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', recommendations that are risky on the short term are never worth it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Formally, this translates as: P(𝑅(𝑠, 𝜋1(𝑠)) = 0) ⩾ P(𝑅(𝑠, 𝜋2(𝑠)) = 0) ⇒ 𝑉 𝜋1 (𝑠) ⩽ 𝑉 𝜋2 (𝑠) for any policies 𝜋1, 𝜋2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Note that it partly defeats the purpose of long-term optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In Table 1, we summarize the assumptions made by each baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In comparison to prior work, our proposed framework has a single assumption: the availability of logged data with slates and asso- ciated clicks (LD), as Table 1 indicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' This assumption is by no means restrictive as such logged data is readily available in common industrial recommendation settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' On top of these baselines, we also include a random policy and a short-term oracle as reference points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The short-term oracle has access to the true user and item embeddings, enabling it to select the items with the highest relevance probability in each slate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Therefore, at each turn of interaction, it gives an upper bound on the immediate reward but it is unable to cope with boredom and influence phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 5 EXPERIMENTAL SETUP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='1 Simulator We design a simulator that allows us to observe the effect of lifting the assumptions required by the baselines, and we experiment with several simulator variants to ensure generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' We summarize our main design choices below and refer the reader to our code available online3 for a more detailed description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Item and user embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Following scenario (S), our simula- tor includes 1, 000 items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' We consider a cold-start situation where users are generated on-the-fly for each new trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Items and users are randomly assigned embeddings of size 20, corresponding to ten 2-dimensional topics: e = (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' , e10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Each 2-dimensional vector e𝑡 is meant to capture the existence of subtopics within topic 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The embedding of a user or item 𝑥 is generated using the following process: (i) sample topic propensities 𝑤𝑡𝑥 ∼ U(0, 1) and normalize such that � 𝑡 𝑤𝑡𝑥 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' (ii) sample topic-specific compo- nents 𝜖𝑡𝑥 ∼ N (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='4 · I2) and rescale as e𝑡𝑥 = 𝑤𝑡𝑥 · min(|𝜖𝑡𝑥 |, 1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' and (iii) normalize the embedding e𝑥 = (e1𝑥, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' , e10 𝑥 ) such that ∥e𝑥 ∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Each item is associated to a main topic, defined as 𝑡(𝑖) = arg max1⩽𝑡⩽10 ∥e𝑡 𝑖 ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' To accomodate different types of content and platforms, we derive two variants of item embeddings in the simulator: one with embeddings obtained as described above, and one with embeddings for which we square and re-normalize each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In Section 6, we highlight this difference in peakedness by referring to the former as diffuse embeddings and the latter as focused embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Relevance computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The relevance probability of item 𝑖 for user 𝑢 is a monotonically increasing function of the dot-product between their respective embeddings: rel(𝑖,𝑢) = 𝜎(e𝑖𝑇 e𝑢), where 𝜎 is a sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Boredom and influence effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' User embeddings can be af- fected by two mechanisms: boredom and influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Each item 𝑖 clicked by user 𝑢 influences the user embedding in the next interac- tion turn as: e𝑢 ← 𝜔e𝑢 +(1−𝜔)e𝑖, where we set 𝜔 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='9 in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Additionally, if in the last 10 items clicked by user 𝑢 five have the same main topic 𝑡𝑏, then 𝑢 gets bored with this topic, meaning we put e𝑡𝑏 𝑢 = 0 for 5 turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' These mechanisms have been defined to penalize myopic behavior and encourage long-term strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Click model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Users click on recommended items according to a position-based model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', the click probability is the product of item-specific attractiveness and rank-specific examination probabil- ities: P(𝑐|𝑖,𝑟) = 𝐴𝑖 × 𝐸𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Specifically, we define for an item located at rank 𝑟: 𝐸𝑟 = 𝜈𝜀𝑟 + (1 − 𝜈)𝜀𝑘+1−𝑟 with 𝜀 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' It is a mixture of the terms 𝜀𝑟 and 𝜀𝑘+1−𝑟, which respectively capture the top-down and bottom-up browsing behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' We use two variants of this click model in our experiments: TopDown with 𝜈 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='0 and Mixed with 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The attractiveness of an item is set to its relevance in TopDown and Mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In addition, we consider a third variant DivPen which also penalizes slates that lack diversity: 𝐴𝑖 is down- weighted by a factor of 3 if more than 4 items from the slate have the same main topic (as in Mixed, we also set 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5 for DivPen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In summary, our experiments are performed on 6 simulator variants defined by the choice of item embedding peakedness (diffuse item 3https://naver/github/gems WSDM ’23, February 27-March 3, 2023, Singapore, Singapore Romain Deffayet, Thibaut Thonet, Jean-Michel Renders, & Maarten de Rijke Table 2: Average cumulative number of clicks on the test set for our 6 simulated environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Bold: best method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' underlined: 2nd-best method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' †: statistically significantly better than all other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 95% confidence intervals are given in parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Methods grouped under “Disclosed env.” have access to privileged information about the environment and can therefore not be fairly compared with “Undisclosed env.” methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Focused item embeddings Diffuse item embeddings Method TopDown Mixed DivPen TopDown Mixed DivPen Disclosed env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' \uf8f1\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f3 Short-term oracle SAC+TopK (ideal) SlateQ 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='7 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='6 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='4 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='7 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='6 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='8 429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='0 (±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='9) 384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='1 (±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5) 386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3 (±15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5) 373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='9 (±25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='0) 371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='9 (±36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='4) 341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3 (±55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3) 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5 (±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='1) 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='7 (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='4) 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='0 (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='9) 209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5 (±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='4) 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='7 (±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='1) 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='8 (±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='8) Undisclosed env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 Random REINFORCE+SoftMax SAC+WkNN SAC+TopK (MF) SAC+GeMS (Ours) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='8 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='2) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='9 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='2) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='6 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='2) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='2) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='2 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='2) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='9 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='2) 248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='1 (±19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3) 233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5 (±18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5) 249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='1 (±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='6) 249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5 (±15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3) 214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='7 (±25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='0) 213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='8 (±27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='1) 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5 (±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='9) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='7 (±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='8) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5 (±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='9) 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='2 (±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='9) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='8 (±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='4) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5 (±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='0) 254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='4 (±17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='1) 232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='7 (±19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='4) 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='2 (±15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='4) 249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='7 (±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3) 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='1 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3) 231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='4 (±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3) 305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3†(±21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='9) 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='6 (±21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5) 254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='1 (±27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='7) 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='0†(±42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='8) 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='6†(±27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='2) 249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='6 (±37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='6) embeddings or focused item embeddings) and the choice of click model (TopDown, Mixed, or DivPen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='2 Implementation and evaluation details Our implementation aims to be as standard as possible, considering the literature on RL, in order to ensure reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' All base- lines are paired with SAC [15], except SlateQ which is based on Q-Learning [34], and SoftMax, which we pair with REINFORCE [32] because it requires a discrete action space and a discretized variant of SAC led to lower performance in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' We implement all agents using two-layer neural networks as function approxima- tors, and use target networks for Q-functions in Slate-Q and SAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' For hyperparameters common to baselines and our method, we first performed a grid search over likely regions of the space on baselines, and re-used the selected values for our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' For all methods we use the Adam optimizer with learning rates of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='001 for Q-networks and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='003 for policy networks when applicable, as well as a discount factor 𝛾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='8 and a polyak averaging parameter 𝜏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' For the hyperparameters specific to our method (𝑑, 𝛽 and 𝜆), we perform a grid search on the TopDown environment with focused item embeddings and select the combination with the highest validation return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' This combination is then re-used on all other environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The searched ranges were defined as 𝑑 ∈ {16, 32}, 𝛽 ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='0} and 𝜆 ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' For methods making the (LD) assumption, we generated a dataset of 100K user trajectories (with 100 interactions turns each) from an 𝜖-greedy oracle policy with 𝜖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', each recommended item is selected either uniformly randomly or by an oracle, with equal prob- abilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The VAE in GeMS is trained on this dataset for 10 epochs with a batch size of 256 and a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' For approaches requiring pre-trained item embeddings (TopK and WkNN), we learn a simple matrix factorization model on the generated dataset by considering as positive samples the pairs composed of the user in the trajectory and each clicked item in their recommended slates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In all of our experiments, we compare average cumulative re- wards over 10 seeded runs, corresponding to ten initializations of the agent’s parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In the case of GeMS, the seed also controls the initialization of the VAE model during pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' We train agents for 100K steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Each step corresponds to a user trajectory, composed of 100 interaction turns (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', 100 slates successively pre- sented to the user) for a unique user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Every 1, 000 training steps, we also evaluate the agents on 200 validation user trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Finally, the agents are tested by selecting the checkpoint with the highest validation return and applying it on 500 test user trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Con- fidence intervals use Student’s 𝑡-distribution, and statistical tests are Welch’s 𝑡-test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Both are based on a 95% confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 6 RESULTS In our experiments, we investigate the following research ques- tions: (RQ1) How does our slate recommendation framework based on GeMS compare to previous methods when the underlying as- sumptions of the latter are lifted?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' (RQ2) Does the proposed GeMS framework effectively balance immediate and future rewards to avoid boredom?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' (RQ3) How do the balancing hyperparameters 𝛽 and 𝜆 in GeMS impact the downstream RL performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='1 Comparison of our method against baselines (RQ1) In this section, we compare the performance of our method and baselines on a wide array of simulated environments, corresponding to the six environments described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Overview of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Table 2 shows the average test return (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', cumulated reward or cumulated number of clicks) after train- ing on 100K user trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' We group methods into two cate- gories: Disclosed env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', methods leveraging hidden environment information, and Undisclosed env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', methods that consider the environment as a black-box and are therefore practically applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' A first observation we can draw, regardless of the specific environ- ment used, is that the short-term oracle is easily beaten by most approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Indeed, the simulator penalizes short-sighted recom- mendations that lead to boredom: in these environments, diversity is required to reach higher returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' We can also observe the superi- ority of SAC+TopK (Ideal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' This is not surprising, as this method benefits from an unfair advantage – access to true item embed- dings – but it suggests that practically applicable methods could be augmented with domain knowledge to improve their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' However, despite having access to privileged information, SlateQ’s performance is subpar, especially in DivPen environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Its lower Generative Slate Recommendation with Reinforcement Learning WSDM ’23, February 27-March 3, 2023, Singapore, Singapore (a) Short-term oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' (b) SAC+GeMS with 𝛾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' (c) SAC+GeMS with 𝛾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Figure 2: Distribution of the relevance scores of items recommended by (a) a short-term oracle, (b) SAC+GeMS with 𝛾 = 0 and (c) SAC+GeMS with 𝛾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Boredom penalizes item scores and is visualized by orange areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The myopic approaches (left, center) lead to more boredom than the long-term approach (right), and therefore to lower average item scores (solid red lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' performance might be explained by its approximate optimization strategy and restrictive single-click assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Overall comparison of methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The proposed SAC+GeMS com- pares favorably to baselines across the range of environments we sim- ulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Out of the 6 tested environments, SAC+GeMS obtained the best average results on all of them, among which 3 show a statisti- cally significant improvement over all other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' SAC+WkNN performs very poorly: we hypothesize that the approach suffers from the curse of dimensionality due to the larger action space (200 dimensions in our experiments) and the assumption made by the approach that candidate items need to be close to target item embeddings according to the Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' SAC+TopK (MF) is more competitive, but the large difference with SAC+TopK (ideal) suggests that TopK is very sensitive to the quality of item embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Despite its very restrictive assumptions and lack of the- oretical guarantees in our setup, REINFORCE+SoftMax was a very competitive baseline overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' However, while its best checkpoint had high return, its training was unstable and failed to converge in our experiments, which suggests it may be unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Comparisons across environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The TopDown environ- ment is the easiest for most methods, regardless of the type of item embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' This is not surprising as all methods besides Random either assume a top-down click model, sample items in a top-down fashion or rely on data from a top-down logging pol- icy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' However, it is worth noting that other factors can dominate the performance, such as sub-optimality of item embeddings for SAC+TopK (MF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Conversely, DivPen was harder for most methods, because it requires a strong additional constraint to obtain high returns: intra-slate diversity must be high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' SAC+GeMS was also af- fected by these dynamics, but remained able to beat other methods by generating diverse slates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Finally, the use of diffused item embed- dings does not appear to cause lower returns for GeMS, compared with focused ones, but is associated with larger confidence intervals for SAC+GeMS: indeed, pivot items spanning multiple topics are more likely to be attractive, at the expense of more fine-grained strategies, making the training process uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='2 GeMS overcomes boredom to improve its return (RQ2) In Section 1 we highlighted that long-term optimization with RL can penalize myopic behavior such as recommending only highly relevant but similar items, which may lead to boredom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In this sec- tion, we verify that SAC+GeMS is able to adapt its slate selection to cope with boredom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' We recall that in our simulated environ- ments (detailed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='1), users get bored of a particular topic whenever 5 of their latest 10 clicks were on items from that topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' When a topic is saturated, its corresponding dimensions in the user embedding are set to 0, which has the effect of diminishing the attractiveness of future items presented to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' It is therefore necessary to avoid boredom in order to reach higher returns, even if it comes at the cost of lower immediate rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In this section, we compare three approaches on the TopDown environment with focused item embeddings: (i) the short-term ora- cle (STO) always maximizing the immediate reward, (ii) SAC+GeMS with 𝛾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='8 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', our proposed method) where 𝛾 is the discount factor of the RL algorithm, and (iii) SAC+GeMS with 𝛾 = 0 which does not explicitly include future rewards in its policy gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In this environment, SAC+GeMS𝛾=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='8 achieves an average test return of 305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3, while SAC+GeMS𝛾=0 reaches 194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3, and STO only ob- tains 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' These results suggest that long-term optimization is indeed required to reach higher returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' It may seem surprising that SAC+GeMS𝛾=0 gets better returns than STO, but its training objective incentivizes average immediate rewards, which implicitly encourages it to avoid low future rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' However, adopting an explicit mechanism to account for its causal effect on the user (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', setting 𝛾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='8) allows SAC+GeMS to improve its decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In Figure 2, we plot the distribution of item scores (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', the dot- product between internal user and item embeddings as defined in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='1) for the items recommended in slates by each of the three methods, with the same seed for all three plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The dashed vertical line shows the score threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='28 needed to reach a relevance probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Therefore, items on the left of this line have a lower click probability while items on the right have a higher click probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The color indicates how many topics were saturated when the agent recommended that particular item whose score is plotted: one can see that when the user is bored of at least one topic, items become less attractive as scores are reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' When no topic is saturated (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', yellow distribution), STO rec- average score ithreshold 12 1 Number of 10 saturated topics 0 1 8 2 1 PDF 6 4 2 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5 Scoreaverage score ithreshold 12 Number of saturated topics 10 0 1 2 8 PDF 6 4 2 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5 Scorethreshold .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' average score 18 Number of 16 saturated topics 14 0 1 12 2 PDF 10 8 6 4 2 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5 ScoreWSDM ’23, February 27-March 3, 2023, Singapore, Singapore Romain Deffayet, Thibaut Thonet, Jean-Michel Renders, & Maarten de Rijke (a) Impact of 𝛽 for 𝜆 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' (b) Impact of 𝜆 for 𝛽 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Figure 3: Average cumulative number of clicks on the validation set obtained by SAC+GeMS with its best validation checkpoint, for different values of 𝛽 and 𝜆 (defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' We also display 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' ommends items with excellent scores (above the threshold and up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='45): as a consequence, STO gets high immediate rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' However, by doing so it incurs a lot of boredom (large orange areas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Overall, it leads to lower expected scores (solid red line) and therefore fewer clicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Conversely, SAC+GeMS𝛾=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='8 sacrifices some immediate reward (yellow distribution shifted to the left) but causes very little boredom (small orange area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Overall, by trading off relevance and diversity, SAC+GeMS𝛾=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='8 yields good immediate rewards while limiting boredom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' It therefore gets higher average scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' SAC+GeMS𝛾=0 exhibits an intermediate behavior due to its limited capabilities: it recommends items of varying relevance, yet leads to substantial boredom (larger orange area than for 𝛾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3 Balancing hyperparameters 𝛽 and 𝜆 (RQ3) In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3, we suggested that the choice of 𝛽 and 𝜆 leads to trade- offs that may impact the downstream performance of SAC+GeMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' As a reminder, 𝛽 adjusts the importance of accurate reconstruction versus smoothness and structure in the latent space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', controlla- bility), while 𝜆 weights the click reconstruction with respect to the slate reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Next, we verify our intuition on the importance of these trade-offs by reporting (in Figure 3) the best validation return obtained for different values of said hyperparameters, on the TopDown environment with focused item embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Figure 3a suggests that, indeed, there exists a “sweet spot” in the selection of 𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' It confirms the intuition described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='3 and the observation of Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [25]: 𝛽 must be appropriately balanced in order to ensure high performance on the downstream RL task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Specifically, we found that choosing 𝛽 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='0 leads to the highest return overall, regardless of whether a latent dimension of 16 or 32 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The impact on the downstream performance of the trade-off between slate and click reconstruction (Figure 3b) is less prominent but can still be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' It justifies our choice to add the click reconstruction term in the loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 1), even though clicks output by GeMS’ decoder are not used during RL training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' This also confirms the importance of introducing and adjusting the hyperparameter 𝜆: modeling clicks jointly with slates improves the final performance of SAC+GeMS, but properly weighting the click reconstruction objective with respect to the slate reconstruction objective is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 7 CONCLUSION We have presented GeMS, a slate representation learning method based on variational auto-encoders for slate recommendation with reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' This method has the notable advantage of being flexible, allowing full-slate modeling and lightweight as- sumptions, in contrast with existing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Findings and broader impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Our experiments across a wide array of environments demonstrate that GeMS compares favor- ably against existing slate representation methods in practical set- tings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Moreover, our empirical analysis highlights that it effectively balances immediate and future rewards, and that the trade-offs imposed by 𝛽 and 𝜆 significantly impact the RL downstream perfor- mance, indicating that properly balancing these hyperparameters is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Our work suggests that generative models are a promising direction for representing rich actions such as slates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Our simulated experiments demonstrate the effec- tiveness of GeMS for representing slates in an RL framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' How- ever, it is well-known that online training of RL agents is too expen- sive and risky, and that in practice agents must be trained offline, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=', directly from logged data [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' We did not address here the specific challenges of offline RL, as we wished to isolate the contribution of the slate representation to downstream performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In future work, we will investigate how generative models can be leveraged in the offline setting, in different scenarios, or with even richer actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' We also plan to look into improvements of the architectures used for structured action representations, for example by using domain knowledge and user models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' ACKNOWLEDGMENTS This research was (partially) funded by the Hybrid Intelligence Center, a 10-year program funded by the Dutch Ministry of Educa- tion, Culture and Science through the Netherlands Organisation for Scientific Research, https://hybrid-intelligence-centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='nl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' All con- tent represents the opinion of the authors, which is not necessarily shared or endorsed by their respective employers and/or sponsors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' REFERENCES [1] Ashton Anderson, Lucas Maystre, Ian Anderson, Rishabh Mehrotra, and Mounia Lalmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Algorithmic Effects on the Diversity of Consumption on Spotify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In WWW ’20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2155–2165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [2] Susan Athey, Raj Chetty, Guido W Imbens, and Hyunseung Kang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The 325 T↑ 300 ks click 275 number of 250 Latent dim Cumulative ( 225 16 32 200 T 175 150 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='0 beta300 1 T clicks T 250 number of Latent dim Cumulative 200 16 32 150 100 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='0 lambdaGenerative Slate Recommendation with Reinforcement Learning WSDM ’23, February 27-March 3, 2023, Singapore, Singapore Surrogate Index: Combining Short-Term Proxies to Estimate Long-Term Treatment Effects More Rapidly and Precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Technical Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' National Bureau of Economic Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [3] Xueying Bai, Jian Guan, and Hongning Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' A Model-Based Rein- forcement Learning with Adversarial Training for Online Recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In NeurIPS ’19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 10734–10745.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [4] Eytan Bakshy, Solomon Messing, and Lada Adamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Exposure to Ideo- logically Diverse News and Opinion on Facebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Science 348, 6239 (2015), 1130–1132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [5] Nicolò Botteghi, Mannes Poel, Beril Sirmaçek, and Christoph Brune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Low- Dimensional State and Action Representation Learning with MDP Homomor- phism Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='01677 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [6] Yash Chandak, Georgios Theocharous, James Kostas, Scott Jordan, and Philip Thomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Learning Action Representations for Reinforcement Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In ICML ’19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 941–950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [7] Praveen Chandar, Brian St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Thomas, Lucas Maystre, Vijay Pappu, Roberto Sanchis- Ojeda, Tiffany Wu, Ben Carterette, Mounia Lalmas, and Tony Jebara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Using Survival Models to Estimate User Engagement in Online Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In WWW ’22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 3186–3195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [8] Minmin Chen, Alex Beutel, Paul Covington, Sagar Jain, Francois Belletti, and Ed H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Chi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Top-K Off-Policy Correction for a REINFORCE Recommender System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In WSDM ’19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 456–464.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [9] Xinshi Chen, Shuang Li, Hui Li, Shaohua Jiang, Yuan Qi, and Le Song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Generative Adversarial User Model for Reinforcement Learning Based Recom- mendation System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In ICML ’19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 1052–1061.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [10] Kyunghyun Cho, Bart van Merrienboer, Çaglar Gülçehre, Dzmitry Bahdanau, Fethi Bougares, Holger Schwenk, and Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In EMNLP ’14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 1724–1734.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [11] Van Dang, Michael Bendersky, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Bruce Croft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Two-Stage Learning to Rank for Information Retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In ECIR ’13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 423–434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [12] Gabriel Dulac-Arnold, Richard Evans, Hado van Hasselt, Peter Sunehag, Timothy Lillicrap, Jonathan Hunt, Timothy Mann, Theophane Weber, Thomas Degris, and Ben Coppin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Deep Reinforcement Learning in Large Discrete Action Spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' arXiv:1512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='07679 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [13] Seth R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Flaxman, Sharad Goel, and Justin M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Rao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Filter Bubbles, Echo Chambers, and Online News Consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Public Opinion Quarterly 80, S1 (2016), 298–320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [14] David Ha and Jürgen Schmidhuber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Recurrent World Models Facilitate Policy Evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In NeurIPS ’18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2455–2467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [15] Tuomas Haarnoja, Aurick Zhou, Pieter Abbeel, and Sergey Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In ICML ’18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 1856–1865.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [16] Christian Hansen, Rishabh Mehrotra, Casper Hansen, Brian Brost, Lucas Maystre, and Mounia Lalmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Shifting Consumption towards Diverse Content on Music Streaming Platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In WSDM ’21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 238–246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [17] Henning Hohnhold, Deirdre O’Brien, and Diane Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Focusing on the Long-Term: It’s Good for Users and Business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In KDD ’15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 1849–1858.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [18] Eugene Ie, Vihan Jain, Jing Wang, Sanmit Narvekar, Ritesh Agarwal, Rui Wu, Heng-Tze Cheng, Tushar Chandra, and Craig Boutilier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' SlateQ: A Tractable Decomposition for Reinforcement Learning with Recommendation Sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In IJ- CAI ’19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2592–2599.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [19] Dietmar Jannach, Pearl Pu, Francesco Ricci, and Markus Zanker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Recom- mender Systems: Past, Present, Future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' AI Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 42, 3 (2021), 3–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [20] Ray Jiang, Sven Gowal, Yuqiu Qian, Timothy A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Mann, and Danilo J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Rezende.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Beyond Greedy Ranking: Slate Optimization via List-CVAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In ICLR ’19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [21] Leslie Pack Kaelbling, Michael L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Littman, and Anthony R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Cassandra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Planning and Acting in Partially Observable Stochastic Domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Artificial Intelligence 101, 1 (1998), 99–134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [22] Diederik Kingma and Max Welling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Auto-Encoding Variational Bayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In ICLR ’14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [23] Yehuda Koren, Robert M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Bell, and Chris Volinsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Matrix Factorization Techniques for Recommender Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Computer 42, 8 (2009), 30–37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [24] Solomon Kullback and Richard A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Leibler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 1951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' On Information and Sufficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The Annals of Mathematical Statistics 22, 1 (1951), 79–86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [25] Shuchang Liu, Fei Sun, Yingqiang Ge, Changhua Pei, and Yongfeng Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Variation Control and Evaluation for Generative Slate Recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In WWW ’21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 436–448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [26] Farzan Masrour, Tyler Wilson, Heng Yan, Pang-Ning Tan, and Abdol-Hossein Esfahanian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Bursting the Filter Bubble: Fairness-Aware Network Link Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In AAAI ’20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 841–848.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [27] James McInerney, Brian Brost, Praveen Chandar, Rishabh Mehrotra, and Benjamin Carterette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Counterfactual Evaluation of Slate Recommendations with Sequential Reward Interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In KDD ’20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 1779–1788.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [28] Eli Pariser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The Filter Bubble: What the Internet Is Hiding from You.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The Penguin Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [29] Wilbert Samuel Rossi, Jan Willem Polderman, and Paolo Frasca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' The Closed Loop between Opinion Formation and Personalised Recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' IEEE Transactions on Control of Network Systems (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [30] Adam Stooke, Kimin Lee, Pieter Abbeel, and Michael Laskin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Decoupling Representation Learning from Reinforcement Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In ICML ’21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 9870–9879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [31] Peter Sunehag, Richard Evans, Gabriel Dulac-Arnold, Yori Zwols, Daniel Visentin, and Ben Coppin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Deep Reinforcement Learning with Attention for Slate Markov Decision Processes with High-Dimensional States and Actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' arXiv:1512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content='01124 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [32] Richard Sutton and Andrew Barto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Reinforcement Learning: An Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' MIT Press, 326–329.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [33] Isaac Waller and Ashton Anderson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Generalists and Specialists: Using Community Embeddings to Quantify Activity Diversity in Online Platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In WWW ’19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 1954–1964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [34] Christopher Watkins and Peter Dayan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Q-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Machine Learning 8 (1992), 279–292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [35] Wenxuan Zhou, Sujay Bajracharya, and David Held.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' PLAS: Latent Action Space for Offline Reinforcement Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In CoRL ’20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 1719–1735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' [36] Lixin Zou, Long Xia, Zhuoye Ding, Jiaxing Song, Weidong Liu, and Dawei Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' Reinforcement Learning to Optimize Long-Term User Engagement in Recommender Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' In KDD ’19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} +page_content=' 2810–2818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NFAT4oBgHgl3EQfoB2V/content/2301.08632v1.pdf'} diff --git a/7tE3T4oBgHgl3EQfRgnj/content/tmp_files/2301.04423v1.pdf.txt b/7tE3T4oBgHgl3EQfRgnj/content/tmp_files/2301.04423v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..825fccdfe57a65e34146376eac16d1f1e46b9d74 --- /dev/null +++ b/7tE3T4oBgHgl3EQfRgnj/content/tmp_files/2301.04423v1.pdf.txt @@ -0,0 +1,417 @@ +Multi-Scanner Canine Cutaneous Squamous Cell +Carcinoma Histopathology Dataset +Frauke Wilm1,2, Marco Fragoso3, Christof A. Bertram4, Nikolas Stathonikos5, +Mathias Öttl1, Jingna Qiu2, Robert Klopfleisch3, Andreas Maier1, +Katharina Breininger2,†, Marc Aubreville6,† +1Pattern Recognition Lab, Friedrich-Alexander-Universität Erlangen-Nünberg, Germany +2Department AIBE, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany +3Institute of Veterinary Pathology, Freie Universität Berlin, Germany +4Institute of Pathology, University of Veterinary Medicine, Vienna, Austria +5Pathology Department, University Medical Centre Utrecht, The Netherlands +6Technische Hochschule Ingolstadt, Ingolstadt, Germany +†shared senior authors +frauke.wilm@fau.de +Abstract. In histopathology, scanner-induced domain shifts are known to impede +the performance of trained neural networks when tested on unseen data. Multi- +domain pre-training or dedicated domain-generalization techniques can help to +develop domain-agnostic algorithms. For this, multi-scanner datasets with a high +variety of slide scanning systems are highly desirable. We present a publicly +available multi-scanner dataset of canine cutaneous squamous cell carcinoma +histopathology images, composed of 44 samples digitized with five slide scanners. +This dataset provides local correspondences between images and thereby isolates +the scanner-induced domain shift from other inherent, e.g. morphology-induced +domain shifts. To highlight scanner differences, we present a detailed evaluation +of color distributions, sharpness, and contrast of the individual scanner subsets. +Additionally, to quantify the inherent scanner-induced domain shift, we train a +tumor segmentation network on each scanner subset and evaluate the performance +both in- and cross-domain. We achieve a class-averaged in-domain intersection +over union coefficient of up to 0.86 and observe a cross-domain performance +decrease of up to 0.38, which confirms the inherent domain shift of the presented +dataset and its negative impact on the performance of deep neural networks. +1 +Introduction +The digitization of histological specimens with dedicated slide scanning systems has +facilitated machine learning-based image analysis for histopathology. These algorithms +have since assisted pathologists in a variety of routine tasks, e.g. mitotic figure detection +[1], for which they even have been able to outperform trained experts in controlled +settings [1, 2]. Still, their performance is highly dependent on the quality and availabil- +ity of training data [3] and can deteriorate considerably on a test set where the image +characteristics differ from the training data [4]. Such differences commonly referred to +as “domain shift” can originate not only from different staining and tissue preparation +protocols of different pathology laboratories but also from the digitization of histolog- +1 +arXiv:2301.04423v1 [eess.IV] 11 Jan 2023 + +2 +Wilm et al. +ical specimens with different scanning systems. Especially from a clinical perspective, +domain-agnostic models are important for generating accurate and reliable predictions. +Previous work has shown that domain generalization techniques, e.g. domain- +adversarial training, can help to develop domain-agnostic models [5]. For this, a training +dataset composed of a wide range of different domains is highly desirable. So far, the +most extensive publicly available multi-scanner histopathology dataset is the training +set of the MICCAI MItosis DOmain Generalization (MIDOG) 2021 challenge [2]. The +dataset consists of 2 mm2-sized cropped regions of 200 breast cancer cases digitized +with four scanners. However, the cases were divided between the scanners, and perfor- +mance differences can therefore not solely be attributed to the slide scanner but also +to the case selection. The Mitos & Atypia dataset [6] is the only public multi-scanner +histopathology dataset with local image correspondences, i.e. the same case was digi- +tized with multiple slide scanners, however, with 16 cases and two scanners, its extent +is limited and it does not leave room for experiments with hold-out test scanners. +In this work, we present a canine cutaneous histopathology dataset, where each of +the 44 samples was digitized with five different slide scanning systems. This multi- +scanner dataset provides local image correspondences, useful for domain generalization +experiments. Accompanied by an annotation database of 1,243 polygon annotations +for seven histologic classes (tumor, epidermis, dermis, subcutis, bone, cartilage, and +a combined class of inflammation and necrosis), this is the first publicly available +multi-scanner segmentation dataset. For each scanner subset, we provide a detailed +evaluation of color distributions, sharpness, and contrast. To quantify the extent of +the scanner-induced domain shift, we performed a technical validation of the dataset +by training a baseline tumor segmentation algorithm on each single scanner domain +and then testing the algorithm across all scanners. For some scanners, we observed +a considerable performance decrease, which highlights the domain shift inherent in +the dataset. The whole slide images (WSIs) and annotation databases are publicly +available on Zenodo (https://doi.org/10.5281/zenodo.7418555), and code for +implementing the baseline architectures can be obtained from our GitHub repository +(https://github.com/DeepPathology/MultiScanner_SCC). +2 +Materials and methods +The dataset presented in this work extends the publicly available CATCH dataset [7], a +collection of 350 WSIs of seven of the most common canine cutaneous tumor subtypes +(50 WSIs per subtype). For the CATCH dataset, the specimens were digitized with the +Aperio ScanScope CS2 (Leica, Germany) at a resolution of 0.25 µm/pixel using a 40 × +objective lens. Use of these samples was approved by the local governmental authorities +(State Office of Health and Social Affairs of Berlin, approval ID: StN 011/20). For the +multi-scanner dataset, we randomly selected one subtype (squamous cell carcinoma) +and digitized the samples with four additional slide scanners (see Figure 1): +• NanoZoomer S210 (Hamamatsu, Japan), 0.22 µm/pixel +• NanoZoomer 2.0-HT (Hamamatsu, Japan), 0.23 µm/pixel +• Pannoramic 1000 (3DHISTECH, Hungary), 0.25 µm/pixel +• Aperio GT 450 (Leica, Germany), 0.26 µm/pixel + +Multi-Scanner Histopathology Dataset +3 +(a) CS2 +(b) NZ210 +(c) NZ2.0 +(d) P1000 +(e) GT450 +Fig. 1. Exemplary region of interest of the multi-scanner dataset. +Due to severe scanning artifacts in at least one of the scans, six specimens were excluded +from the dataset, resulting in a total of 220 WSIs (44 samples digitized with five +scanners each). The CATCH annotation database provides annotations for the individual +tumor subtypes and six additional skin tissue classes (epidermis, dermis, subcutis, +bone, cartilage, and a combined class of inflammation and necrosis). We transferred all +annotations to the other scanners using the WSI registration algorithm by Marzahl et +al. [8] and visually validated them by overlaying the transformed polygon annotations +onto the scans. We provide public access to the WSIs on Zenodo (https://doi. +org/10.5281/zenodo.7418555), licensed under a Creative Commons Attribution +4.0 International License. However, due to storage restrictions, we have converted them +to lower-resolution pyramidal TIFFs (4 µm/pixel), which has shown to be adequate for +training segmentation tasks on the CATCH dataset [7]. +2.1 +Dataset validation +For each scanner subset, we evaluated the average RGB color distribution, sharpness, +and contrast. For sharpness estimation, we used the cumulative probability of blur +detection (CPBD) metric [9], which is a perceptual-based image sharpness metric. It is +computed via edge detection, followed by a blur estimation at the detected edges. The +CPBD metric then corresponds to the cumulative probability of blur detection, i.e. the +percentage of image edges that fall below a threshold of a perceptually noticeable blur. +For implementation details, we refer to [9]. For the analysis of RGB distributions and +contrast, we used Otsu’s adaptive thresholding to separate foreground tissue from white +background. For each image, we calculated the average intensities of the color channels +𝐼𝑅, 𝐼𝐺, and 𝐼𝐵 in the detected tissue regions. Afterward, we converted the regions to +grayscale and computed the Michelson contrast [10] 𝐶𝑀 as a measure of global contrast. +2.2 +Technical validation +For technical validation of the dataset, we trained a segmentation model on each scan- +ner domain and tested the algorithm across all scanners. For model development, we +performed a slide-level split into training (N=30), validation (N=5), and test (N=9) +cases. We trained a UNet with a ResNet18 encoder pre-trained on ImageNet for the +segmentation into tumor, non-tumor, and background. For this, we combined all skin +tissue classes into one non-tumor class and used the automatically detected background + +4 +Wilm et al. +areas to train the background class. We trained the networks on image patches sized +512 × 512 pixels, extracted at a resolution of 4 µm/pixel. During each epoch, we sampled +50 patches per WSI within the annotated polygons. Due to a high class imbalance, we +randomly sampled the polygons with a class-weighting of 10 % background and 45 % +each of tumor and non-tumor regions. For each scanner, we applied z-score normaliza- +tion with the training set statistics (mean and standard deviation) and performed data +augmentation using random flipping, affine transformations, and random lightning and +contrast change. We used the Adam optimizer and trained the networks with a com- +bination of cross-entropy and Dice loss. We trained the models with a batch size of +8 and a cyclic learning rate of 10−4 for 100 epochs, after which we observed model +convergence. Model selection was guided by the highest intersection over union (mIoU) +on the validation set. +3 +Results +Figure 2 shows the RGB distribution of the non-background areas for the complete +dataset of 44 WSIs per scanner. The distributions match the exemplary patches in +Figure 1, where the patches of the Aperio CS2 and the NanoZoomer 210 appear redder, +which is reflected in a shift of the red pixel distributions to higher values. When looking +at the distributions of the Aperio GT450, all curves are densely located at the higher color +component values, which corresponds to the bright appearance of the patch in Figure 1d. +Table 1 summarizes the channel-wise color averages, sharpness, and contrast of the slide +scanning systems. These results further underline the visual impression of the patches +in Figure 1. When calculating the ratio of the red and the blue color channel 𝐼𝑅/𝐼𝐵, the +NZ210 results in a ratio of 1.12 and the NZ2.0 in a ratio of 1.04, which matches the much +redder appearance of the NZ210 patch and the bluer appearance of the NZ2.0 patch. +Overall, the CS2, NZ210, NZ2.0, and P1000 show comparable sharpness and contrast +values, while the Aperio 450 exhibits a slightly higher sharpness but a considerably lower +contrast. Figure 3 visualizes the mIoU when training the segmentation network on one +scanner, and testing it on all scanners. The results show high in-domain performance +(diagonal) with mIoU values between 0.82 for the P1000 and GT450, and 0.86 for +the NZ210. The cross-domain performance highlights the scanner-induced domain shift +inherent in our dataset. While the networks trained on the CS2 and the NZ210 generalize +considerably well, with performance decreases of up to 0.08 and 0.12 compared to the in- +domain mIoU, the highest cross-domain performance drop was observed when training +0 +50 +100 +150 +200 +250 +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +0.035 +0.040 +Density +(a) CS2 +0 +50 +100 +150 +200 +250 +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +0.035 +0.040 +Density +(b) NZ210 +0 +50 +100 +150 +200 +250 +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +0.035 +0.040 +Density +(c) NZ2.0 +0 +50 +100 +150 +200 +250 +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +0.035 +0.040 +Density +(d) P1000 +0 +50 +100 +150 +200 +250 +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +0.035 +0.040 +Density +(e) GT450 +Fig. 2. Kernel density estimation of RGB values per scanner. + +Multi-Scanner Histopathology Dataset +5 +Tab. 1. Channel-wise color distributions 𝐼𝑅, 𝐼𝐺, and 𝐼𝐵, sharpness 𝑆𝐶𝑃𝐵𝐷 calculated as cumu- +lative probability of blur detection, and Michelson contrast 𝐶𝑀 of the scanners (𝜇 ± 𝜎). +. +𝐼𝑅 +𝐼𝐺 +𝐼𝐵 +𝑆𝐶𝑃𝐵𝐷 +𝐶𝑀 +CS2 +201.84 ± 19.46 +153.18 ± 35.41 +171.54 ± 30.02 +0.80 ± 0.02 +0.74 ± 0.12 +NZ210 +218.88 ± 17.96 +172.64 ± 28.04 +195.26 ± 20.15 +0.82 ± 0.03 +0.81 ± 0.14 +NZ2.0 +192.49 ± 21.63 +153.46 ± 36.72 +184.51 ± 23.90 +0.81 ± 0.02 +0.81 ± 0.13 +P1000 +223.41 ± 18.60 +164.97 ± 41.15 +211.44 ± 21.64 +0.80 ± 0.02 +0.71 ± 0.14 +GT450 +226.59 ± 12.99 +208.18 ± 20.88 +218.80 ± 15.92 +0.84 ± 0.04 +0.53 ± 0.15 +on the P1000, with a decrease of up to 0.38. A visual inspection of segmentation results +showed that the network trained on the P1000 misclassified many background areas +of the other scanners. A reason might be the integrated tissue detection of the P1000, +which sets all pixels outside the tissue bounding box to (255, 255, 255) in order to +reduce scanning times. This artificially removes common artifacts, e.g. dust particles, +and the network might only look for high pixel values and not learn the morphological +characteristics of background areas. +CS2 +NZ210 +NZ2.0 +P1000 +GT450 +test +CS2 +NZ210 +NZ2.0 +P1000 +GT450 +train +0.83 +0.83 +0.82 +0.75 +0.8 +0.79 +0.86 +0.84 +0.74 +0.81 +0.71 +0.7 +0.84 +0.82 +0.79 +0.6 +0.44 +0.65 +0.82 +0.55 +0.81 +0.71 +0.81 +0.7 +0.82 +0.45 +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +Fig. 3. Scanner-wise performance of segmentation net- +works. Matrix entry 𝑚𝑖, 𝑗 is the mean intersection over +union (mIoU) when training on the scanning system +in row 𝑖 and testing on the scanning system in column +𝑗. Diagonal elements indicate in-domain performance, +whereas off-diagonal elements represent cross-domain +performance. +4 +Discussion +Our experiments have demonstrated the negative impact of scanner-induced domain +shifts on the performance of deep neural networks, indicated by a considerable de- +crease in mIoU on unseen scanners. This confirms the observations of previous works +and supports the need for methods that can tackle this domain shift and adequate +datasets to evaluate their generalization capability. The presented dataset exceeds exist- +ing multi-scanner datasets in terms of sample size and scanning systems. Furthermore, +it provides local image correspondences, which isolate the scanner-induced from the +morphology-induced domain shift and allow the development of algorithms dependent +on these correspondences, e.g. WSI registration algorithms. We have implicitly shown +the eligibility of our dataset for this application by successfully transferring our anno- +tation database from the CS2 scanner to the remaining scanner using WSI registration. +The detailed evaluation of our scanner subsets has highlighted considerable differences + +6 +Wilm et al. +regarding color distributions and contrasts present in clinically used scanners. Surpris- +ingly, even though our evaluations resulted in the lowest contrast value for the Aperio +GT450, this did not impede segmentation performance, shown by an in-domain mIoU +of 0.82, which is comparable to the in-domain mIoUs of the remaining scanners. In our +technical validation, we observed a large cross-domain performance decrease, especially +when training on the P1000 scanner. We assume that this can mainly be attributed to the +unique pre-processing steps of the scanner vendor, as the P1000 showed similar image +statistics to the CS2 but their average cross-domain performance differed considerably. +However, we also observed a cross-domain performance decrease for the remaining +scanners, which indicates that some of the learned feature representations did not gen- +eralize well across scanners. Future work could focus on a closer evaluation of which +scanner characteristics hinder the extraction of domain-agnostic features and should +therefore be disregarded, e.g. by using specific filters for data pre-processing or using +adversarial training to punish the extraction of these features. +Acknowledgement. F.W. gratefully acknowledges the financial support received by +Merck Healthcare KGaA and the technical support received by the Clinical Assay +Strategy 1 group at Merck Healthcare KGaA during sample digitization. K.B. gratefully +acknowledges support by d.hip campus - Bavarian aim in form of a faculty endowment. +References +1. Aubreville M, Bertram CA, Marzahl C, Gurtner C, Dettwiler M, Schmidt A et al. Deep +learning algorithms out-perform veterinary pathologists in detecting the mitotically most +active tumor region. Sci Rep 10:16447 (2020), pp. 1–11. +2. Aubreville M, Stathonikos N, Bertram CA, Klopleisch R, Hoeve N ter, Ciompi F et al. Mitosis +domain generalization in histopathology images–The MIDOG challenge. Med Image Anal +84:102699 (2023). +3. Deng S, Zhang X, Yan W, Chang EI, Fan Y, Lai M et al. Deep learning in digital pathology +image analysis: a survey. Front Med 14.4 (2020), pp. 470–487. +4. Stacke K, Eilertsen G, Unger J, Lundström C. Measuring domain shift for deep learning in +histopathology. IEEE J Biomed Health Inform 25.2 (2020), pp. 325–336. +5. Wilm F, Marzahl C, Breininger K, Aubreville M. Domain adversarial RetinaNet as a refer- +ence algorithm for the MItosis DOmain Generalization challenge. Biomedical Image Regis- +tration, Domain Generalisation and Out-of-Distribution Analysis: MICCAI 2021 Challenges. +Springer. 2022, pp. 5–13. +6. Roux L, Racoceanu D, Capron F, Calvo J, Attieh E, Le Naour G et al. Mitos & Atypia. Image +Pervasive Access Lab (IPAL), Agency Sci., Technol. & Res. Inst. Infocom Res., Singapore, +Tech. Rep 1 (2014), pp. 1–8. +7. Wilm F et al. CAnine CuTaneous Cancer Histology dataset (version 1). The Cancer Imaging +Archive (2022). https://doi.org/10.7937/TCIA.2M93-FX66. +8. Marzahl C, Wilm F, F. DF, Tharun L, Perner S, Bertram CA et al. Robust quad-tree based +registration on whole slide images. Comput Pathol (2021). PMLR, 2021, pp. 181–190. +9. Narvekar ND, Karam LJ. A no-reference image blur metric based on the cumulative proba- +bility of blur detection (CPBD). IEEE Trans Image Process 20.9 (2011), pp. 2678–2683. +10. Michelson AA. Studies in optics. Courier Corporation, 1995. + diff --git a/7tE3T4oBgHgl3EQfRgnj/content/tmp_files/load_file.txt b/7tE3T4oBgHgl3EQfRgnj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c1e5948b2932479fae66a20108adf2e9149a0f5 --- /dev/null +++ b/7tE3T4oBgHgl3EQfRgnj/content/tmp_files/load_file.txt @@ -0,0 +1,377 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf,len=376 +page_content='Multi-Scanner Canine Cutaneous Squamous Cell Carcinoma Histopathology Dataset Frauke Wilm1,2, Marco Fragoso3, Christof A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Bertram4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Nikolas Stathonikos5,' metadata={'source': 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+page_content=' Marc Aubreville6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='† 1Pattern Recognition Lab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Friedrich-Alexander-Universität Erlangen-Nünberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Germany 2Department AIBE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Friedrich-Alexander-Universität Erlangen-Nürnberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Germany 3Institute of Veterinary Pathology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Freie Universität Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Germany 4Institute of Pathology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' University of Veterinary Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Austria 5Pathology Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' University Medical Centre Utrecht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' The Netherlands 6Technische Hochschule Ingolstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Ingolstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Germany †shared senior authors frauke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='wilm@fau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='de Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' In histopathology, scanner-induced domain shifts are known to impede the performance of trained neural networks when tested on unseen data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Multi- domain pre-training or dedicated domain-generalization techniques can help to develop domain-agnostic algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' For this, multi-scanner datasets with a high variety of slide scanning systems are highly desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' We present a publicly available multi-scanner dataset of canine cutaneous squamous cell carcinoma histopathology images, composed of 44 samples digitized with five slide scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' This dataset provides local correspondences between images and thereby isolates the scanner-induced domain shift from other inherent, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' morphology-induced domain shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' To highlight scanner differences, we present a detailed evaluation of color distributions, sharpness, and contrast of the individual scanner subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Additionally, to quantify the inherent scanner-induced domain shift, we train a tumor segmentation network on each scanner subset and evaluate the performance both in- and cross-domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' We achieve a class-averaged in-domain intersection over union coefficient of up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='86 and observe a cross-domain performance decrease of up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='38, which confirms the inherent domain shift of the presented dataset and its negative impact on the performance of deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 1 Introduction The digitization of histological specimens with dedicated slide scanning systems has facilitated machine learning-based image analysis for histopathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' These algorithms have since assisted pathologists in a variety of routine tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' mitotic figure detection [1], for which they even have been able to outperform trained experts in controlled settings [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Still, their performance is highly dependent on the quality and availabil- ity of training data [3] and can deteriorate considerably on a test set where the image characteristics differ from the training data [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Such differences commonly referred to as “domain shift” can originate not only from different staining and tissue preparation protocols of different pathology laboratories but also from the digitization of histolog- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='04423v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='IV] 11 Jan 2023 2 Wilm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' ical specimens with different scanning systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Especially from a clinical perspective, domain-agnostic models are important for generating accurate and reliable predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Previous work has shown that domain generalization techniques, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' domain- adversarial training, can help to develop domain-agnostic models [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' For this, a training dataset composed of a wide range of different domains is highly desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' So far, the most extensive publicly available multi-scanner histopathology dataset is the training set of the MICCAI MItosis DOmain Generalization (MIDOG) 2021 challenge [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' The dataset consists of 2 mm2-sized cropped regions of 200 breast cancer cases digitized with four scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' However, the cases were divided between the scanners, and perfor- mance differences can therefore not solely be attributed to the slide scanner but also to the case selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' The Mitos & Atypia dataset [6] is the only public multi-scanner histopathology dataset with local image correspondences, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' the same case was digi- tized with multiple slide scanners, however, with 16 cases and two scanners, its extent is limited and it does not leave room for experiments with hold-out test scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' In this work, we present a canine cutaneous histopathology dataset, where each of the 44 samples was digitized with five different slide scanning systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' This multi- scanner dataset provides local image correspondences, useful for domain generalization experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Accompanied by an annotation database of 1,243 polygon annotations for seven histologic classes (tumor, epidermis, dermis, subcutis, bone, cartilage, and a combined class of inflammation and necrosis), this is the first publicly available multi-scanner segmentation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' For each scanner subset, we provide a detailed evaluation of color distributions, sharpness, and contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' To quantify the extent of the scanner-induced domain shift, we performed a technical validation of the dataset by training a baseline tumor segmentation algorithm on each single scanner domain and then testing the algorithm across all scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' For some scanners, we observed a considerable performance decrease, which highlights the domain shift inherent in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' The whole slide images (WSIs) and annotation databases are publicly available on Zenodo (https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='7418555), and code for implementing the baseline architectures can be obtained from our GitHub repository (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='com/DeepPathology/MultiScanner_SCC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 2 Materials and methods The dataset presented in this work extends the publicly available CATCH dataset [7], a collection of 350 WSIs of seven of the most common canine cutaneous tumor subtypes (50 WSIs per subtype).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' For the CATCH dataset, the specimens were digitized with the Aperio ScanScope CS2 (Leica, Germany) at a resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='25 µm/pixel using a 40 × objective lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Use of these samples was approved by the local governmental authorities (State Office of Health and Social Affairs of Berlin, approval ID: StN 011/20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' For the multi-scanner dataset, we randomly selected one subtype (squamous cell carcinoma) and digitized the samples with four additional slide scanners (see Figure 1): NanoZoomer S210 (Hamamatsu, Japan), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='22 µm/pixel NanoZoomer 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='0-HT (Hamamatsu, Japan), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='23 µm/pixel Pannoramic 1000 (3DHISTECH, Hungary), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='25 µm/pixel Aperio GT 450 (Leica, Germany), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='26 µm/pixel Multi-Scanner Histopathology Dataset 3 (a) CS2 (b) NZ210 (c) NZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='0 (d) P1000 (e) GT450 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Exemplary region of interest of the multi-scanner dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Due to severe scanning artifacts in at least one of the scans, six specimens were excluded from the dataset, resulting in a total of 220 WSIs (44 samples digitized with five scanners each).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' The CATCH annotation database provides annotations for the individual tumor subtypes and six additional skin tissue classes (epidermis, dermis, subcutis, bone, cartilage, and a combined class of inflammation and necrosis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' We transferred all annotations to the other scanners using the WSI registration algorithm by Marzahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' [8] and visually validated them by overlaying the transformed polygon annotations onto the scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' We provide public access to the WSIs on Zenodo (https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='7418555), licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='0 International License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' However, due to storage restrictions, we have converted them to lower-resolution pyramidal TIFFs (4 µm/pixel), which has shown to be adequate for training segmentation tasks on the CATCH dataset [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='1 Dataset validation For each scanner subset, we evaluated the average RGB color distribution, sharpness, and contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' For sharpness estimation, we used the cumulative probability of blur detection (CPBD) metric [9], which is a perceptual-based image sharpness metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' It is computed via edge detection, followed by a blur estimation at the detected edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' The CPBD metric then corresponds to the cumulative probability of blur detection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' the percentage of image edges that fall below a threshold of a perceptually noticeable blur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' For implementation details, we refer to [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' For the analysis of RGB distributions and contrast, we used Otsu’s adaptive thresholding to separate foreground tissue from white background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' For each image, we calculated the average intensities of the color channels 𝐼𝑅, 𝐼𝐺, and 𝐼𝐵 in the detected tissue regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Afterward, we converted the regions to grayscale and computed the Michelson contrast [10] 𝐶𝑀 as a measure of global contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='2 Technical validation For technical validation of the dataset, we trained a segmentation model on each scan- ner domain and tested the algorithm across all scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' For model development, we performed a slide-level split into training (N=30), validation (N=5), and test (N=9) cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' We trained a UNet with a ResNet18 encoder pre-trained on ImageNet for the segmentation into tumor, non-tumor, and background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' For this, we combined all skin tissue classes into one non-tumor class and used the automatically detected background 4 Wilm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' areas to train the background class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' We trained the networks on image patches sized 512 × 512 pixels, extracted at a resolution of 4 µm/pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' During each epoch, we sampled 50 patches per WSI within the annotated polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Due to a high class imbalance, we randomly sampled the polygons with a class-weighting of 10 % background and 45 % each of tumor and non-tumor regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' For each scanner, we applied z-score normaliza- tion with the training set statistics (mean and standard deviation) and performed data augmentation using random flipping, affine transformations, and random lightning and contrast change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' We used the Adam optimizer and trained the networks with a com- bination of cross-entropy and Dice loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' We trained the models with a batch size of 8 and a cyclic learning rate of 10−4 for 100 epochs, after which we observed model convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Model selection was guided by the highest intersection over union (mIoU) on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 3 Results Figure 2 shows the RGB distribution of the non-background areas for the complete dataset of 44 WSIs per scanner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' The distributions match the exemplary patches in Figure 1, where the patches of the Aperio CS2 and the NanoZoomer 210 appear redder, which is reflected in a shift of the red pixel distributions to higher values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' When looking at the distributions of the Aperio GT450, all curves are densely located at the higher color component values, which corresponds to the bright appearance of the patch in Figure 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Table 1 summarizes the channel-wise color averages, sharpness, and contrast of the slide scanning systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' These results further underline the visual impression of the patches in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' When calculating the ratio of the red and the blue color channel 𝐼𝑅/𝐼𝐵, the NZ210 results in a ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='12 and the NZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='0 in a ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='04, which matches the much redder appearance of the NZ210 patch and the bluer appearance of the NZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='0 patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Overall, the CS2, NZ210, NZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='0, and P1000 show comparable sharpness and contrast values, while the Aperio 450 exhibits a slightly higher sharpness but a considerably lower contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Figure 3 visualizes the mIoU when training the segmentation network on one scanner, and testing it on all scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' The results show high in-domain performance (diagonal) with mIoU values between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='82 for the P1000 and GT450, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='86 for the NZ210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' The cross-domain performance highlights the scanner-induced domain shift inherent in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' While the networks trained on the CS2 and the NZ210 generalize considerably well, with performance decreases of up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='08 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='12 compared to the in- domain mIoU, the highest cross-domain performance drop was observed when training 0 50 100 150 200 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='040 Density (a) CS2 0 50 100 150 200 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='040 Density (b) NZ210 0 50 100 150 200 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='040 Density (c) NZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='0 0 50 100 150 200 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='040 Density (d) P1000 0 50 100 150 200 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='040 Density (e) GT450 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Kernel density estimation of RGB values per scanner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Multi-Scanner Histopathology Dataset 5 Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Channel-wise color distributions 𝐼𝑅, 𝐼𝐺, and 𝐼𝐵, sharpness 𝑆𝐶𝑃𝐵𝐷 calculated as cumu- lative probability of blur detection, and Michelson contrast 𝐶𝑀 of the scanners (𝜇 ± 𝜎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 𝐼𝑅 𝐼𝐺 𝐼𝐵 𝑆𝐶𝑃𝐵𝐷 𝐶𝑀 CS2 201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='84 ± 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='46 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='18 ± 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='41 171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='54 ± 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='12 NZ210 218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='88 ± 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='96 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='64 ± 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='04 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='26 ± 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='14 NZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='0 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='49 ± 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='63 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='46 ± 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='72 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='51 ± 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='13 P1000 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='41 ± 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='60 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='97 ± 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='15 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='44 ± 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='14 GT450 226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='59 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='99 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='18 ± 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='88 218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='80 ± 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='15 on the P1000, with a decrease of up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' A visual inspection of segmentation results showed that the network trained on the P1000 misclassified many background areas of the other scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' A reason might be the integrated tissue detection of the P1000, which sets all pixels outside the tissue bounding box to (255, 255, 255) in order to reduce scanning times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' This artificially removes common artifacts, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' dust particles, and the network might only look for high pixel values and not learn the morphological characteristics of background areas.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='85 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Scanner-wise performance of segmentation net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Matrix entry 𝑚𝑖, 𝑗 is the mean intersection over union (mIoU) when training on the scanning system in row 𝑖 and testing on the scanning system in column 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Diagonal elements indicate in-domain performance, whereas off-diagonal elements represent cross-domain performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 4 Discussion Our experiments have demonstrated the negative impact of scanner-induced domain shifts on the performance of deep neural networks, indicated by a considerable de- crease in mIoU on unseen scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' This confirms the observations of previous works and supports the need for methods that can tackle this domain shift and adequate datasets to evaluate their generalization capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' The presented dataset exceeds exist- ing multi-scanner datasets in terms of sample size and scanning systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Furthermore, it provides local image correspondences, which isolate the scanner-induced from the morphology-induced domain shift and allow the development of algorithms dependent on these correspondences, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' WSI registration algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' We have implicitly shown the eligibility of our dataset for this application by successfully transferring our anno- tation database from the CS2 scanner to the remaining scanner using WSI registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' The detailed evaluation of our scanner subsets has highlighted considerable differences 6 Wilm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' regarding color distributions and contrasts present in clinically used scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Surpris- ingly, even though our evaluations resulted in the lowest contrast value for the Aperio GT450, this did not impede segmentation performance, shown by an in-domain mIoU of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='82, which is comparable to the in-domain mIoUs of the remaining scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' In our technical validation, we observed a large cross-domain performance decrease, especially when training on the P1000 scanner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' We assume that this can mainly be attributed to the unique pre-processing steps of the scanner vendor, as the P1000 showed similar image statistics to the CS2 but their average cross-domain performance differed considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' However, we also observed a cross-domain performance decrease for the remaining scanners, which indicates that some of the learned feature representations did not gen- eralize well across scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Future work could focus on a closer evaluation of which scanner characteristics hinder the extraction of domain-agnostic features and should therefore be disregarded, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' by using specific filters for data pre-processing or using adversarial training to punish the extraction of these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' gratefully acknowledges the financial support received by Merck Healthcare KGaA and the technical support received by the Clinical Assay Strategy 1 group at Merck Healthcare KGaA during sample digitization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' gratefully acknowledges support by d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='hip campus - Bavarian aim in form of a faculty endowment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Aubreville M, Bertram CA, Marzahl C, Gurtner C, Dettwiler M, Schmidt A et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Deep learning algorithms out-perform veterinary pathologists in detecting the mitotically most active tumor region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Sci Rep 10:16447 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 1–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Aubreville M, Stathonikos N, Bertram CA, Klopleisch R, Hoeve N ter, Ciompi F et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Mitosis domain generalization in histopathology images–The MIDOG challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Med Image Anal 84:102699 (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Deng S, Zhang X, Yan W, Chang EI, Fan Y, Lai M et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Deep learning in digital pathology image analysis: a survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Front Med 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='4 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 470–487.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Stacke K, Eilertsen G, Unger J, Lundström C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Measuring domain shift for deep learning in histopathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' IEEE J Biomed Health Inform 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='2 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 325–336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Wilm F, Marzahl C, Breininger K, Aubreville M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Domain adversarial RetinaNet as a refer- ence algorithm for the MItosis DOmain Generalization challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Biomedical Image Regis- tration, Domain Generalisation and Out-of-Distribution Analysis: MICCAI 2021 Challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 5–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Roux L, Racoceanu D, Capron F, Calvo J, Attieh E, Le Naour G et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Mitos & Atypia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Image Pervasive Access Lab (IPAL), Agency Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=', Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' & Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Infocom Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=', Singapore, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Rep 1 (2014), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Wilm F et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' CAnine CuTaneous Cancer Histology dataset (version 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' The Cancer Imaging Archive (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='7937/TCIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='2M93-FX66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Marzahl C, Wilm F, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' DF, Tharun L, Perner S, Bertram CA et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Robust quad-tree based registration on whole slide images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Comput Pathol (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' PMLR, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 181–190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Narvekar ND, Karam LJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' A no-reference image blur metric based on the cumulative proba- bility of blur detection (CPBD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' IEEE Trans Image Process 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content='9 (2011), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 2678–2683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Michelson AA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Studies in optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} +page_content=' Courier Corporation, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'} diff --git 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However, +sharing data can be an expensive and risky endeavour. Exist- +ing sharing systems like Distributed File Systems provide full +read, write, and execute Role-based Access Control (RBAC) +for sharing data, but can be expensive and difficult to scale. +Likewise such systems operate on a binary access model for +their data, either a user can read all the data or read none of +the data. This approach is not necessary for a more read-only +oriented data landscape, and one where data contains many +dimensions that represent a risk if overshared. In order to +encourage users to share data and smooth out the process +of accessing such data a new approach is needed. This new +approach must simplify the RBAC of older DFS approaches +to something more read-only and something that integrates +redaction for user protections. +To accomplish this we present CageCoach, a simple sharing- +oriented Distributed Cryptographic File System (DCFS). Cage- +Coach leverages the simplicity and speed of basic HTTP, +linked data concepts, and automatic redaction systems to +facilitate safe and easy sharing of user data. The implemen- +tation of CageCoach is available at https://github.umn.edu/ +CARPE415/CageCoach. +1 +INTRODUCTION +User-generated data drives the modern world. Everything +from Uber driver rides and Google search queries to video +game experiences and Amazon purchase patterns feed user +data back into these systems to provide insights for improve- +ment. Additionally, users sharing their data as part of crowd +sourcing solutions has proven key to reverse engineering gig +working applications such as Uber[5, 16, 18], Lyft[16, 18], +and Shipt[3, 18]. Further these efforts help solve civic and +national needs such as with Atlanta’s Data Dashboard[13], +Minneapolis’s Opendata program[7], or the United State’s +Citizen Science initiative[8]. +However, users providing their data to these initiatives +often comes with a level of risk and a loss of control over the +data they provide. Once a user has handed over information +the safety considerations, redaction approaches, and man- +agement decisions are out of their control. Further, should +any shared user data become dangerous to a user, the user +has no more sway to alleviate this risk other than ask the +current data holder to act, a practice often fruitless. +In order to further encourage users to share their data, a +new sharing oriented data hosting system is required. Such a +platform must be simple to implement, easy to request data +from, but still provide some assurances of privacy and safety +for users involved. Crucially it should remain in the user’s +control, and not be subject to control by others even those +hosting data such as on public hosting systems. The privacy +capability must be granular not just in who can access data +but what specific data is accessible. For example, for some +users, sharing their full name to everyone who asks is un- +reasonable. Thus they should be able to share with some a +partial redaction of their name. Existing works such as Dis- +tributed File Systems (DFS) are promising, but require exten- +sive implementation, Role-based Access Control (RBAC) en- +forcement, and do not implement granular redaction. Other +platforms like Google Drive, Dropbox, and Kaggle are great +for sharing bulk data but also do not provide granular redac- +tion and require trusting of the platform holders to not share +otherwise redacted user data. +In this work, we introduce CageCoach a sharing oriented +distributed cryptographic file system. CageCoach’s notable +features are: +• Simple Trustless DCFS built over HTTP GET/POST +• Customizable RBAC and Datatype Granular Redac- +tion Pipeline +• Easier sharing with Decentralized data access and +centralized user control +CageCoach streamlines the older RBAC based models of +DFSs and decentralizes the data hosting approaches of plat- +forms making for an overall simpler means of sharing data +with others while retaining granular privacy control for users. +This system is leverages simple HTTP GET/POST operations +to interact with symmetrically encrypted files hosted on any +HTTP platform to achieve decentralized hosting. These files +point back to their owners, represented by a controlling +server, that can facilitate redacted data access for a data re- +quester providing user control of data access. Finally, the +1 +arXiv:2301.04214v1 [cs.CR] 10 Jan 2023 + +, +user’s controlling server applies user defined redaction oper- +ations from a suite of modules CageCoach provides to reduce +sensitive data leakage. +CageCoach’s code can be found at https://github.umn.edu/ +CARPE415/CageCoach. +2 +RELATED WORK +Distributed File Systems (DFS) and cryptographic file sys- +tems (DCFS) have been around for a long time with some +works as early as 1993[1] and as recent as 2020[2]. These are +mature fields with well-developed and commercial products +we see every day, such as Dropbox, GoogleDrive, Hadoop, +Ceph, and others[22]. Despite this, the changing data land- +scape and changing usage behaviors with data invite re- +examinations of existing systems to better fit them for a new +era. The work must relevant in the current data landscape, +data redaction, is an old field but with a renewed interest in +the face of big data breaches, data privacy concerns, and ma- +chine learning for data protection. In this section we outline +these two related areas and contrast them with our proposed +system. +2.1 +Distributed & Cryptographic File +Systems +Distributed File Systems (DFS) are systems for maintaining +coherent file management across desperate hosting devices. +Examples include standard file hosting such as Google Drive, +Dropbox, and InRupt’s Solid[19]. Such systems have a long +history and continued relevance in the modern era. DFS also +manifest as cloud storage systems, albeit with looser file sys- +tem format adherence to mesh with the more diverse Internet +access environment. Extending DFSs into privacy and secu- +rity oriented spaces yields the Distributed Cryptographic +File System (DCFS) domain. Works such as UPSS[2] focus +on creating a sharing-oriented and protective DFS with full +RBAC and mutable verifiable histories of each file involved as +a check against malicious behavior. Further other works such +as [10, 12] aim to utilize the blockchain to achieve the same +RBAC with a more decentralized approach. Finally, other +approaches aim to refine key management in encryption for +DFS[14].These systems while powerful, rely on relatively +expensive RBAC and infrastructure or require significant +trust for the platform holders. In the former case, simplify- +ing the RBAC with the mostly read-only reality of user data +can lower RBAC complexity significantly. In the latter case, +hosting infrastructure is still necessary, but one must create +a trustless environment in order to retain control of one’s +data even on such hosting platforms. +Our work focuses on streamlining data sharing by creat- +ing a middlepoint between strong, rigid, and RBAC focused +approaches such as DCFSs and trust-oriented data platforms +and services like Uber, Kaggle, and Gridwise. +2.2 +Data Redaction +Data redaction is not a new field, but has gained vigor in the +last decade or so as the data economy has shaped. Redac- +tion provides the means for which sensitive data can be +made less sensitive and thus less dangerous in the event of +leaks, breaches, or theft. Likewise, redaction has its place in +academic publications when such publications may contain +in themselves dangerous or sensitive information[4]. Many +existing tools provide a user the quick means of redacting +a document such as [6] and [20]. A handful of commercial +products, such as [21], [15], and [17], apply machine learning +to identify and remove automatically sensitive data. Finally, +other work such as [11] highlight an interesting scenario +where redaction itself must be transparent enough such that +the redaction doesn’t mislead the information. These systems +as implemented are not part of a sharing pipeline and are +applied ad-hoc to data. A system such as the one outlined by +UPSS[2], envisions such technologies are part of a pipeline +of data requests but did not implement or specify beyond +such designs. +Our work applies the concepts behind these redaction +systems, but crucially, as part of a standard granular access +pipeline and not as a one-off and static redaction. This in +effect realizes some aspects of the UPSS[2] pipeline, but with- +out the more complex full RBAC suite. +3 +PROBLEM AND DESIGN GOALS +In order to build a system that encourages users to share their +data two primary problems and design considerations must +be achieved: Simplification of access control for accessing +and requesting data and automatic policy informed data +redaction. With these two aspects a sharing-oriented DFS +will lower the cost of sharing and accessing data and provide +a wide net of protections for users who choose to share. +3.1 +Simplify Access Control For Data +Existing DFS systems utilize a full suite of RBAC function- +ality to provide read, write, and execute functionality for +shared files. These provisions while useful, require signif- +icant infrastructure such as certificates and user profiles +registered within the computational structure of the data +host. This full suite of RBAC is necessary if the group of +users intended to read, write, and/or execute the shared data, +but costly if sharing (read only) is the intention. By removing +the write and execute assumptions of RBAC we can in turn +simplify the operating infrastructure required for accessing +data and making sharing a lower cost effort. This lower cost +is necessary for encouraging users to share their data, as it +2 + +CageCoach: Sharing-Oriented Redaction-Capable Distributed Cryptographic File System +, +will be easier to host for consumption, and for consumers of +data as it will be easier to access. +3.2 +Provide Integrated Automatic User +Data Redaction +Regardless of ease of access, users must be given some as- +surances of safety, privacy, and proper use for their data. +Traditional RBAC focuses on binary access models for data, +either a user can read all the data or none of the data in a typi- +cally hosted file. This approach is not adequate for data items +that contain core sensitive fields. For example, a typical sales +receipt is useful for inventory systems and market trending +services, as they provide insights into purchases and sales +trends, however, these same receipts may contain the pur- +chaser’s name, credit card information, and/or address and +location. Such fields are not important for the overall trend, +but present a security risk for the user. In a binary RBAC +model, such fields would available if the receipt is available. +A more granular approach to access is needed. Such an ap- +proach is outlined but not realized or specified by UPSS[2]. +Such an approach would require that when a user’s data is re- +quested by another, a trusted middle system acquires the raw +full set of data, and then redacts and removes information +that is included in the data but not allowed for that partic- +ular user. For example, removing the name, address, and +credit fields from the sales receipt scenario. This approach +is required to provide granular and safer exposure of user’s +data for general consumption. Further, this process can be +handled by user-defined policy thus providing guidelines +for any user data added in the future thus lowering sharing +costs further. +4 +CAGECOACH SYSTEM +We realize the goals of a sharing-oriented DFS with our +system CageCoach. CageCoach simplifies the RBAC and +infrastructure of existing DFSs and integrates redaction tech- +nologies into a data request pipeline. All of this together +creates a simple and easy means for users to safely and eas- +ily share their data. CageCoach is organized around several +concepts and a flow, outlined in fig. 1. Requesters, who re- +quest user data. Data hosts, which host encrypted data files +and some attached meta data files. Finally, a Data Control +Server (DCS) which manages the owner’s data, processes +requests made by requesters, and redacts outgoing sensitive +data. CageCoach’s operational use-case is: +(1) A owner uploads some data (video, text, audio, etc) to +a hosting system after encrypting and creating a meta +file for the data. +(2) A requester sees this data and examines the meta file +(using GET for example) for information as to where +the owner’s DCS operates. +Figure 1: CageCoach System, providing a streamlined +means for requestors to ask for data and receive useful +but protected data. +(3) The requester sends a POST request to the owner’s +DCS server, asking to view the original data item. +(4) The DCS receives this request, verifies the requester’s +identity through asymmetric key phrase decryption, +and then uses GET to retrieve the remotely hosted +encrypted data file. +(5) The DCS decrypts the file with its own internal sym- +metric key and then applies a series of redaction oper- +ations on the data. +(6) The DCS forwards the remaining unredacted data to +the requester, completing the request and preventing +unnecessary or forbidden data from leaving encrypt- +ed/controlled space. +The details for how the RBAC is simplified and how the +redaction is integrated is detailed in the following sections. +4.1 +Simplifying RBAC Using HTTP And +Read-Only Assumptions +CageCoach simplifies the primary RBAC and infrastructure +of other DFSs by assuming that user data need only be read, +not written too or executed collaboratively. Additionally, +unlike UPSS[2], since there is no write permissions data +versions are no longer necessary thus can relax the assump- +tion UPSS makes for needing a transparent modifications +tree. With this simplification in mind, CageCoach utilizes +the most common means of read-only operation on the Inter- +net: HTTP GET. This means that user data can be hosted on +any system that facilitates HTTP GET, such as open source +systems like Apache2. The data that gets hosted is the user’s +encrypted file and a plain text meta data file. Using some con- +cepts of linked data, the meta data file points to the owner’s +DCS to actually facilitate the request for data among other +fields. The total definition for this meta data file is: +3 + +(2)Directrequestertodataowner +B +(1)Requestaccesstodata +000 +(3) Downloads encrypted file +Data Host +(Dropbox,GDrive,Apache) +000 +000 +HTTP Data Control +Requester +Server (Dcs) +AccessControl(ACL)And +DataCensoringRules(DCR) +(5)Alloweddatais returned +(4)Decryptsandprocessesfile, +• owner-url: URL indicating where the owner’s DCS is. The +place where any request will be processed. +• meta-data: User filled info tags about the data, such as +what format it is, overall context. All of this information is +optional. +• description: A more textual description of the data, op- +tional if an owner wishes to provide more than just tags of +information. +• data-url: The URL indicating where the data this meta file +belongs to is. This is important for providing some backup if +the meta file is moved elsewhere or if it must live elsewhere +in hosting. +• data-hash-sha1: A sha1 of the encrypted file to provide a +minimal check for any requester that wishes to double check +the file they are asking about. +Despite our overall read-only approach, some computa- +tional efforts are still required. Namely the decryption of +the requested file and the granular redaction of information +within this file. The purpose of redirecting the requester +from the data host is to provide a centralized response by +the owner and the computational space for redaction poli- +cies. The requester will send an HTTP POST request to the +DCS indicated by the owner-url and receive a decrypted +and redacted data file. The DCS’s process is implemented as +a basic python HTTP server. The process involves several +steps: 1) Receive a POST request with the URL of the data +being requested and optionally an ID and asymmetrically +encrypted phrase to verify the requester’s identity. Cage- +Coach implements this with RSA public/private key pairs. +2) Locate the data profile for the requested data on the DCS +server, itself a simple text file containing pointers to decrypt +and identify the requested data. Additionally, if the user is +registered with the DCS (registry comprised of a private key +for decrypting phrases, the plain text passphrase, and a id +name) it will load their profile. We implement this as simply +a separate json file containing each requester’s information. +Our approach assumes this registry happens outside of the +CageCoach architecture but can utilize it. 3) The DCS will +download the encrypted file from its host using HTTP GET. +After reception, the DCS will decrypt the data file and load +the redaction policies that match the specific data item (by +its name), the data type (json, mp3, etc), and finally the poli- +cies for the requester (if provided). CageCoach implements +this encryption with symmetric keys using pythons Fernet +library. 4) The DCS will apply these redaction operations, +gradually chipping away data until left with whatever is al- +lowed to pass. 5) The remaining data is sent to the requester +in the POST response. The specifics of how the redaction is +applied is outlined in the next section. +Figure 2: CageCoach Redaction Pipeline, providing a +generalized measure of privacy assurance. +4.2 +Access Control and Redaction +Pipelines +CageCoach’s read-only assumption for user data is not a +binary, like older models of RBAC based system, but granu- +lar. By using a series of redaction operations over requested +data, CageCoach can allow partial access to data. These op- +erations, dividable by datatype as outlined in Fig. 2, provide +for blurring faces in images, redacting text in jsons and csvs, +and muting specific words or background noises recognized +in audio. In the overall data request pipeline after a user +has requested data and the DCS has downloaded the target +data, it will apply these redaction operations according to +the specific user, datatype, and data item. This provides three +levels of granularity for controlling data flow outwards to re- +questers: by datatype (all jsons, csvs, mp3s, etc), by data item +(ex: specific files like example-1.json hosted on Google Drive +or example-2.json hosted on dropbox), and by requester id +(ex: John Doe can access the user’s name, but Jane Doe can +only see the user’s first name). However, such operations +that would be specific to an owner, such as blurring only +the owner’s face, require the owner provide their own data +to the redacting DCS. Our implementation we provide does +general redaction such as blurring all faces and removing a +handful of well known text fields such as social security and +street addresses. We do not implement an audio redaction +approach as there isn’t a general python capable pre-built +audio redaction library nor a common set of what "words" +should be auto removed, unlike faces in images. CageCoach +does support extensions to these operations to tailor to spe- +cific users. Our implementation uses the Haar cascade and +OpenCV2 [9] python libraries for blurring faces (illustrated +with the blurring of photo of American Union Army General +Benjamin Butler fig. 3), and python Pandas to redact textual +data (example of such in fig. 4). +4 + +010110 +RedactionProcessesByDataTypes +010001 +111011 +Blob +Age:43 +Name:"JohnDoe" +SSN:"999-99-9999" +File +CensorFields +Encrypted +8871 +File +Image +CensorFaces/Persons +4) +Audio +CensorAudio SegmentsCageCoach: Sharing-Oriented Redaction-Capable Distributed Cryptographic File System +, +Figure 3: CageCoach Redaction Pipeline example blur- +ring a specific image. +Figure 4: CageCoach Redaction Pipeline example +redacting specific text and fields. +5 +CONCLUSION +In this work, we introduced a new sharing oriented imple- +mentation of DCFS: CageCoach. CageCoach streamlines the +older RBAC heavy and trust-necessary hosting models of +DFS, while using the simpler HTTP GET/POST ecosystem to +facilitate easier data sharing. All of this is possible while still +respecting the privacy of users through granular customize- +able redaction pipelines that handle removal of sensitive user +information. +6 +LIMITATIONS AND FUTURE WORK +CageCoach has a set of drawbacks and limitations. Cage- +Coach is implemented as a demonstration of a new inter- +pretation of sharing-oriented DCFS and not intended for +industrial or commercial use. Future implementations would +need to provide better integration with hosting services like +Google and Dropbox, and provide tougher and more robust +security checks and infrastructure. Likewise future work +improvements would be needed to make the redaction oper- +ations more capable and workable on a wider set of diverse +data. Notably there are two non-implementation limitations +that stunt CageCoach and the broader goal of safe sharing +oriented DFS: +• No system can stop external data reconstruction. +No matter if a user is using CageCoach, Google Drive, +or any other hosting system, external actors with access to +pieces of separate data can always reassemble it together. +For example, an actor A has access to a subset of data 1, and +an actor B has access to another subset of data 1. These two +actors are not allowed access to either subset of data by the +policies of the user whose data it is. However, this does not +stop nor disincentivise actor A and B from simply sharing +with each other the user’s data. Each filling in the other’s gap +of missing data. No system can solve this if the requesting +actors are able to observe data. +• Leakage is still possible through indirect implicating +fields. +CageCoach’s redaction pipeline is quite rudimentary, in +some cases data may be leaked through a combination of un- +related fields. For example, with a street address, a malicious +user may be able to correctly guess a zip code when paired +with other information. This is due to CageCoach’s inability +to understand the connections between data. +CageCoach’s unique sharing-oriented DCFS structure pro- +vides several new areas of exploration. CageCoach itself can +be expanded to cover more datatypes, and work can be done +to integrate the ingress of user’s data to the data hosts that +CageCoach manages. +6.1 +Collective Redaction Rules For +Multi-Owner Data +Given our system’s usage of a redaction pipeline, one could +envision a scenario where data that is collected by one user, +but contains multiple other users’ data is pass around each +impacted user’s DCS for specific group based redaction. This +would facilitate greater granularity of redaction and a sense +of group ownership over data and its privacy implications. +6.2 +Enhanced ACL And Redaction +Through Impact Trees +A future work could examine how to enhance the redaction +rules to include field implications to provide greater coverage +of privacy in the event a user misses these concepts them- +selves. This would fill in the gaps that leaking implicating +fields create. +REFERENCES +[1] Matt Blaze. 1993. A Cryptographic File System for UNIX. In Proceedings +of the 1st ACM Conference on Computer and Communications Security +(Fairfax, Virginia, USA) (CCS ’93). Association for Computing Machin- +ery, New York, NY, USA, 9–16. https://doi.org/10.1145/168588.168590 +5 + +"Name":"John Doe" +"Age""24" +"Height": "4 feet, 2 inches" +1 +"Age": +"24" +"Height": +"4 feet, x", +[2] Arastoo Bozorgi, Mahya Soleimani Jadidi, and Jonathan Anderson. +2020. Challenges in Designing a Distributed Cryptographic File System. +In Security Protocols XXVII, Jonathan Anderson, Frank Stajano, Bruce +Christianson, and Vashek Matyáš (Eds.). Springer International Pub- +lishing, Cham, 177–192. https://link.springer.com/chapter/10.1007/ +978-3-030-57043-9_17 +[3] Dan Calacci and Alex Pentland. 2022. Bargaining with the Black-Box: +Designing and Deploying Worker-Centric Tools to Audit Algorithmic +Management. Proc. ACM Hum.-Comput. Interact. 6, CSCW2, Article +428 (nov 2022), 24 pages. https://doi.org/10.1145/3570601 +[4] Arturo Casadevall, Lynn Enquist, Michael Imperiale, Paul Keim, +Michael Osterholm, and David Relman. 2013. Redaction of Sensi- +tive Data in the Publication of Dual Use Research of Concern. mBio 5 +(12 2013). https://doi.org/10.1128/mBio.00991-13 +[5] Le Chen, Alan Mislove, and Christo Wilson. 2015. Peeking Beneath the +Hood of Uber. In Proceedings of the 2015 Internet Measurement Confer- +ence (Tokyo, Japan) (IMC ’15). Association for Computing Machinery, +New York, NY, USA, 495–508. https://doi.org/10.1145/2815675.2815681 +[6] extract team. [n.d.]. Automated Data Redaction Software. https://www. +extractsystems.com/automated-data-redaction-software accessed on +Dec 2022. +[7] Minneapolis Government. [n.d.]. Minneapolis Open Data. +https: +//opendata.minneapolismn.gov/ accessed on Sun 18 Dec 2022. +[8] United States Government. [n.d.]. Citizen Science. +https://www. +citizenscience.gov/# accessed on Sun 18 Dec 2022. +[9] Olli-Pekka Heinisuo. [n.d.]. OpenCV on Wheels. +https://pypi.org/ +project/opencv-python/ accessed on Dec 2022. +[10] Hsiao-Shan Huang, Tian-Sheuan Chang, and Jhih-Yi Wu. 2020. A Se- +cure File Sharing System Based on IPFS and Blockchain. In Proceedings +of the 2020 2nd International Electronics Communication Conference. +ACM. https://doi.org/10.1145/3409934.3409948 +[11] Jinhua Ma, Xinyi Huang, Yi Mu, and Robert H. Deng. 2022. Authen- +ticated Data Redaction With Accountability and Transparency. IEEE +Transactions on Dependable and Secure Computing 19, 1 (2022), 149–160. +https://doi.org/10.1109/TDSC.2020.2998135 +[12] Muqaddas Naz, Fahad A. Al-zahrani, Rabiya Khalid, Nadeem Javaid, +Ali Mustafa Qamar, Muhammad Khalil Afzal, and Muhammad Shafiq. +2019. A Secure Data Sharing Platform Using Blockchain and Inter- +planetary File System. Sustainability 11, 24 (2019). https://doi.org/10. +3390/su11247054 +[13] Firaz Peer and Carl DiSalvo. 2022. +The Work of Infrastructural +Bricoleurs in Building Civic Data Dashboards. +Proc. ACM Hum.- +Comput. Interact. 6, CSCW1, Article 124 (apr 2022), 25 pages. https: +//doi.org/10.1145/3512971 +[14] K. V. Pradeep, V. Vijayakumar, V. Subramaniyaswamy, and Arash H. +Lashkari. 2019. An Efficient Framework for Sharing a File in a Secure +Manner Using Asymmetric Key Distribution Management in Cloud +Environment. J. Comput. Netw. Commun. 2019 (jan 2019), 8. https: +//doi.org/10.1155/2019/9852472 +[15] redacted.ai team. [n.d.]. Redacted.ai. https://redacted.ai/ accessed on +Dec 2022. +[16] Todd W. Schneider. 2020. Reverse Engineering Uber and Lyft Surge +Pricing in Chicago. +https://toddwschneider.com/posts/chicago- +ridehail-surge-pricing/ accessed on Dec 2022. +[17] Document.Redact team. [n.d.]. Document.Redact. +https://super.ai/ +blog/redacting-information-from-documents-automatically accessed +on Dec 2022. +[18] Gridwise Team. [n.d.]. Gridwise. https://gridwise.io/ access on Mon +12 Dec 2022. +[19] InRupt Team. [n.d.]. Solid. https://www.inrupt.com/solid +[20] Objective Team. [n.d.]. Objective Redact. https://www.objective.com/ +products/objective-redact accessed on Dec 2022. +[21] DOMA Technologies. [n.d.]. DOMA. https://www.domaonline.com/ +solutions/digitalservices/data-redaction/ accessed on Dec 2022. +[22] Mahmut Ünver and Atilla Erguzen. 2016. A STUDY ON DISTRIBUTED +FILE SYSTEMS: An example of NFS. +6 + diff --git a/8NE2T4oBgHgl3EQf8Ag-/content/tmp_files/load_file.txt b/8NE2T4oBgHgl3EQf8Ag-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a872f767f8d84f120456ebab7ea40f3f7755e096 --- /dev/null +++ b/8NE2T4oBgHgl3EQf8Ag-/content/tmp_files/load_file.txt @@ -0,0 +1,364 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf,len=363 +page_content='CageCoach: Sharing-Oriented Redaction-Capable Distributed Cryptographic File System Jason Carpenter CARPE415@umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='edu University of Minnesota Minneapolis, MN Zhi-Li Zhang zhzhang@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='edu University of Minnesota Minneapolis, MN ABSTRACT The modern data economy is built on sharing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' However, sharing data can be an expensive and risky endeavour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Exist- ing sharing systems like Distributed File Systems provide full read, write, and execute Role-based Access Control (RBAC) for sharing data, but can be expensive and difficult to scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Likewise such systems operate on a binary access model for their data, either a user can read all the data or read none of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' This approach is not necessary for a more read-only oriented data landscape, and one where data contains many dimensions that represent a risk if overshared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' In order to encourage users to share data and smooth out the process of accessing such data a new approach is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' This new approach must simplify the RBAC of older DFS approaches to something more read-only and something that integrates redaction for user protections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' To accomplish this we present CageCoach, a simple sharing- oriented Distributed Cryptographic File System (DCFS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Cage- Coach leverages the simplicity and speed of basic HTTP, linked data concepts, and automatic redaction systems to facilitate safe and easy sharing of user data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' The implemen- tation of CageCoach is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='edu/ CARPE415/CageCoach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 1 INTRODUCTION User-generated data drives the modern world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Everything from Uber driver rides and Google search queries to video game experiences and Amazon purchase patterns feed user data back into these systems to provide insights for improve- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Additionally, users sharing their data as part of crowd sourcing solutions has proven key to reverse engineering gig working applications such as Uber[5, 16, 18], Lyft[16, 18], and Shipt[3, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Further these efforts help solve civic and national needs such as with Atlanta’s Data Dashboard[13], Minneapolis’s Opendata program[7], or the United State’s Citizen Science initiative[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' However, users providing their data to these initiatives often comes with a level of risk and a loss of control over the data they provide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Once a user has handed over information the safety considerations, redaction approaches, and man- agement decisions are out of their control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Further, should any shared user data become dangerous to a user, the user has no more sway to alleviate this risk other than ask the current data holder to act, a practice often fruitless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' In order to further encourage users to share their data, a new sharing oriented data hosting system is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Such a platform must be simple to implement, easy to request data from, but still provide some assurances of privacy and safety for users involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Crucially it should remain in the user’s control, and not be subject to control by others even those hosting data such as on public hosting systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' The privacy capability must be granular not just in who can access data but what specific data is accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' For example, for some users, sharing their full name to everyone who asks is un- reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Thus they should be able to share with some a partial redaction of their name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Existing works such as Dis- tributed File Systems (DFS) are promising, but require exten- sive implementation, Role-based Access Control (RBAC) en- forcement, and do not implement granular redaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Other platforms like Google Drive, Dropbox, and Kaggle are great for sharing bulk data but also do not provide granular redac- tion and require trusting of the platform holders to not share otherwise redacted user data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' In this work, we introduce CageCoach a sharing oriented distributed cryptographic file system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' CageCoach’s notable features are: Simple Trustless DCFS built over HTTP GET/POST Customizable RBAC and Datatype Granular Redac- tion Pipeline Easier sharing with Decentralized data access and centralized user control CageCoach streamlines the older RBAC based models of DFSs and decentralizes the data hosting approaches of plat- forms making for an overall simpler means of sharing data with others while retaining granular privacy control for users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' This system is leverages simple HTTP GET/POST operations to interact with symmetrically encrypted files hosted on any HTTP platform to achieve decentralized hosting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' These files point back to their owners, represented by a controlling server, that can facilitate redacted data access for a data re- quester providing user control of data access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Finally, the 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='04214v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='CR] 10 Jan 2023 , user’s controlling server applies user defined redaction oper- ations from a suite of modules CageCoach provides to reduce sensitive data leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' CageCoach’s code can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='edu/ CARPE415/CageCoach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 2 RELATED WORK Distributed File Systems (DFS) and cryptographic file sys- tems (DCFS) have been around for a long time with some works as early as 1993[1] and as recent as 2020[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' These are mature fields with well-developed and commercial products we see every day, such as Dropbox, GoogleDrive, Hadoop, Ceph, and others[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Despite this, the changing data land- scape and changing usage behaviors with data invite re- examinations of existing systems to better fit them for a new era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' The work must relevant in the current data landscape, data redaction, is an old field but with a renewed interest in the face of big data breaches, data privacy concerns, and ma- chine learning for data protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' In this section we outline these two related areas and contrast them with our proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='1 Distributed & Cryptographic File Systems Distributed File Systems (DFS) are systems for maintaining coherent file management across desperate hosting devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Examples include standard file hosting such as Google Drive, Dropbox, and InRupt’s Solid[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Such systems have a long history and continued relevance in the modern era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' DFS also manifest as cloud storage systems, albeit with looser file sys- tem format adherence to mesh with the more diverse Internet access environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Extending DFSs into privacy and secu- rity oriented spaces yields the Distributed Cryptographic File System (DCFS) domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Works such as UPSS[2] focus on creating a sharing-oriented and protective DFS with full RBAC and mutable verifiable histories of each file involved as a check against malicious behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Further other works such as [10, 12] aim to utilize the blockchain to achieve the same RBAC with a more decentralized approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Finally, other approaches aim to refine key management in encryption for DFS[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='These systems while powerful, rely on relatively expensive RBAC and infrastructure or require significant trust for the platform holders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' In the former case, simplify- ing the RBAC with the mostly read-only reality of user data can lower RBAC complexity significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' In the latter case, hosting infrastructure is still necessary, but one must create a trustless environment in order to retain control of one’s data even on such hosting platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Our work focuses on streamlining data sharing by creat- ing a middlepoint between strong, rigid, and RBAC focused approaches such as DCFSs and trust-oriented data platforms and services like Uber, Kaggle, and Gridwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='2 Data Redaction Data redaction is not a new field, but has gained vigor in the last decade or so as the data economy has shaped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Redac- tion provides the means for which sensitive data can be made less sensitive and thus less dangerous in the event of leaks, breaches, or theft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Likewise, redaction has its place in academic publications when such publications may contain in themselves dangerous or sensitive information[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Many existing tools provide a user the quick means of redacting a document such as [6] and [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' A handful of commercial products, such as [21], [15], and [17], apply machine learning to identify and remove automatically sensitive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Finally, other work such as [11] highlight an interesting scenario where redaction itself must be transparent enough such that the redaction doesn’t mislead the information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' These systems as implemented are not part of a sharing pipeline and are applied ad-hoc to data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' A system such as the one outlined by UPSS[2], envisions such technologies are part of a pipeline of data requests but did not implement or specify beyond such designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Our work applies the concepts behind these redaction systems, but crucially, as part of a standard granular access pipeline and not as a one-off and static redaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' This in effect realizes some aspects of the UPSS[2] pipeline, but with- out the more complex full RBAC suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 3 PROBLEM AND DESIGN GOALS In order to build a system that encourages users to share their data two primary problems and design considerations must be achieved: Simplification of access control for accessing and requesting data and automatic policy informed data redaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' With these two aspects a sharing-oriented DFS will lower the cost of sharing and accessing data and provide a wide net of protections for users who choose to share.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='1 Simplify Access Control For Data Existing DFS systems utilize a full suite of RBAC function- ality to provide read, write, and execute functionality for shared files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' These provisions while useful, require signif- icant infrastructure such as certificates and user profiles registered within the computational structure of the data host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' This full suite of RBAC is necessary if the group of users intended to read, write, and/or execute the shared data, but costly if sharing (read only) is the intention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' By removing the write and execute assumptions of RBAC we can in turn simplify the operating infrastructure required for accessing data and making sharing a lower cost effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' This lower cost is necessary for encouraging users to share their data, as it 2 CageCoach: Sharing-Oriented Redaction-Capable Distributed Cryptographic File System , will be easier to host for consumption, and for consumers of data as it will be easier to access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='2 Provide Integrated Automatic User Data Redaction Regardless of ease of access, users must be given some as- surances of safety, privacy, and proper use for their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Traditional RBAC focuses on binary access models for data, either a user can read all the data or none of the data in a typi- cally hosted file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' This approach is not adequate for data items that contain core sensitive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' For example, a typical sales receipt is useful for inventory systems and market trending services, as they provide insights into purchases and sales trends, however, these same receipts may contain the pur- chaser’s name, credit card information, and/or address and location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Such fields are not important for the overall trend, but present a security risk for the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' In a binary RBAC model, such fields would available if the receipt is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' A more granular approach to access is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Such an ap- proach is outlined but not realized or specified by UPSS[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Such an approach would require that when a user’s data is re- quested by another, a trusted middle system acquires the raw full set of data, and then redacts and removes information that is included in the data but not allowed for that partic- ular user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' For example, removing the name, address, and credit fields from the sales receipt scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' This approach is required to provide granular and safer exposure of user’s data for general consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Further, this process can be handled by user-defined policy thus providing guidelines for any user data added in the future thus lowering sharing costs further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 4 CAGECOACH SYSTEM We realize the goals of a sharing-oriented DFS with our system CageCoach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' CageCoach simplifies the RBAC and infrastructure of existing DFSs and integrates redaction tech- nologies into a data request pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' All of this together creates a simple and easy means for users to safely and eas- ily share their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' CageCoach is organized around several concepts and a flow, outlined in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Requesters, who re- quest user data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Data hosts, which host encrypted data files and some attached meta data files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Finally, a Data Control Server (DCS) which manages the owner’s data, processes requests made by requesters, and redacts outgoing sensitive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' CageCoach’s operational use-case is: (1) A owner uploads some data (video, text, audio, etc) to a hosting system after encrypting and creating a meta file for the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' (2) A requester sees this data and examines the meta file (using GET for example) for information as to where the owner’s DCS operates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Figure 1: CageCoach System, providing a streamlined means for requestors to ask for data and receive useful but protected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' (3) The requester sends a POST request to the owner’s DCS server, asking to view the original data item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' (4) The DCS receives this request, verifies the requester’s identity through asymmetric key phrase decryption, and then uses GET to retrieve the remotely hosted encrypted data file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' (5) The DCS decrypts the file with its own internal sym- metric key and then applies a series of redaction oper- ations on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' (6) The DCS forwards the remaining unredacted data to the requester, completing the request and preventing unnecessary or forbidden data from leaving encrypt- ed/controlled space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' The details for how the RBAC is simplified and how the redaction is integrated is detailed in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='1 Simplifying RBAC Using HTTP And Read-Only Assumptions CageCoach simplifies the primary RBAC and infrastructure of other DFSs by assuming that user data need only be read, not written too or executed collaboratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Additionally, unlike UPSS[2], since there is no write permissions data versions are no longer necessary thus can relax the assump- tion UPSS makes for needing a transparent modifications tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' With this simplification in mind, CageCoach utilizes the most common means of read-only operation on the Inter- net: HTTP GET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' This means that user data can be hosted on any system that facilitates HTTP GET, such as open source systems like Apache2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' The data that gets hosted is the user’s encrypted file and a plain text meta data file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Using some con- cepts of linked data, the meta data file points to the owner’s DCS to actually facilitate the request for data among other fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' The total definition for this meta data file is: 3 (2)Directrequestertodataowner B (1)Requestaccesstodata 000 (3) Downloads encrypted file Data Host (Dropbox,GDrive,Apache) 000 000 HTTP Data Control Requester Server (Dcs) AccessControl(ACL)And DataCensoringRules(DCR) (5)Alloweddatais returned (4)Decryptsandprocessesfile, owner-url: URL indicating where the owner’s DCS is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' The place where any request will be processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' meta-data: User filled info tags about the data, such as what format it is, overall context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' All of this information is optional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' description: A more textual description of the data, op- tional if an owner wishes to provide more than just tags of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' data-url: The URL indicating where the data this meta file belongs to is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' This is important for providing some backup if the meta file is moved elsewhere or if it must live elsewhere in hosting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' data-hash-sha1: A sha1 of the encrypted file to provide a minimal check for any requester that wishes to double check the file they are asking about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Despite our overall read-only approach, some computa- tional efforts are still required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Namely the decryption of the requested file and the granular redaction of information within this file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' The purpose of redirecting the requester from the data host is to provide a centralized response by the owner and the computational space for redaction poli- cies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' The requester will send an HTTP POST request to the DCS indicated by the owner-url and receive a decrypted and redacted data file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' The DCS’s process is implemented as a basic python HTTP server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' The process involves several steps: 1) Receive a POST request with the URL of the data being requested and optionally an ID and asymmetrically encrypted phrase to verify the requester’s identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Cage- Coach implements this with RSA public/private key pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 2) Locate the data profile for the requested data on the DCS server, itself a simple text file containing pointers to decrypt and identify the requested data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Additionally, if the user is registered with the DCS (registry comprised of a private key for decrypting phrases, the plain text passphrase, and a id name) it will load their profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' We implement this as simply a separate json file containing each requester’s information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Our approach assumes this registry happens outside of the CageCoach architecture but can utilize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 3) The DCS will download the encrypted file from its host using HTTP GET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' After reception, the DCS will decrypt the data file and load the redaction policies that match the specific data item (by its name), the data type (json, mp3, etc), and finally the poli- cies for the requester (if provided).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' CageCoach implements this encryption with symmetric keys using pythons Fernet library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 4) The DCS will apply these redaction operations, gradually chipping away data until left with whatever is al- lowed to pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 5) The remaining data is sent to the requester in the POST response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' The specifics of how the redaction is applied is outlined in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Figure 2: CageCoach Redaction Pipeline, providing a generalized measure of privacy assurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='2 Access Control and Redaction Pipelines CageCoach’s read-only assumption for user data is not a binary, like older models of RBAC based system, but granu- lar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' By using a series of redaction operations over requested data, CageCoach can allow partial access to data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' These op- erations, dividable by datatype as outlined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 2, provide for blurring faces in images, redacting text in jsons and csvs, and muting specific words or background noises recognized in audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' In the overall data request pipeline after a user has requested data and the DCS has downloaded the target data, it will apply these redaction operations according to the specific user, datatype, and data item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' This provides three levels of granularity for controlling data flow outwards to re- questers: by datatype (all jsons, csvs, mp3s, etc), by data item (ex: specific files like example-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='json hosted on Google Drive or example-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='json hosted on dropbox), and by requester id (ex: John Doe can access the user’s name, but Jane Doe can only see the user’s first name).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' However, such operations that would be specific to an owner, such as blurring only the owner’s face, require the owner provide their own data to the redacting DCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Our implementation we provide does general redaction such as blurring all faces and removing a handful of well known text fields such as social security and street addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' We do not implement an audio redaction approach as there isn’t a general python capable pre-built audio redaction library nor a common set of what "words" should be auto removed, unlike faces in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' CageCoach does support extensions to these operations to tailor to spe- cific users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Our implementation uses the Haar cascade and OpenCV2 [9] python libraries for blurring faces (illustrated with the blurring of photo of American Union Army General Benjamin Butler fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 3), and python Pandas to redact textual data (example of such in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 4 010110 RedactionProcessesByDataTypes 010001 111011 Blob Age:43 Name:"JohnDoe" SSN:"999-99-9999" File CensorFields Encrypted 8871 File Image CensorFaces/Persons 4) Audio CensorAudio SegmentsCageCoach: Sharing-Oriented Redaction-Capable Distributed Cryptographic File System , Figure 3: CageCoach Redaction Pipeline example blur- ring a specific image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Figure 4: CageCoach Redaction Pipeline example redacting specific text and fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 5 CONCLUSION In this work, we introduced a new sharing oriented imple- mentation of DCFS: CageCoach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' CageCoach streamlines the older RBAC heavy and trust-necessary hosting models of DFS, while using the simpler HTTP GET/POST ecosystem to facilitate easier data sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' All of this is possible while still respecting the privacy of users through granular customize- able redaction pipelines that handle removal of sensitive user information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 6 LIMITATIONS AND FUTURE WORK CageCoach has a set of drawbacks and limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Cage- Coach is implemented as a demonstration of a new inter- pretation of sharing-oriented DCFS and not intended for industrial or commercial use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Future implementations would need to provide better integration with hosting services like Google and Dropbox, and provide tougher and more robust security checks and infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Likewise future work improvements would be needed to make the redaction oper- ations more capable and workable on a wider set of diverse data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Notably there are two non-implementation limitations that stunt CageCoach and the broader goal of safe sharing oriented DFS: No system can stop external data reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' No matter if a user is using CageCoach, Google Drive, or any other hosting system, external actors with access to pieces of separate data can always reassemble it together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' For example, an actor A has access to a subset of data 1, and an actor B has access to another subset of data 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' These two actors are not allowed access to either subset of data by the policies of the user whose data it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' However, this does not stop nor disincentivise actor A and B from simply sharing with each other the user’s data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Each filling in the other’s gap of missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' No system can solve this if the requesting actors are able to observe data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Leakage is still possible through indirect implicating fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' CageCoach’s redaction pipeline is quite rudimentary, in some cases data may be leaked through a combination of un- related fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' For example, with a street address, a malicious user may be able to correctly guess a zip code when paired with other information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' This is due to CageCoach’s inability to understand the connections between data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' CageCoach’s unique sharing-oriented DCFS structure pro- vides several new areas of exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' CageCoach itself can be expanded to cover more datatypes, and work can be done to integrate the ingress of user’s data to the data hosts that CageCoach manages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='1 Collective Redaction Rules For Multi-Owner Data Given our system’s usage of a redaction pipeline, one could envision a scenario where data that is collected by one user, but contains multiple other users’ data is pass around each impacted user’s DCS for specific group based redaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' This would facilitate greater granularity of redaction and a sense of group ownership over data and its privacy implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='2 Enhanced ACL And Redaction Through Impact Trees A future work could examine how to enhance the redaction rules to include field implications to provide greater coverage of privacy in the event a user misses these concepts them- selves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' This would fill in the gaps that leaking implicating fields create.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' REFERENCES [1] Matt Blaze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' A Cryptographic File System for UNIX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' In Proceedings of the 1st ACM Conference on Computer and Communications Security (Fairfax, Virginia, USA) (CCS ’93).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Association for Computing Machin- ery, New York, NY, USA, 9–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='1145/168588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='168590 5 "Name":"John Doe" "Age""24" "Height": "4 feet, 2 inches" 1 "Age": "24" "Height": "4 feet, x", [2] Arastoo Bozorgi, Mahya Soleimani Jadidi, and Jonathan Anderson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Challenges in Designing a Distributed Cryptographic File System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' In Security Protocols XXVII, Jonathan Anderson, Frank Stajano, Bruce Christianson, and Vashek Matyáš (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Springer International Pub- lishing, Cham, 177–192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='com/chapter/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='1007/ 978-3-030-57043-9_17 [3] Dan Calacci and Alex Pentland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Bargaining with the Black-Box: Designing and Deploying Worker-Centric Tools to Audit Algorithmic Management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' ACM Hum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='-Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 6, CSCW2, Article 428 (nov 2022), 24 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='1145/3570601 [4] Arturo Casadevall, Lynn Enquist, Michael Imperiale, Paul Keim, Michael Osterholm, and David Relman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Redaction of Sensi- tive Data in the Publication of Dual Use Research of Concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' mBio 5 (12 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='1128/mBio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='00991-13 [5] Le Chen, Alan Mislove, and Christo Wilson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Peeking Beneath the Hood of Uber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' In Proceedings of the 2015 Internet Measurement Confer- ence (Tokyo, Japan) (IMC ’15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 495–508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='1145/2815675.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='2815681 [6] extract team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' [n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Automated Data Redaction Software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' extractsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='com/automated-data-redaction-software accessed on Dec 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' [7] Minneapolis Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' [n.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' ACM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='1145/3409934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content='3409948 [11] Jinhua Ma, Xinyi Huang, Yi Mu, and Robert H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'} +page_content=' Deng.' metadata={'source': 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many domains, such as eco- +nomics, health care, public policy, web mining, online ad- +vertising, and marketing campaigns. Although significant +advances have been made to overcome the challenges in +causal effect estimation with observational data, such as +missing counterfactual outcomes and selection bias between +treatment and control groups, the existing methods mainly +focus on source-specific and stationary observational data. +Such learning strategies assume that all observational data +are already available during the training phase and from only +one source. +Along with the fast-growing segments of industrial appli- +cations, this assumption is unsubstantial in practice. Taking +Alipay as an example, which is one of the world’s largest +mobile payment platforms and offers financial services to +billion-scale users, a tremendous amount of data containing +much privacy-related information is produced daily and col- +lected from different sources. In conclusion, the following +two points are summed up. The first one is based on the +characteristics of observational data, which are incremen- +tally available from non-stationary data distributions. For +instance, the electronic financial records for one marketing +campaign are growing every day and they may be collected +from different cities or even other countries. This character- +istic implies that one cannot have access to all observational +data at a one-time point and from one single source. The sec- +ond reason is based on the realistic consideration of accessi- +bility. For example, when new observational data are avail- +able, if we want to refine the model previously trained by +original data, maybe the original training data are no longer +accessible due to a variety of reasons, e.g., legacy data may +be unrecorded, proprietary, the sensitivity of financial data, +too large to store, or subject to privacy constraint of personal +information (Zhang et al. 2020). This practical concern of +accessibility is ubiquitous in various academic and indus- +trial applications. That’s what it boiled down to: in the era +of big data, we face new challenges in causal inference with +observational data, i.e., the extensibility for incrementally +available observational data, the adaptability for extra do- +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +main adaptation problem except for the imbalance between +treatment and control groups, and the accessibility for an +enormous amount of data. +In this position paper, we formally define the problem of +continual treatment effect estimation, describe its research +challenges, and then present possible solutions to this prob- +lem. Moreover, we will discuss future research directions on +this topic. +Related Work +Instead of randomized controlled trials, observational data +is obtained by the researcher simply observing the subjects +without any interference. That means that the researchers +have no control over the treatment assignments, and they just +observe the subjects and record data based on their obser- +vations (Yao et al. 2021). Therefore, from the observational +data, directly estimating the treatment effect is challenging +due to the missing counterfactual outcomes and the exis- +tence of confounders. Recently, powerful machine learning +methods such as tree-based methods (Athey and Imbens +2016; Wager and Athey 2018), representation learning +(Li and Fu +2017; +Shalit, Johansson, and Sontag +2017; +Yao et al. 2018; Chu, Rathbun, and Li 2022), meta-learning +(K¨unzel et al. 2019; Nie and Wager 2021), generative mod- +els (Louizos et al. 2017; Yoon, Jordon, and van der Schaar +2018) have achieved prominent progress in treatment effect +estimation task. +In addition, the combination of causal inference and +other research fields also exhibits complementary strengths, +such as computer vision (Tang et al. 2020; Liu et al. 2022a), +graph learning (Ma et al. 2022; Chu, Rathbun, and Li 2021), +natural language processing (Feder et al. 2022; Liu et al. +2022b), and so on. The involved causal analysis helps to im- +prove the model’s capability of discovering and resolving +the underlying system beyond the statistical relationships +learned from observational data. +Problem Definition +Suppose that the observational data contain n units collected +from d different domains and the d-th dataset Dd contains +the data {(x, y, t)|x ∈ X, y ∈ Y, t ∈ T } collected from d-th +domain, which contains nd units. Let X denote all observed +variables, Y denote the outcomes in the observational data, + +and T be a binary variable. Let D1:d = {D1, D2, ..., Dd} +be the set of combination of d datasets, separately collected +from d different domains. For d datasets {D1, D2, ..., Dd}, +they have the commonly observed variables, but due to the +fact that they are collected from different domains, they have +different distributions with respect to X, Y , and T in each +dataset. Each unit in the observational data received one of +two treatments. Let ti denote the treatment assignment for +unit i; i = 1, ..., n. For binary treatments, ti = 1 is for +the treatment group and ti = 0 for the control group. The +outcome for unit i is denoted by yi +t when treatment t is ap- +plied to unit i. For observational data, only one of the poten- +tial outcomes is observed. The observed outcome is called +the factual outcome, and the remaining unobserved poten- +tial outcomes are called counterfactual outcomes. +This task can follow the potential outcome frame- +work for estimating treatment effects +(Rubin 1974; +Splawa-Neyman, Dabrowska, and Speed 1990). The indi- +vidual treatment effect (ITE) for unit i is the difference be- +tween the potential treated and control outcomes and is de- +fined as +ITEi = yi +1 − yi +0. +(1) +The average treatment effect (ATE) is the difference be- +tween the mean potential treated and control outcomes, +which is defined as +ATE = 1 +n +n +� +i=1 +(yi +1 − yi +0). +(2) +The success of the potential outcome framework is based +on the following assumptions (Imbens and Rubin 2015), +which ensure that the treatment effect can be identified. +Assumption 1 Stable Unit Treatment Value Assumption +(SUTVA): The potential outcomes for any unit do not vary +with the treatments assigned to other units, and, for each +unit, there are no different forms or versions of each treat- +ment level, which lead to different potential outcomes. +Assumption 2 Consistency: The potential outcome of treat- +ment t is equal to the observed outcome if the actual treat- +ment received is t. +Assumption 3 Positivity: For any value of x, treatment as- +signment is not deterministic, i.e.,P(T = t|X = x) > 0, for +all t and x. +Assumption 4 Ignorability: Given covariates, treatment +assignment is independent of the potential outcomes, i.e., +(y1, y0) ⊥⊥ t|x. +Our goal is to develop a novel continual causal inference +framework to estimate the causal effect for all available data, +including new data Dd and the previous data D1:(d−1), with- +out having access to previous data D1:(d−1). +Research Challenges +Existing causal effect inference methods, however, are un- +able to deal with the aforementioned new challenges, i.e., +extensibility, adaptability, and accessibility. Although it is +possible to adapt existing causal inference methods to cater +to these issues, these adjusted methods still have inevitable +defects. Three straightforward adaptation strategies are de- +scribed as follows: +1. If we directly apply the model previously trained based +on original data to new observational data, the perfor- +mance on new tasks will be very poor due to the domain +shift issues among different data sources; +2. Suppose we utilize newly available data to re-train the +previously learned model for adapting changes in the +data distribution. In that case, old knowledge will be +completely or partially overwritten by the new one, +which can result in severe performance degradation on +old tasks. This is the well-known catastrophic forgetting +problem (McCloskey and Cohen 1989; French 1999); +3. To overcome the catastrophic forgetting problem, we +may rely on the storage of old data and combine the old +and new data together, and then re-train the model from +scratch. However, this strategy is memory inefficient and +time-consuming, and it brings practical concerns such as +copyright or privacy issues when storing data for a long +time (Samet, Miri, and Granger 2013). +Any of these three strategies, in combination with the exist- +ing causal effect inference methods, is deficient. +Potential Solution +To address the continual treatment effect estimation prob- +lem, we propose a Continual Causal Effect Representation +Learning framework (CERL) for estimating causal effect +with incrementally available observational data. Instead of +having access to all previous observational data, we only +store a limited subset of feature representations learned from +previous data. Combining selective and balanced represen- +tation learning, feature representation distillation, and fea- +ture transformation, our framework preserves the knowl- +edge learned from previous data and updates the knowledge +by leveraging new data so that it can achieve the continual +causal effect estimation for incrementally new data without +compromising the estimation capability for previous data. +Framework Overview. To estimate the incrementally +available observational data, the framework of CERL is +mainly composed of two components: (1) the baseline +causal effect learning model is only for the first available +observational data, and thus we don’t need to consider the +domain shift issue among different data sources. This com- +ponent is equivalent to the traditional causal effect estima- +tion problem; (2) the continual causal effect learning model +is for the sequentially available observational data, where +we need to handle more complex issues, such as knowledge +transfer, catastrophic forgetting, global representation bal- +ance, and memory constraint. +Baseline Causal Effect Learning Model. We first train +the baseline causal effect learning model for the initial obser- +vational dataset and then bring in subsequent datasets. The +task on the initial dataset can be converted to a traditional +causal effect estimation problem. Owing to the success +of deep learning for counterfactual inference, we propose + +to learn the selective and balanced feature representations +(Shalit, Johansson, and Sontag 2017; Chu, Rathbun, and Li +2020) for units in treatment and control groups and then in- +fer the potential outcomes based on learned representation +space. +Sustainability of Model Learning. We have built the +baseline model for causal effect estimation with observa- +tional data from a single source. To avoid catastrophic for- +getting when learning new data, we propose to preserve a +subset of lower-dimensional feature representations rather +than all original covariates. We also can adjust the number +of preserved feature representations according to the mem- +ory constraint. +Continual Causal Effect Learning. We have stored mem- +ory and the baseline model. To continually estimate the +causal effect for incrementally available observational data, +we incorporate feature representation distillation and feature +representation transformation (Chu et al. 2023) to estimate +the causal effect for all seen data based on a balanced global +feature representation space. +Research Opportunities +Although significant advances have been made to over- +come the challenges in causal effect estimation from an aca- +demic perspective, industrial applications based on obser- +vational data are always more complicated and harder. Un- +like source-specific and stationary observational data, most +real-world data are incrementally available and from non- +stationary data distributions. Significantly, we also face the +realistic consideration of accessibility. This work is the first +attempt to investigate the continual lifelong causal effect in- +ference problem and propose the corresponding evaluation +criteria. However, constructing the comprehensive analyt- +ical tools and the theoretical framework derived from this +brand-new problem requires non-trivial efforts. Specifically, +there are several potential directions for continual causal in- +ference: +• In addition to the distribution shift of the covariates +among different domains, there are other potential tech- +nical issues for continual effect estimation: for example, +perhaps we do not initially observe all the necessary con- +founding variables and may get access to increasingly +more confounders. +• Compared with homogeneous treatment effects (the +magnitude and direction of the treatment effect are the +same for all patients, regardless of any other patient char- +acteristics), heterogeneous causal effects could differ for +different individuals. This could be another candidate +to consider for the continual treatment effect estimation +model. +• The basic assumptions for traditional causal effect esti- +mation may not be completely applicable. New assump- +tions may be supplemented, or previous assumptions +need to be relaxed. +• There exists a natural connection with continual domain +adaptation among different times or domains (“contin- +ual” causal inference) and between treatment and control +groups (continual “causal inference”). +• Compared to traditional causal effect estimation tasks +based on a small amount of medical data, the continual +causal inference method will face big data computing or +cloud computing due to its objective task. +• With the increasing public concern over privacy leakage +in data, federated learning, which collaboratively trains +the machine learning model without directly sharing the +raw data among the data holders, may become a potential +solution for continual causal inference. +References +Athey, S.; and Imbens, G. 2016. Recursive partitioning for +heterogeneous causal effects. Proceedings of the National +Academy of Sciences, 113(27): 7353–7360. +Chu, Z.; Li, R.; Rathbun, S. L.; and Li, S. 2023. Continual +Causal Inference with Incremental Observational Data. In +The 39th IEEE International Conference on Data Engineer- +ing. +Chu, Z.; Rathbun, S. L.; and Li, S. 2020. Matching in se- +lective and balanced representation space for treatment ef- +fects estimation. In Proceedings of the 29th ACM Interna- +tional Conference on Information & Knowledge Manage- +ment, 205–214. +Chu, Z.; Rathbun, S. L.; and Li, S. 2021. Graph infomax +adversarial learning for treatment effect estimation with net- +worked observational data. In Proceedings of the 27th ACM +SIGKDD Conference on Knowledge Discovery & Data Min- +ing, 176–184. +Chu, Z.; Rathbun, S. L.; and Li, S. 2022. Learning Info- +max and Domain-Independent Representations for Causal +Effect Inference with Real-World Data. In Proceedings of +the 2022 SIAM International Conference on Data Mining +(SDM), 433–441. SIAM. +Feder, A.; Keith, K. A.; Manzoor, E.; Pryzant, R.; Sridhar, +D.; Wood-Doughty, Z.; Eisenstein, J.; Grimmer, J.; Reichart, +R.; Roberts, M. E.; et al. 2022. Causal inference in natural +language processing: Estimation, prediction, interpretation +and beyond. Transactions of the Association for Computa- +tional Linguistics, 10: 1138–1158. +French, R. M. 1999. Catastrophic forgetting in connectionist +networks. Trends in cognitive sciences, 3(4): 128–135. +Imbens, G. W.; and Rubin, D. B. 2015. Causal inference +in statistics, social, and biomedical sciences. +Cambridge +University Press. +K¨unzel, S. R.; Sekhon, J. S.; Bickel, P. J.; and Yu, B. +2019. Metalearners for estimating heterogeneous treatment +effects using machine learning. Proceedings of the national +academy of sciences, 116(10): 4156–4165. +Li, S.; and Fu, Y. 2017. Matching on balanced nonlinear +representations for treatment effects estimation. Advances +in Neural Information Processing Systems, 30. +Liu, B.; Wang, D.; Yang, X.; Zhou, Y.; Yao, R.; Shao, Z.; and +Zhao, J. 2022a. Show, Deconfound and Tell: Image Caption- +ing With Causal Inference. In Proceedings of the IEEE/CVF +Conference on Computer Vision and Pattern Recognition, +18041–18050. + +Liu, J.; Wei, W.; Chu, Z.; Gao, X.; Zhang, J.; Yan, T.; and +Kang, Y. 2022b. Incorporating Causal Analysis into Diversi- +fied and Logical Response Generation. In Proceedings of the +29th International Conference on Computational Linguis- +tics. International Committee on Computational Linguistics. +Louizos, C.; Shalit, U.; Mooij, J. M.; Sontag, D.; Zemel, R.; +and Welling, M. 2017. Causal effect inference with deep +latent-variable models. In Advances in Neural Information +Processing Systems, 6446–6456. +Ma, J.; Wan, M.; Yang, L.; Li, J.; Hecht, B.; and Teevan, J. +2022. Learning causal effects on hypergraphs. In Proceed- +ings of the 28th ACM SIGKDD Conference on Knowledge +Discovery and Data Mining, 1202–1212. +McCloskey, M.; and Cohen, N. J. 1989. Catastrophic inter- +ference in connectionist networks: The sequential learning +problem. In Psychology of learning and motivation, vol- +ume 24, 109–165. Elsevier. +Nie, X.; and Wager, S. 2021. +Quasi-oracle estimation of +heterogeneous treatment effects. Biometrika, 108(2): 299– +319. +Rubin, D. B. 1974. Estimating causal effects of treatments +in randomized and nonrandomized studies. Journal of edu- +cational Psychology, 66(5): 688. +Samet, S.; Miri, A.; and Granger, E. 2013. +Incremental +learning of privacy-preserving Bayesian networks. Applied +Soft Computing, 13(8): 3657–3667. +Shalit, U.; Johansson, F. D.; and Sontag, D. 2017. Estimat- +ing individual treatment effect: generalization bounds and +algorithms. In International Conference on Machine Learn- +ing, 3076–3085. PMLR. +Splawa-Neyman, J.; Dabrowska, D. M.; and Speed, T. 1990. +On the application of probability theory to agricultural ex- +periments. Essay on principles. Section 9. Statistical Sci- +ence, 465–472. +Tang, K.; Niu, Y.; Huang, J.; Shi, J.; and Zhang, H. 2020. +Unbiased scene graph generation from biased training. In +Proceedings of the IEEE/CVF conference on computer vi- +sion and pattern recognition, 3716–3725. +Wager, S.; and Athey, S. 2018. Estimation and inference of +heterogeneous treatment effects using random forests. Jour- +nal of the American Statistical Association, 113(523): 1228– +1242. +Yao, L.; Chu, Z.; Li, S.; Li, Y.; Gao, J.; and Zhang, A. 2021. +A survey on causal inference. ACM Transactions on Knowl- +edge Discovery from Data (TKDD), 15(5): 1–46. +Yao, L.; Li, S.; Li, Y.; Huai, M.; Gao, J.; and Zhang, A. 2018. +Representation learning for treatment effect estimation from +observational data. Advances in Neural Information Pro- +cessing Systems, 31. +Yoon, J.; Jordon, J.; and van der Schaar, M. 2018. GANITE: +Estimation of individualized treatment effects using genera- +tive adversarial nets. In International Conference on Learn- +ing Representations. +Zhang, J.; Zhang, J.; Ghosh, S.; Li, D.; Tasci, S.; Heck, L.; +Zhang, H.; and Kuo, C.-C. J. 2020. Class-incremental learn- +ing via deep model consolidation. In The IEEE Winter Con- +ference on Applications of Computer Vision, 1131–1140. + diff --git a/B9AzT4oBgHgl3EQfGPvM/content/tmp_files/load_file.txt b/B9AzT4oBgHgl3EQfGPvM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..11f959c4e2a337aee854632ef9ef20f0006d6fbe --- /dev/null +++ b/B9AzT4oBgHgl3EQfGPvM/content/tmp_files/load_file.txt @@ -0,0 +1,420 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf,len=419 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content='01026v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content='LG] 3 Jan 2023 Continual Treatment Effect Estimation: Challenges and Opportunities Zhixuan Chu1, Sheng Li2 1Ant Group, Hangzhou, China 2University of Virginia, Charlottesville, USA chuzhixuan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content='czx@alibaba-inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content='com, shengli@virginia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content='edu Introduction A further understanding of cause and effect within obser- vational data is critical across many domains, such as eco- nomics, health care, public policy, web mining, online ad- vertising, and marketing campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Although significant advances have been made to overcome the challenges in causal effect estimation with observational data, such as missing counterfactual outcomes and selection bias between treatment and control groups, the existing methods mainly focus on source-specific and stationary observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Such learning strategies assume that all observational data are already available during the training phase and from only one source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Along with the fast-growing segments of industrial appli- cations, this assumption is unsubstantial in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Taking Alipay as an example, which is one of the world’s largest mobile payment platforms and offers financial services to billion-scale users, a tremendous amount of data containing much privacy-related information is produced daily and col- lected from different sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' In conclusion, the following two points are summed up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' The first one is based on the characteristics of observational data, which are incremen- tally available from non-stationary data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' For instance, the electronic financial records for one marketing campaign are growing every day and they may be collected from different cities or even other countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' This character- istic implies that one cannot have access to all observational data at a one-time point and from one single source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' The sec- ond reason is based on the realistic consideration of accessi- bility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' For example, when new observational data are avail- able, if we want to refine the model previously trained by original data, maybe the original training data are no longer accessible due to a variety of reasons, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=', legacy data may be unrecorded, proprietary, the sensitivity of financial data, too large to store, or subject to privacy constraint of personal information (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' This practical concern of accessibility is ubiquitous in various academic and indus- trial applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' That’s what it boiled down to: in the era of big data, we face new challenges in causal inference with observational data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=', the extensibility for incrementally available observational data, the adaptability for extra do- Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' main adaptation problem except for the imbalance between treatment and control groups, and the accessibility for an enormous amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' In this position paper, we formally define the problem of continual treatment effect estimation, describe its research challenges, and then present possible solutions to this prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Moreover, we will discuss future research directions on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Related Work Instead of randomized controlled trials, observational data is obtained by the researcher simply observing the subjects without any interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' That means that the researchers have no control over the treatment assignments, and they just observe the subjects and record data based on their obser- vations (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Therefore, from the observational data, directly estimating the treatment effect is challenging due to the missing counterfactual outcomes and the exis- tence of confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Recently, powerful machine learning methods such as tree-based methods (Athey and Imbens 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Wager and Athey 2018), representation learning (Li and Fu 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Shalit, Johansson, and Sontag 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Chu, Rathbun, and Li 2022), meta-learning (K¨unzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Nie and Wager 2021), generative mod- els (Louizos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Yoon, Jordon, and van der Schaar 2018) have achieved prominent progress in treatment effect estimation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' In addition, the combination of causal inference and other research fields also exhibits complementary strengths, such as computer vision (Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2022a), graph learning (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Chu, Rathbun, and Li 2021), natural language processing (Feder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2022b), and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' The involved causal analysis helps to im- prove the model’s capability of discovering and resolving the underlying system beyond the statistical relationships learned from observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Problem Definition Suppose that the observational data contain n units collected from d different domains and the d-th dataset Dd contains the data {(x, y, t)|x ∈ X, y ∈ Y, t ∈ T } collected from d-th domain, which contains nd units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Let X denote all observed variables, Y denote the outcomes in the observational data, and T be a binary variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Let D1:d = {D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=', Dd} be the set of combination of d datasets, separately collected from d different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' For d datasets {D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=', Dd}, they have the commonly observed variables, but due to the fact that they are collected from different domains, they have different distributions with respect to X, Y , and T in each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Each unit in the observational data received one of two treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Let ti denote the treatment assignment for unit i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' For binary treatments, ti = 1 is for the treatment group and ti = 0 for the control group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' The outcome for unit i is denoted by yi t when treatment t is ap- plied to unit i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' For observational data, only one of the poten- tial outcomes is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' The observed outcome is called the factual outcome, and the remaining unobserved poten- tial outcomes are called counterfactual outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' This task can follow the potential outcome frame- work for estimating treatment effects (Rubin 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Splawa-Neyman, Dabrowska, and Speed 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' The indi- vidual treatment effect (ITE) for unit i is the difference be- tween the potential treated and control outcomes and is de- fined as ITEi = yi 1 − yi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' (1) The average treatment effect (ATE) is the difference be- tween the mean potential treated and control outcomes, which is defined as ATE = 1 n n � i=1 (yi 1 − yi 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' (2) The success of the potential outcome framework is based on the following assumptions (Imbens and Rubin 2015), which ensure that the treatment effect can be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Assumption 1 Stable Unit Treatment Value Assumption (SUTVA): The potential outcomes for any unit do not vary with the treatments assigned to other units, and, for each unit, there are no different forms or versions of each treat- ment level, which lead to different potential outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Assumption 2 Consistency: The potential outcome of treat- ment t is equal to the observed outcome if the actual treat- ment received is t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Assumption 3 Positivity: For any value of x, treatment as- signment is not deterministic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=',P(T = t|X = x) > 0, for all t and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Assumption 4 Ignorability: Given covariates, treatment assignment is independent of the potential outcomes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=', (y1, y0) ⊥⊥ t|x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Our goal is to develop a novel continual causal inference framework to estimate the causal effect for all available data, including new data Dd and the previous data D1:(d−1), with- out having access to previous data D1:(d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Research Challenges Existing causal effect inference methods, however, are un- able to deal with the aforementioned new challenges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=', extensibility, adaptability, and accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Although it is possible to adapt existing causal inference methods to cater to these issues, these adjusted methods still have inevitable defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Three straightforward adaptation strategies are de- scribed as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' If we directly apply the model previously trained based on original data to new observational data, the perfor- mance on new tasks will be very poor due to the domain shift issues among different data sources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Suppose we utilize newly available data to re-train the previously learned model for adapting changes in the data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' In that case, old knowledge will be completely or partially overwritten by the new one, which can result in severe performance degradation on old tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' This is the well-known catastrophic forgetting problem (McCloskey and Cohen 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' French 1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' To overcome the catastrophic forgetting problem, we may rely on the storage of old data and combine the old and new data together, and then re-train the model from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' However, this strategy is memory inefficient and time-consuming, and it brings practical concerns such as copyright or privacy issues when storing data for a long time (Samet, Miri, and Granger 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Any of these three strategies, in combination with the exist- ing causal effect inference methods, is deficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Potential Solution To address the continual treatment effect estimation prob- lem, we propose a Continual Causal Effect Representation Learning framework (CERL) for estimating causal effect with incrementally available observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Instead of having access to all previous observational data, we only store a limited subset of feature representations learned from previous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Combining selective and balanced represen- tation learning, feature representation distillation, and fea- ture transformation, our framework preserves the knowl- edge learned from previous data and updates the knowledge by leveraging new data so that it can achieve the continual causal effect estimation for incrementally new data without compromising the estimation capability for previous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Framework Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' To estimate the incrementally available observational data, the framework of CERL is mainly composed of two components: (1) the baseline causal effect learning model is only for the first available observational data, and thus we don’t need to consider the domain shift issue among different data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' This com- ponent is equivalent to the traditional causal effect estima- tion problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' (2) the continual causal effect learning model is for the sequentially available observational data, where we need to handle more complex issues, such as knowledge transfer, catastrophic forgetting, global representation bal- ance, and memory constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Baseline Causal Effect Learning Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' We first train the baseline causal effect learning model for the initial obser- vational dataset and then bring in subsequent datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' The task on the initial dataset can be converted to a traditional causal effect estimation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Owing to the success of deep learning for counterfactual inference, we propose to learn the selective and balanced feature representations (Shalit, Johansson, and Sontag 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Chu, Rathbun, and Li 2020) for units in treatment and control groups and then in- fer the potential outcomes based on learned representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Sustainability of Model Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' We have built the baseline model for causal effect estimation with observa- tional data from a single source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' To avoid catastrophic for- getting when learning new data, we propose to preserve a subset of lower-dimensional feature representations rather than all original covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' We also can adjust the number of preserved feature representations according to the mem- ory constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Continual Causal Effect Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' We have stored mem- ory and the baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' To continually estimate the causal effect for incrementally available observational data, we incorporate feature representation distillation and feature representation transformation (Chu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2023) to estimate the causal effect for all seen data based on a balanced global feature representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Research Opportunities Although significant advances have been made to over- come the challenges in causal effect estimation from an aca- demic perspective, industrial applications based on obser- vational data are always more complicated and harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Un- like source-specific and stationary observational data, most real-world data are incrementally available and from non- stationary data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Significantly, we also face the realistic consideration of accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' This work is the first attempt to investigate the continual lifelong causal effect in- ference problem and propose the corresponding evaluation criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' However, constructing the comprehensive analyt- ical tools and the theoretical framework derived from this brand-new problem requires non-trivial efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Specifically, there are several potential directions for continual causal in- ference: In addition to the distribution shift of the covariates among different domains, there are other potential tech- nical issues for continual effect estimation: for example, perhaps we do not initially observe all the necessary con- founding variables and may get access to increasingly more confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Compared with homogeneous treatment effects (the magnitude and direction of the treatment effect are the same for all patients, regardless of any other patient char- acteristics), heterogeneous causal effects could differ for different individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' This could be another candidate to consider for the continual treatment effect estimation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' The basic assumptions for traditional causal effect esti- mation may not be completely applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' New assump- tions may be supplemented, or previous assumptions need to be relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' There exists a natural connection with continual domain adaptation among different times or domains (“contin- ual” causal inference) and between treatment and control groups (continual “causal inference”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Compared to traditional causal effect estimation tasks based on a small amount of medical data, the continual causal inference method will face big data computing or cloud computing due to its objective task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' With the increasing public concern over privacy leakage in data, federated learning, which collaboratively trains the machine learning model without directly sharing the raw data among the data holders, may become a potential solution for continual causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' References Athey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and Imbens, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Recursive partitioning for heterogeneous causal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences, 113(27): 7353–7360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Chu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Rathbun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Continual Causal Inference with Incremental Observational Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' In The 39th IEEE International Conference on Data Engineer- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Chu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Rathbun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Matching in se- lective and balanced representation space for treatment ef- fects estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' In Proceedings of the 29th ACM Interna- tional Conference on Information & Knowledge Manage- ment, 205–214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Chu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Rathbun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Graph infomax adversarial learning for treatment effect estimation with net- worked observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' In Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Min- ing, 176–184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Chu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Rathbun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Learning Info- max and Domain-Independent Representations for Causal Effect Inference with Real-World Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' In Proceedings of the 2022 SIAM International Conference on Data Mining (SDM), 433–441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' SIAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Feder, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Keith, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Manzoor, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Pryzant, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Sridhar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Wood-Doughty, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Eisenstein, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Grimmer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Reichart, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Roberts, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Causal inference in natural language processing: Estimation, prediction, interpretation and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Transactions of the Association for Computa- tional Linguistics, 10: 1138–1158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' French, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Catastrophic forgetting in connectionist networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Trends in cognitive sciences, 3(4): 128–135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Imbens, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and Rubin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Causal inference in statistics, social, and biomedical sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' K¨unzel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Sekhon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Bickel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and Yu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Metalearners for estimating heterogeneous treatment effects using machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Proceedings of the national academy of sciences, 116(10): 4156–4165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and Fu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Matching on balanced nonlinear representations for treatment effects estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Liu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Zhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Yao, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Shao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and Zhao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2022a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Show, Deconfound and Tell: Image Caption- ing With Causal Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 18041–18050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Wei, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Chu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Gao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Yan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and Kang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2022b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Incorporating Causal Analysis into Diversi- fied and Logical Response Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' In Proceedings of the 29th International Conference on Computational Linguis- tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' International Committee on Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Louizos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Shalit, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Mooij, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Sontag, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Zemel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and Welling, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Causal effect inference with deep latent-variable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, 6446–6456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Ma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Wan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Yang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Hecht, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and Teevan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Learning causal effects on hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' In Proceed- ings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, 1202–1212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' McCloskey, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and Cohen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Catastrophic inter- ference in connectionist networks: The sequential learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' In Psychology of learning and motivation, vol- ume 24, 109–165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Nie, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and Wager, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Quasi-oracle estimation of heterogeneous treatment effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Biometrika, 108(2): 299– 319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Rubin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Estimating causal effects of treatments in randomized and nonrandomized studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Journal of edu- cational Psychology, 66(5): 688.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Samet, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Miri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and Granger, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Incremental learning of privacy-preserving Bayesian networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Applied Soft Computing, 13(8): 3657–3667.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Shalit, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Johansson, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and Sontag, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Estimat- ing individual treatment effect: generalization bounds and algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' In International Conference on Machine Learn- ing, 3076–3085.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Splawa-Neyman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Dabrowska, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and Speed, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' On the application of probability theory to agricultural ex- periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Essay on principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Statistical Sci- ence, 465–472.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Tang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Niu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Unbiased scene graph generation from biased training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF conference on computer vi- sion and pattern recognition, 3716–3725.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Wager, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and Athey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Estimation and inference of heterogeneous treatment effects using random forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Jour- nal of the American Statistical Association, 113(523): 1228– 1242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Yao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Chu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Gao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' A survey on causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' ACM Transactions on Knowl- edge Discovery from Data (TKDD), 15(5): 1–46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Yao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Huai, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Gao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Representation learning for treatment effect estimation from observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Advances in Neural Information Pro- cessing Systems, 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Yoon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Jordon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and van der Schaar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' GANITE: Estimation of individualized treatment effects using genera- tive adversarial nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' In International Conference on Learn- ing Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Ghosh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Li, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Tasci, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Heck, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' and Kuo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' Class-incremental learn- ing via deep model consolidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} +page_content=' In The IEEE Winter Con- ference on Applications of Computer Vision, 1131–1140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'} diff --git a/BNE2T4oBgHgl3EQfRgdU/content/2301.03781v1.pdf 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Groves1★, K. Kreckel2, F. Santoro3, F. Belfiore4, E. Zavodnik2, E. Congiu5, O. V. Egorov2, +E. Emsellem6, K. Grasha7, A. Leroy8, F. Scheuermann2, E. Schinnerer3, E. J. Watkins2, A. T. Barnes9, +F. Bigiel9, D. A. Dale10, S. C. O. Glover11, I. Pessa3, P. Sanchez-Blazquez12,13, +and T. G. Williams3 +1International Centre for Radio Astronomy Research, University of Western Australia, 7 Fairway, Crawley, 6009 WA, Australia +2Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Mönchhofstraße 12-14, 69120 Heidelberg, Germany +3Max Planck Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany +4INAF – Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50157 Firenze, Italy +5 Departamento de Astronomía, Universidad de Chile, Camino del Observatorio 1515, Las Condes, Santiago, Chile +6European Southern Observatory, Karl-Schwarzschild Straße 2, D-85748 Garching bei München, Germany +7Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia +8Department of Astronomy, The Ohio State University, 140 West 18th Avenue, Columbus, Ohio 43210, USA +9Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany +10Department of Physics & Astronomy, University of Wyoming, Laramie, WY, 82071, USA +11Universität Heidelberg, Zentrum für Astronomie, Institut für theoretische Astrophysik, Albert-Ueberle-Str. 2, 69120 Heidelberg, Germany +12Departamento de Física de la Tierra y Astrofísica, Universidad Complutense de Madrid, E-28040 Madrid, Spain +13Instituto de Física de partículas y del Cosmos, IPARCOS, E-28040 Madrid, Spain +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +Ionized nebulae provide critical insights into the conditions of the interstellar medium (ISM). Their bright emission lines enable +the measurement of physical properties, such as the gas-phase metallicity, across galaxy disks and in distant galaxies. The +PHANGS–MUSE survey has produced optical spectroscopic coverage of the central star-forming discs of 19 nearby main- +sequence galaxies. Here, we use the H𝛼 morphology from this data to identify 30,790 distinct nebulae, finding thousands of +nebulae per galaxy. For each nebula, we extract emission line fluxes and, using diagnostic line ratios, identify the dominant +excitation mechanism. A total of 23,244 nebulae (75%) are classified as H ii regions. The dust attenuation of every nebulae is +characterised via the Balmer decrement and we use existing environmental masks to identify their large scale galactic environment +(centre, bar, arm, interarm and disc). Using strong-line prescriptions, we measure the gas-phase oxygen abundances (metallicity) +and ionization parameter for all H ii regions. With this new catalogue, we measure the radial metallicity gradients and explore +second order metallicity variations within each galaxy. By quantifying the global scatter in metallicity per galaxy, we find a weak +negative correlation with global star formation rate and stronger negative correlation with global gas velocity dispersion (in both +ionized and molecular gas). With this paper we release the full catalogue of strong line fluxes and derived properties, providing +a rich database for a broad variety of ISM studies. +Key words: galaxies:ISM – H ii Regions – galaxies: abundances +1 INTRODUCTION +Emission lines from ionized nebulae play a fundamental role in our +understanding of galaxy evolution. Apart from their use in deter- +mining spectroscopic redshifts, emission lines have been used to +determine star-formation rates, the presence of active galactic nuclei +(AGN), galaxy dynamics, gas-phase metal abundances and more (see +e.g. Kewley et al. 2019). However, in most extragalactic studies, the +emission lines have been typically measured from a single spectrum +of the entire galaxy (e.g. SDSS, Abazajian et al. 2009; GAMA, Driver +et al. 2009; VVDS, Le Fevre et al. 2003). Resolved maps of nearby +★ email:brent.groves@uwa.edu.au +emission line galaxies in strong spectral lines has been available +with narrow filters, slit spectra on bright H ii regions (e.g. Croxall +et al. 2009; Moustakas et al. 2010) and even spectrally resolved with +Fabry-Perot surveys (e.g. Veilleux 2002; Epinat et al. 2008; Moiseev +et al. 2015; Sil’chenko et al. 2019), but it is only with the advent +of integral field spectrographs (IFS) that maps of multiple emission +lines across thousands of galaxies have become common, with most +achieving kpc scale sampling (e.g. CALIFA, Sánchez et al. 2012; +VENGA, Blanc et al. 2013; MaNGA, Bundy et al. 2015; SAMI, +Croom et al. 2021). +However, the individual ionized nebulae that the emission lines +originate from (e.g. H ii regions, supernova remnants, planetary neb- +ulae) are typically <100 pc in size. This means that in these large +© 2022 The Authors +arXiv:2301.03811v1 [astro-ph.GA] 10 Jan 2023 + +2 +Groves et al. +surveys, the nebulae cannot be spatially distinguished from each other +or the faint surrounding diffuse ionized gas (DIG; Reynolds 1990). +While there have been efforts to account for the DIG (e.g. Zhang +et al. 2017; Espinosa-Ponce et al. 2020), only in nearby galaxies (or +rare lensed systems) are we able to achieve the 10–100 pc resolution +required to separate out the individual ionized nebula (i.e. H ii re- +gions; Kennicutt 1989). Once imaged in multiple emission lines, it is +then possible to use line ratios to distinguish nebular emission in H ii +regions from other nebular sources (e.g. supernovae (SNe), planetary +nebulae (PNe) Ciardullo et al. 2002; Smith et al. 2005; Long et al. +2010). Long-slit studies have compiled growing samples of emission +line spectroscopy for H ii regions (e.g. Pilyugin et al. 2014; Berg et al. +2020), yet generally these studies pre-select for the brightest regions, +which introduces biases in the sampling across galaxy disks. +With the advent of new instruments and approaches, wide-area +and high angular resolution spectral maps are now available. This +is made possible through long-slit spectral stepping (e.g. the TY- +PHOON survey Ho et al. 2017), imaging Fourier transform spec- +trographs (e.g. the SIGNALS survey with SITELLE on the CFHT; +Rousseau-Nepton et al. 2019) or IFS (e.g. MAD or TIMER on the +MUSE/VLT; Erroz-Ferrer et al. 2019; Gadotti et al. 2019). Such +maps not only make it possible for individual nebulae to be identi- +fied, but the integrated emission line fluxes within each nebula mean +that they can be classified and their key physical properties measured. +Furthermore, sensitive data can also examine the properties of the +intervening diffuse ionized gas. Full spectral maps can also be used +to understand the properties of the underlying stellar populations, +including the stars that are the potential sources of ionizing photons. +Surveys using spectral maps have already identified leaking radiation +from H ii regions as the dominant ionizing source of the DIG (Della +Bruna et al. 2020; Belfiore et al. 2022), quantified the impact of the +DIG on the measurement of the gas-phase metallicity (Poetrodjojo +et al. 2019), and determined that metallicity variations exist within +galaxies on top of the well known radial gradients (Ho et al. 2017; +Kreckel et al. 2019; Metha et al. 2021; Sánchez-Menguiano et al. +2020; Williams et al. 2022). +It is within this context that we describe here the PHANGS–MUSE +Nebulae Catalogue of over 30,000 nebulae across 19 galaxies, the +largest catalogue of high-resolution (< 100 pc) extragalactic nebulae +with homogeneous optical spectroscopic coverage currently avail- +able. First introduced by Santoro et al. (2022) when fitting for the +H ii region luminosity functions, the nebula catalogue has already +been used in a number of papers for a range of science topics, in- +cluding; quantifying the pre-supernova feedback within 6000 of the +H ii regions (Barnes et al. 2021), modelling of line ratios for diffuse +ionized gas surrounding these H ii regions (Belfiore et al. 2022), and +interpolating metallicities measured at each H ii region to construct +full coverage metallicity maps (Williams et al. 2022). With this paper +we release a full catalogue of emission line properties and derived +physical properties associated with the objects. +This catalogue is based on the mosaicked MUSE IFS observations +from the Physics at High-Angular resolution in Nearby GalaxieS +program (PHANGS–MUSE survey Emsellem et al. 2022), which +are drawn from the larger PHANGS survey (Leroy et al. 2021, +PHANGS)1. We summarise this survey and the specific subsam- +ple targeted with MUSE in Section 2. We present our methods of +constructing our catalogue of nebular emitting objects in Section 3. +We describe how derived properties are obtained and included in the +catalogue as value-added products in Section 4. We present results +1 http://www.phangs.org +focused on the metallicity measurements in those objects classified as +H ii regions in Section 5. We discuss the interpretation of the metal- +licity variation we observe in Section 6, as well as other technical +aspects of our catalogue, and conclude in Section 7. +2 THE PHANGS–MUSE SURVEY +The PHANGS survey was designed specifically to resolve galaxies +into the individual elements of the star-formation process: molec- +ular clouds, H ii regions, and stellar clusters. Driven by this aim, +the full PHANGS sample was originally determined by select- +ing southern-sky accessible (−75◦ ≤ 𝛿 ≤ +25◦, for ALMA & +MUSE), low inclination (𝑖 < 75◦), massive star-forming galax- +ies (log(𝑀∗/𝑀⊙) > 9.75 and log(𝑠𝑆𝐹𝑅/yr−1) > −11) within +∼ 23 Mpc, such that 1” < 100 pc (described in full detail Leroy +et al. 2021, as the PHANGS–ALMA sample). In addition to ALMA +data, there is a wealth of data from other telescopes such as the Hub- +ble Space Telescope (the PHANGS-HST survey Lee et al. 2022), +and the MUSE IFS on ESO’s VLT (Bacon et al. 2010), known as the +PHANGS–MUSE survey (Emsellem et al. 2022). +The PHANGS–MUSE survey is an ESO large program (∼ 170 h, +PI Schinnerer) aimed at spectroscopically mapping the discs of 19 +nearby star-forming galaxies. This subsample of PHANGS (orig- +inally selected to align with the PHANGS–ALMA pilot surveys) +covers a broad range in stellar mass, but is biased somewhat to main- +sequence massive galaxies (Table 1). Further details on the sample, +observations, reduction and MUSE data products are described in +Emsellem et al. (2022). Here we summarise the sample and data, and +refer the reader to Emsellem et al. (2022) for the full details. Public +data products, including data cubes and line maps, are available at +the ESO archive 2 and CADC3. +The physical properties of these galaxies are listed in Table 1. As +all our galaxies are within ∼20 Mpc, the typical seeing (0.′′91) of +our MUSE observations means that all structures down to 100 pc +can be isolated within the disk environment, with a median physi- +cal resolution of ∼70 pc. The galaxy distances we use are the latest +compilation from Anand et al. (2021), including new tip of the red +giant branch distances from the PHANGS–HST observations and +new planetary nebula luminosity function distances measured from +the PHANGS–MUSE data itself (Scheuermann et al. 2022). The in- +clination and position angle of the galaxies were determined by Lang +et al. (2020) from the PHANGS ALMA CO rotation curves analysis, +or near-IR imaging when the CO data was not available or the fit +to the CO velocity field was deemed unreliable. The position angle +and inclinations were used to determine the deprojected radial dis- +tances used in this paper. The listed stellar masses and star formation +rates are global measures from UV and IR photometry (Leroy et al. +2019). Representative disc scale lengths are provided, both the 25th +magnitude B-band isophotal radius (R25) from RC3 (de Vaucouleurs +et al. 1991 via HyperLEDA; Makarov et al. 2014) and the effective +radius containing half of the stellar mass of the galaxy (reff). These +quantities are compiled and computed in Leroy et al. (2021). +Due to their proximity, the stellar structures of the galaxies, such +as spiral arms and bars, are clearly resolved. Within this work we +use the structural morphology masks created by Querejeta et al. +(2021) using Spitzer 3.6 𝜇m imaging. Querejeta et al. (2021) used +2 https://archive.eso.org/scienceportal/home?data_ +collection=PHANGS +3 https://www.canfar.net/storage/vault/list/phangs/ +RELEASES/PHANGS-MUSE +MNRAS 000, 1–27 (2022) + +PHANGS Nebular Catalogue +3 +photometric fitting that decomposes the galaxies into bulges and +discs (Salo et al. 2015), then further divided the structures visually +and through fitting into centres, bars (with the bar mask defined +by a fitted ellipse), rings and lenses, and spiral arms. Spiral arms +were fitted as logarithmic spiral curves with widths fitted to both the +stellar and molecular gas surface density, and the inter-arm region +was considered to be any region in the disc outside of these. We +follow Querejeta et al. (2021) in distinguishing between inner bar +and outer disc regions when considering the arm and inter-arms and +use their notations (their Figure 2 and Table 1, respectively). These +morphological masks allow us to determine the influence of local +environment on the nebulae. Given our bias towards massive main +sequence galaxies, spiral or disk features are seen in all, and only +four out of the 19 galaxies are not barred. +The PHANGS–MUSE large program was observed over several +semesters and includes data from other programs and includes MUSE +observations in both ground layer AO (adaptive optics) and non AO +mode. Combined with variations in seeing, this means that the point +spread function (PSF) varied between galaxies and among point- +ings within the same galaxy. To account for this variation between +pointings in the same galaxy, we created mosaicked datacubes with a +consistent PSF, where all pointings in a single galaxy were convolved +to a single Gaussian PSF, whose size was determined by the pointing +with the worst (largest) PSF. We used this optimised convolution data +(copt) to identify the nebulae for the catalogue. The consistent PSF +across the mosaic avoids issues with variable nebulae sizes across a +single galaxy. +The mosaicked MUSE datacubes were then passed through a data +analysis pipeline (DAP) to provide maps of value-added products +such as emission lines, mean stellar properties, gas and stellar kine- +matics and more (as detailed in Emsellem et al. 2022). As the emis- +sion lines form the key data for this paper, we briefly describe the +analysis here. The DAP uses the penalised pixel fitting method (pPXF +Cappellari 2017) to derive both the stellar continuum and emission +lines properties within the spectral range 4850–7000 Å. F Before any +fitting, the MUSE data is corrected for foreground Galactic extinc- +tion, using the Cardelli et al. (1989) extinction law and the 𝐸(𝐵 −𝑉) +attributed to the Milky Way foreground from Schlafly & Finkbeiner +(2011). +To fit the stellar continuum and derive the stellar properties, the +datacubes are first spatially Voronoi-binned (using the vorbin pack- +age Cappellari & Copin 2003) to achieve a minimum S/N of 35 +in the 5300 − 5500 Å wavelength range. The continuum between +4850−7000 Å in each bin is then fit with a combination of E-MILES +simple stellar population model templates (Vazdekis et al. 2016) +generated with a Chabrier (2003) initial mass function and BaSTI +isochrones (Pietrinferni et al. 2004). The Na I D absorption dou- +blet (already removed in AO observations) are masked. The higher +spectral resolution templates are convolved to the resolution of the +MUSE data before fitting. The spectra is first fit to determine the +stellar kinematics using a smaller set of model templates sampled at +eight ages (0.15 − 14 Gyr, logarithmically sampled in steps of 0.22 +dex), and four metallicities ([Z/H] = −1.5; −0.35; −0.06; 0.4]). To +the fit for the stellar population parameters we fix the kinematics and +use a larger set of templates sampled at ages = [0.03, 0.05, 0.08, +0.15, 0.25, 0.40, 0.60, 1.0, 1.75, 3.0, 5.0, 8.5, 13.5] Gyr and [Z/H] = +[−1.49, −0.96, −0.35, +0.06, +0.26, +0.4]. When fitting for the stel- +lar population properties we also constrain the average attenuation +of the stellar continuum, parametrized by the Calzetti (2001) curve. +To fit the emission lines we rerun pPXF on the mosaicked cubes +at an individual spaxel level, with the emission lines treated as ad- +ditional Gaussian components. The underlying stellar continuum is +fit using the smaller set of E-MILES templates and the derived kine- +matics of the Voronoi bin that contained the individual spaxel, with +the inclusion of an 8th-order multiplicative polynomial. We fit all +strong emission lines and tie the kinematics (velocity and velocity +dispersion) in three groups; Hydrogen lines (H𝛼, H𝛽), low ioniza- +tion lines (e.g., [N ii]𝜆6583, [S ii]𝜆𝜆6716, 6731), and high ionization +lines (e.g., [O iii]𝜆5007, [S iii]𝜆6312). These maps of stellar kine- +matics and mean properties, emission line fluxes, and gas kinematics +form the key part of the data analysis pipeline and the PHANGS– +MUSE release, as described in Emsellem et al. (2022). +3 METHODS +3.1 Nebular catalogue construction +The PHANGS–MUSE galaxies are replete with emission lines, with +more than 95% of our 0.′′2 spaxels within 0.5𝑅25 containing H𝛼 +emission at a > 3𝜎 level (see figure 20 in Emsellem et al. 2022). +With such filled maps, distinguishing individual nebulae from each +other and the diffuse ionized gas is difficult, even with a median +physical resolution of 70 pc. Therefore, to identify the nebulae we +require an unbiased and robust region identifier. While several such +methods exist and have been applied previously (e.g., Clumpfind; +Williams et al. (1994); Kreckel et al. (2016) or pyHIIExtractor; Lugo- +Aranda et al. (2022)), we chose to use HIIphot, a code specifically +built to identify and characterise H ii regions with their irregular +morphology (Thilker et al. 2000). We use a slightly altered version +HIIphot to work on the H𝛼 maps created from IFS data, first used in +Kreckel et al. (2019). +Originally designed to be applied to narrow band imaging data +centred on the H𝛼 line, HIIphot used the associated broad band +data used for continuum subtraction from the narrow band data to +determine the significance of the H𝛼 detection. However, IFS can +spectrally resolve any underlying stellar continuum and subtract this +as done within the data analysis pipeline. Therefore HIIphot was +altered to work on H𝛼 maps alone with the associated fitting error +map to identify the nebulae and determine their boundaries. However +the main algorithm in nebulae identification is still as described in +Thilker et al. (2000). +The key to nebulae identification is to first distinguish individual +nebulae, then grow these up to a given termination criterion defining +the edges of the nebulae. While a classical photoionized nebula has +a clear boundary defined by the edge of the Strömgren sphere, real +nebulae may be centrally concentrated or appear as rings, or have +several peaks and a diffuse boundary due to density variations within +the ISM. The angular resolution of our observations means that we +only resolve the largest of H ii region complexes. In most H ii regions +our resolution smooths any features and boundaries and, a potentially +larger problem we discuss in Section 6.4, merge proximate nebulae. +Therefore the choice of controlling parameters is driven by both the +dataset and the physics of nebulae. +As described in Santoro et al. (2022), to identify the nebulae we +first require to detect the peaks in H𝛼 emission, or ‘seed regions’, +above the diffuse background. We set this background for each galaxy +to be the median of all H𝛼 pixels within the MUSE FoV with ΣH𝛼 < +1 × 10−17 erg s−1 cm−2 arcsec−2. This ranges from ΣH𝛼 = 1 to +3 × 10−18 erg s−1 cm−2 arcsec−2 across our sample. The detection +threshold within HIIphot was set to 3𝜎 above this background, where +𝜎 is the standard deviation of the background pixels and typically +around the same level as the background. +Given the diverse morphologies of H ii regions (and other ionized +MNRAS 000, 1–27 (2022) + +4 +Groves et al. +Table 1. General properties of the PHANGS-MUSE galaxies. +Name +Distance𝑎 +𝑣𝑏 +sys +𝑃𝐴𝑐 +𝑖𝑐 +log10 𝑀 𝑑 +∗ +𝑅𝑏 +25 +𝑟eff +𝐸 (𝐵 − 𝑉 )𝑒 +MW +resolution +Mpc +km s−1 +deg +deg +M⊙ +arcmin +arcmin +mag +pc +IC5332 +9.0 +699 +74.4 +26.9 +9.67 +3.0 +1.4 +0.014 +45 +NGC0628 +9.8 +651 +20.7 +8.9 +10.34 +4.9 +1.4 +0.061 +42 +NGC1087 +15.9 +1502 +359.1 +42.9 +9.93 +1.5 +0.7 +0.030 +71 +NGC1300 +19.0 +1545 +278.0 +31.8 +10.62 +3.0 +1.2 +0.026 +62 +NGC1365∗ +19.6 +1613 +201.1 +55.4 +10.99 +6.0 +3.3 𝑓 +0.018 +84 +NGC1385 +17.2 +1477 +181.3 +44.0 +9.98 +1.7 +0.7 +0.017 +96 +NGC1433∗ +18.6 +1057 +199.7 +28.6 +10.87 +3.1 +0.8 +0.008 +83 +NGC1512 +18.8 +871 +261.9 +42.5 +10.71 +4.2 +0.9 +0.009 +96 +NGC1566∗ +17.7 +1483 +214.7 +29.5 +10.78 +3.6 +0.6 +0.008 +76 +NGC1672∗ +19.4 +1318 +134.3 +42.6 +10.73 +3.1 +0.6 +0.020 +73 +NGC2835 +12.2 +867 +1.0 +41.3 +10.00 +3.2 +0.9 +0.086 +33 +NGC3351 +10.0 +775 +193.2 +45.1 +10.36 +3.6 +1.0 +0.024 +43 +NGC3627∗ +11.3 +715 +173.1 +57.3 +10.83 +5.1 +1.1 +0.029 +69 +NGC4254 +13.1 +2388 +68.1 +34.4 +10.42 +2.5 +0.6 +0.033 +61 +NGC4303∗ +17.0 +1560 +312.4 +23.5 +10.52 +3.4 +0.7 +0.019 +96 +NGC4321 +15.2 +1572 +156.2 +38.5 +10.75 +3.0 +1.2 +0.023 +59 +NGC4535 +15.8 +1954 +179.7 +44.7 +10.53 +4.1 +1.4 +0.017 +80 +NGC5068 +5.2 +667 +342.4 +35.7 +9.40 +3.7 +1.3 +0.090 +23 +NGC7496∗ +18.7 +1639 +193.7 +35.9 +10.00 +1.7 +0.7 +0.008 +104 +𝑎From the compilation of Anand et al. (2021). 𝑏From LEDA (Makarov et al. 2014). 𝑐From Lang et al. (2020), based on CO(2–1) +kinematics. 𝑑Derived by Leroy et al. (2021), using GALEX UV and WISE IR photometry. 𝑒From Schlafly & Finkbeiner (2011). +𝑓 Due to AGN bias, derived from the scale length (l∗) as reff = 1.41 l∗ following Equation 5 in Leroy et al. (2021). ∗Classified as +an AGN by Véron-Cetty & Véron (2010) +nebulae), HIIphot performs iterative Gaussian smoothing on the H𝛼 +maps, merging connecting features to create the nebulae ‘footprints’. +To avoid the detection of regions with unphysical sizes, we limit +the spatial smoothing to three iterations, each time increasing the +smoothing kernel (starting from the original resolution image) by +10%. These footprints are then further trimmed to a ‘seed’ with a +consistent isophotal boundary defined by 50% of the median within +the footprint. Once detected, we further cleaned the seed sample to +avoid artefacts due to noise by imposing a S/N cut of 50 above the +H𝛼 error maps for the integrated flux values. +Defining the boundary edges of nebulae is challenging, with many +criteria existing in the literature (e.g. H𝛼 surface brightness, line +ratios, H𝛼 equivalent widths). By using HIIphot we chose to use the +spatial gradient of the H𝛼 surface brightness to define the boundaries +of our nebulae. As discussed in Thilker et al. (2000), the choice of +terminal gradient is ambiguous, with flatter values leading to larger +H ii regions that can include the diffuse ionized gas directly associated +with the H ii region (see, e.g. Belfiore et al. 2022, for the level of +association) but also lead to a more contiguous map of nebulae +(Figure 5 in Thilker et al. 2000). The spatial resolution of the data +also impacts the exact boundaries, smoothing edges and potentially +merging adjoining nebulae. We chose to use a single termination +gradient of 5.0 EM pc−1 (where the emission measure, EM, is in +cm−6 pc) for all galaxies (corresponding to 2.43 × 10−16 erg s−1 +arcsec−2 pc−1). This value is similar to that used in other nearby +galaxy studies (Oey et al. 2007; Zhang et al. 2017, e.g.), and visually +provided the best balance in terms of capturing the total H𝛼 flux for +each nebula, while limiting the size growth. We chose to use a single +termination gradient rather than one for each galaxy for consistency +in nebulae identification, even given the factor of ∼4 difference in +physical resolution across the PHANGS–MUSE sample. +These steps lead to 31,497 identified nebulae with defined bound- +aries across our sample of 19 galaxies. For each nebula we report +the central position in both RA and Dec, weighted by H𝛼 intensity, +and their position relative to the galaxy centres. We quantify the area +encompassed by the nebulae in pixels, however as most of our re- +gions are unresolved or only marginally resolved (see also Section +3.2), we do not provide size measurements (though see Section 4.2 +where we do present 10%, 50%, and 90% circularized radial sizes +for the overall distributions in each galaxy). As described in Em- +sellem et al. (2022, particularly §5.3), foreground star masks were +generated for all PHANGS–MUSE galaxies based on the Gaia DR2 +catalogue (Gaia Collaboration et al. 2018). We exclude 98 sources +whose footprint falls within the star masks and are likely impacted +by artefacts from incorrect stellar continuum subtraction. We also +flag 609 nebulae with centres within 1 PSF FWHM of the edges of +our PHANGS–MUSE galaxy footprints. While the emission lines in +these regions are likely correctly measured, their proximity to the +edges mean that their boundaries are potentially incorrectly defined +and that their integrated line fluxes may not represent the total emis- +sion of the nebulae. For larger nebulae where the distinct ionized +zones can be distinguished (i.e. the S++ and S+ zones are resolved), +the emission line ratios measured for these regions are potentially +incorrect. While these flagged nebulae are included in the full cat- +alogue, we exclude them from our further analysis. For the results +presented in this paper, we focus on the remaining 30,790 nebulae +(Figure 1). +Spatial masks corresponding to the locations of each identified +nebula are released as data products accompanying this paper, and +can also be found via the PHANGS webpage4. An image atlas show- +ing the footprints of all nebulae in each galaxy is included in Ap- +pendix A. +3.2 Emission line measurements +With the footprints of all nebulae defined, we integrated the original +MUSE spectra within each nebula and re-fit using the same data +4 https://sites.google.com/view/phangs/home/data +MNRAS 000, 1–27 (2022) + +PHANGS Nebular Catalogue +5 +Figure 1. A visualisation of the two dimensional spatial extent and distribution of all nebulae in the galaxy NGC 1433. The background greyscale image shows +the H𝛼 emission in log scale, and the colour of the nebulae indicates their intrinsic (dust corrected) H𝛼 luminosity. A full atlas of each galaxy is available in +Appendix A. +analysis pipeline (DAP) described in Emsellem et al. (2022) used +to create the original H𝛼 maps. We do this to increase the signal- +to-noise of our emission lines and to detect faint spectral features, +such as the temperature sensitive auroral lines. The only changes +to the pipeline are to extend the wavelength range fitted to include +the [S iii]𝜆9069 Å emission line and, when integrating the spectra, +we use the unconvolved, native resolution mosaicked data cubes +to minimise the impact of PSF smearing at nebula boundaries. The +latter only has a small impact due to the consistency in seeing between +observations, but for some galaxies (e.g. NGC1365, as seen in Table +A1 in Emsellem et al. 2022) the PSF variation can be a factor of two +within the different pointings of the mosaic. +As with the global PHANGS–MUSE DAP, we fit all emis- +sion lines simultaneously, but also include lines that are fainter +(e.g. [N ii]𝜆5754) and at longer wavelengths (i.e. [S iii]𝜆9069). The +full list of lines released in this catalogue is given in Table 2. Sim- +ilarly, when fitting lines we assume single Gaussian profiles and tie +the kinematics (velocity and velocity dispersion) in three groups: +hydrogen lines, low ionization lines, and high ionization lines. Line +velocities are reported relative to the systemic velocity of the galaxy, +provided in Table 1. In our fit we account for a Milky Way foreground +extinction, assuming the 𝐸(𝐵 − 𝑉) values provided by Schlafly & +Finkbeiner (2011) (also listed in Table 1) and an O’Donnell (1994) +extinction law. +In Figure 2, we show a typical nebular spectrum from IC5332 +(ID:38, 𝐿H𝛼 = 1036.8 erg s−1), along with the best fitted spectrum +from the data analysis pipeline overlaid, and the relative residual +from the fit at the bottom. What is clear from this spectrum are the +emission lines, and how well we reproduce these. Also clear are the +strong sky line residuals (especially beyond 6800Å). The underlying +continuum is dominated by stellar light, although at the scaling in +this figure, only certain stellar absorption features are visible. +As an insert we show a zoom-in of the H𝛼 region of the spec- +trum with the strong [N ii] and [S ii] lines. Also shown is the stellar +continuum fit in green, clearly demonstrating the H𝛼 absorption +feature (H𝛼𝑎𝑏𝑠). In the brightest nebulae, weaker lines such as the +[O ii]𝜆7319, 7330Å doublet are clearly visible. In some nebulae, faint +residuals around bright lines are visible, suggesting more complex +kinematics than can be modelled by a single Gaussian component. +However typically these residuals are still at the level of the spectral +uncertainty propagated from the MUSE spectral cubes (cyan lines in +residual plot). In the example shown in Figure 2, 89% of all pixels +shown have residuals within 3𝜎 of the errors, with sky residuals +dominating the outlying pixels. +We also determine the Balmer emission line equivalent width +(EW), as both an indication of the relative brightness of the nebula +to the underlying stellar population, and a proxy for the local spe- +cific star formation rate surface density. We calculate EW(H𝛼) and +EW(H𝛽) following the procedure described in Westfall et al. (2019), +as applied within the MaNGA survey for calculation of emission-line +moments. For EW(H𝛼), we integrate the flux over a central band from +6557.6 – 6571.35Å in the rest frame. We calculate the continuum flux +using determining the median over a blue (6483.0–6513.0Å) and red +(6623.0–6653.0Å) channel and then determine the mean between +these. For EW(H𝛽), we use the same approach, with the H𝛽 line +determined over the 4847.9 – 4876.6Å band and the continuum flux +using 4827.9 – 4847.9Å and 4876.6 – 4891.6Å for the blue and red +channels. We calculate the equivalent widths in two ways. The first +approach is to use the direct nebulae spectrum, with the line flux +simply the integral over the central band minus the determined con- +tinuum, what we call here the ‘raw’ EW. However, this ignores the +impact of the underlying stellar absorption feature visible in Figure +2. The second method is to integrate over the spectrum once the +best-fitting continuum fit from pPXF has been subtracted or the ‘fit’ +MNRAS 000, 1–27 (2022) + +-47°12'00" +39 +30" +13'00" +38 +Dec +30" +37 +14'00" +0.C +30" +36 +3h42m12s +06s +00s +41m54s +48s +R.A. +356 +Groves et al. +Table 2. Wavelengths and ionisation potential of the relevant ion for each emission line included in the public catalog. All lines are corrected for the Milky +Way foreground dust extinction. Wavelengths are taken from the National Institute of Standards and Technology (NIST; https://physics.nist.gov/ +PhysRefData/ASD/lines_form.html), and are Ritz wavelengths in air (consist with wavelengths in the public data release) except for the H Balmer lines, in +which case we use the ‘observed ’ wavelength in air as reported in NIST. The DAP string name is used to identify the correct extension in the PHANGS-MUSE +MAPS files. Ionisation potentials are taken from Draine (2011). +Line name +Wavelength +String ID +Ionisation potential +Fixed ratio +(air) [Å] +[eV] +Hydrogen Balmer lines +H𝛽 +4861.35 +HB4861 +13.60 +no +H𝛼 +6562.79 +HA6562 +13.60 +no +Low ionisation lines +[O i]𝜆6300 +6300.30 +OI6300 +— +no +[N ii]𝜆6548 +6548.05 +NII6548 +14.53 +0.34 [N ii]𝜆6584 +[N ii]𝜆6584 +6583.45 +NII6583 +14.53 +no +[S ii]𝜆6717 +6716.44 +SII6716 +10.36 +no +[S ii]𝜆6731 +6730.82 +SII6730 +10.36 +no +High ionization lines +[O iii]𝜆4959 +4958.91 +OIII4958 +35.12 +0.35 [O iii]𝜆5007 +[O iii]𝜆5007 +5006.84 +OIII5006 +35.12 +no +[S iii]𝜆9068 +9068.6 +SIII9068 +23.34 +no +EW. This accounts for the underlying absorption feature, however +requires sufficient S/N in the data to get a good fit to the continuum +and is poorly determined in low spectral resolution data, such as +narrow-band imaging. In both cases the EW is then the determined +line flux over the mean continuum. +The median value across our full catalogue is EW(H𝛼)raw ∼ 20Å +and EW(H𝛽)raw ∼ 1Å, with ∼ 43% of nebulae having EW(H𝛽)raw < +0 due to the underlying stellar absorption feature. While the median +difference between EW(H𝛼)raw−EW(H𝛼)fit is typically only −2.2Å, +the difference is relatively stronger for the weaker H𝛽 line at −4.1Å. +While this presents the EW in a standard format, suitable for com- +parison with previous work, it is clear that our nebulae are sitting +within the central stellar disk of each galaxy. Due to this, our stel- +lar continuum band naturally suffers from a significant contribution +of light from old stellar population, which is not associated with +the young nebulae. The impact of this background contribution is +explored in Scheuermann et al. (submitted). +Given that nebular objects can be marginally resolved in our data, +with H ii regions displaying a variety of H𝛼 morphologies, deter- +mination of the completeness of our catalogue by quantifying the +recovery rate of artificial source injection is not straightforward. In +Santoro et al. (2022), the completeness for our catalogue was es- +timated in an empirical way by considering the H𝛼 line emission +outside of the region masks and measuring the H𝛼 surface bright- +ness at the 90th percentile level of the surface brightness distribution. +This surface brightness was then converted to a luminosity assum- +ing an unresolved point source. By this metric, typical completeness +limits are 1036 – 1037 erg s−1, which are roughly equivalent to the +ionizing flux of a single O7V star (Vacca 1994). We refer the reader +to Santoro et al. (2022) for further details and a complete table. For +objects classified as H ii regions (see Section 4.2), in Table 3 we +quantify the 10th, 50th and 90th percentile extinction corrected H𝛼 +luminosities and physical sizes. +4 VALUE-ADDED PRODUCTS +Given the large suite of emission lines measured within our nebulae, +there are multiple physical properties that can be determined. We in- +clude these value-added properties in the nebulae catalogue, though +note that different calibrations can be used for many of these prop- +Table 3. Distribution of H ii region attenuation-corrected H𝛼 luminosities +(L𝐻 𝛼) and sizes (𝑟circ) for each galaxy, listing the 10%, 50% and 90% values. +Galaxy +log(L𝐻 𝛼 [erg s−1]) +𝑟circ [pc] +10% +50% +90% +10% +50% +90% +IC5332 +35.9 +36.4 +37.2 +27 +33 +46 +NGC0628 +36.2 +36.7 +37.8 +31 +35 +58 +NGC1087 +36.7 +37.5 +38.5 +51 +55 +98 +NGC1300 +36.6 +37.2 +38.1 +58 +62 +88 +NGC1365 +37.0 +37.7 +38.9 +76 +82 +126 +NGC1385 +36.7 +37.5 +38.8 +45 +51 +103 +NGC1433 +36.5 +37.1 +37.9 +58 +62 +85 +NGC1512 +36.8 +37.4 +38.1 +79 +83 +115 +NGC1566 +36.6 +37.3 +38.6 +48 +53 +100 +NGC1672 +37.0 +37.6 +38.9 +64 +70 +121 +NGC2835 +36.5 +37.1 +38.1 +48 +52 +84 +NGC3351 +36.3 +36.8 +37.7 +36 +39 +58 +NGC3627 +37.0 +37.7 +38.9 +41 +49 +96 +NGC4254 +36.7 +37.5 +38.7 +41 +48 +91 +NGC4303 +36.9 +37.6 +38.7 +46 +56 +107 +NGC4321 +36.9 +37.6 +38.5 +59 +64 +96 +NGC4535 +36.4 +37.0 +38.1 +32 +40 +71 +NGC5068 +35.8 +36.3 +37.5 +19 +23 +45 +NGC7496 +36.6 +37.2 +38.3 +56 +61 +96 +erties resulting in systematically different results. For all properties +below we only present the results where the relevant lines have a +S/N greater than 5, and when the nebular classification is appropriate +(e.g. metallicities are only calculated for H ii regions). A complete +list of all columns contained in our nebular catalogue is provided in +Table 4. +4.1 Dust Attenuation +All line fluxes are provided in our catalogue as observed values, yet +the derivations of physical quantities (e.g., metallicity, ionization pa- +rameter) are typically based on intrinsic line fluxes. Therefore, before +deriving any quantities, the measured fluxes need to be corrected for +reddening due to dust. We assume here an O’Donnell (1994) extinc- +tion curve with an 𝑅𝑉 = 3.1, that represents a small modification +of the Cardelli et al. (1989) extinction curve. We then derive the +reddening, 𝐸(𝐵 − 𝑉), based on this curve and assume an intrinsic +MNRAS 000, 1–27 (2022) + +PHANGS Nebular Catalogue +7 +[NII] +[SII] +Hα +Hαabs +[NII] +Hα+[NII] +[SII] +CaT +Hβabs +[ΟΙII] +Ηβ +Figure 2. A nebula spectrum from IC5332 (region ID 38, 𝐿H𝛼 = 1036.8 erg s−1), showing a typical integrated spectrum (gold), including the underlying stellar +(and weak nebula) continuum with clear strong emission and absorption lines (with the strong lines labelled) and visible residuals from the sky background +subtraction. Overlaid is the resulting analysis pipeline fit from pPXF (blue dashed line), from which we extract the emission line fluxes and kinematics. The +lower panel shows the relative residual of the spectral fit in indigo and relative spectral error in cyan ((Nebula-Fit)/Fit and error/fit, respectively). In the insert we +show a zoom in of the H𝛼 and [N ii] region, where the spectral lines can be seen more clearly. We also show the underlying continuum fit (dotted green line) +revealing the underlying H𝛼abs feature. +Balmer ratio of H𝛼/H𝛽= 2.86. In practice the choice of extinction +curve has little impact on the corrections, as extinction curves do +not deviate significantly across the MUSE wavelength range and the +H𝛼 and H𝛽 emission lines are bracketing most of the emission lines +of interest. However the derived 𝐴𝑉 , and hence line luminosity, is +directly dependent upon our assumed value of 𝑅𝑉 . By using the +O’Donnell (1994) extinction curve we are assuming that the nebula +itself only experiences attenuation from a uniform foreground dust +layer. In reality, complex dust geometries within the nebulae (as seen +in nearby H ii regions like the Tarantula; De Marchi et al. 2016) as +well as blending of multiple regions along our line of sight might +bias our inferred extinction. However, we believe at our <100 pc +scales with distinguished nebulae and the thin star-forming disk the +foreground screen assumption is more justified than a mixed-media +model for the majority of our nebulae. +In Figure 3, we show the distribution of V-band attenuations, +𝐴𝑉 , experienced by the nebulae with significant detections of H𝛽 +(S/N > 5; 31,377 objects; 99.9% of the sample). We find a median +𝐴𝑉 = 0.72 mag (16%–84% range is 0.34–1.2 mag) and a tail of +highly attenuated nebulae (5% of objects have 𝐴𝑉 > 1.7 mag). +When we weight the 𝐴𝑉 ’s linearly by the intrinsic H𝛼 luminosity +(attenuation corrected), we find a median 𝐴𝑉 = 1.4 mag (16%–84% +range is 0.8–2.5 mag), a significant increase. While it does appear +in the sample (and as suggested by Figure 3) that the brightest H ii +regions are more attenuated, this difference in median 𝐴𝑉 is also +caused by the highly obscured faint H ii regions being undetected +in H𝛽. Also visible in this figure are a small subset (<3%) with +unphysical attenuation (𝐴(𝑉) < 0, meaning H𝛼/H𝛽 < 2.86) even +with a S/N > 5 in both H𝛼 and H𝛽. 80% of these are consistent within +the 3𝜎 line flux uncertainties with a value of H𝛼/H𝛽 = 2.86. The +remainder are typically found in nebulae with low H𝛼 equivalent +widths, as can be seen in the central plot in Figure 3, suggesting +that for most of these nebulae the underlying Balmer absorption +features are incorrectly subtracted leading to an overestimated H𝛽 +flux. However, it may also be that the intrinsic Balmer ratio for some +of these nebulae is less than our fiducial value of 2.86 due to physical +MNRAS 000, 1–27 (2022) + +8 +Groves et al. +Table 4. Columns in the catalogue +Column +Unit +Description +gal_name +galaxy name +region_ID +region ID +cen_ra +deg +RA (J2000) center, weighted by H𝛼 intensity +cen_dec +deg +Dec (J2000), weighted by H𝛼 intensity +flag_edge +flag set to 1 if within one PSF of the field edge +flag_star +flag set to 1 if overlapping with a star +deproj_r_R25 +R25 +Deprojected distance from galaxy center in units of R25 +deproj_r_reff +reff +Deprojected distance from galaxy center in units of reff +deproj_phi +deg +Deprojected position angle within galaxy disk +region_area +pixels +H ii region area +emline*_FLUX† +1e-20 erg/s/cm2 +emission line fluxes (see Table 2) +emline*_FLUX_CORR† +1e-20 erg/s/cm2 +attenuation-corrected emission line fluxes (see Table 2) +assuming an O’Donnell (1994) extinction curve and R𝑉 = 3.1 +emline*_VEL† +km/s +line velocity relative to v𝑠𝑦𝑠 (Table 1) +emline*_SIGMA† +km/s +line velocity dispersion, corrected for instrumental broadening +AV† +mag +A𝑉 , V-band attenuation derived from the Balmer decrement +assuming an O’Donnell (1994) extinction curve and R𝑉 = 3.1 +EW_HA6562_raw† +Å +Equivalent width of H𝛼, measured directly +EW_HB4861_raw† +Å +Equivalent width of H𝛽, measured directly +EW_HA6562_fit† +Å +Equivalent width of H𝛼, measured after stellar continuum subtracted +EW_HB4861_fit† +Å +Equivalent width of H𝛽, measured after stellar continuum subtracted +HA6562_LUM_CORR +erg/s +attenuation corrected H𝛼 luminosity +BPT_NII +BPT flag, see Table 5 +BPT_SII +BPT flag, see Table 5 +BPT_OI +BPT flag, see Table 5 +met_scal† +Metallicities determined using the Scal prescription (Pilyugin & Grebel 2016) +Delta_met_scal +Offset in metallicity relative to the radial gradient (Table 9) +logU† +Ionization parameter derived from [S iii]/[S ii] using the prescription in Diaz et al. 1991 +Environment +Environment flag, as in Table 8 +∗emission lines are listed in Table 2 +†Note that corresponding errors are included as *_ERR +reasons associated with the nebula itself (e.g. Planetary Nebulae +are both faint and typically several thousand Kelvin hotter than H ii +regions and therefore have an intrinsically lower ratio). For these +unphysical attenuations we set 𝐴𝑉 = 0 mag when considering the +reddening correction of the line ratios. All emission lines included +in our catalogue are also provided as corrected values (’*_CORR’; +see Table 4) by applying our determined 𝐴𝑉 and chosen extinction +curve. +4.2 Emission line diagnostic classifications and H ii region +catalogue construction +In defining the nebulae catalogue, we have used the HIIphot code +(Thilker et al. 2000). However, within the PHANGS–MUSE data +we also clearly see H𝛼 emission associated with supernova rem- +nants, planetary nebulae, and regions ionized by active galactic nuclei +(AGN). As a first pass at separating these regions we use emission +line ratio diagrams (commonly called BPT diagrams after their in- +troduction in Baldwin et al. 1981) and use the diagnostic curves +described in Kewley et al. (2001) and Kauffmann et al. (2003) to +classify the nebulae. We note that while the Kauffmann et al. (2003) +curve is derived empirically from global spectra, it still provides a +useful constraint on whether ionization by processes other than UV +photons from OB-stars are playing a role in the nebulae (e.g. shocks, +AGN, etc.; Law et al. 2021; Belfiore et al. 2022) +We use the three strong line diagnostic diagrams (Figure 4) to +classify our nebulae; [O iii]/H𝛽 versus [N ii]𝜆6584/H𝛼, [O iii]/H𝛽 +versus [S ii]𝜆6717, 6731/H𝛼, and [O iii]/H𝛽 vs [O i]𝜆6300/H𝛼. We +note that different galaxies follow tracks that are slightly offset and +correlate with the total stellar mass of the galaxy (and presumably +its metallicity), with all galaxies shown individually in Appendix B. +For each diagnostic, we flag the nebulae with S/N < 5 in any of the +lines used in the diagnostic, then mark the remaining as H ii regions, +composites, clear AGN impact or LINER-like (indicative of shocks +or strong contributions from more diffuse ionized gas) spectra (Table +5). We construct an H ii region sample from those objects classified +as H ii regions by all three diagnostics (20,577 objects), as well as +objects where [O i] is not detected with S/N > 5 but they are otherwise +consistent with the H ii region BPT diagnostics (2,667 objects). This +results in a total of 23,244 (74.0%) of objects that are consistent with +photoionization by massive stars, and we consider this to be our full +H ii region catalogue. This sample would increase by about 1500 +objects if we reduced our S/N requirement to 3, and it would only +increase by about ∼100 objects if we included objects that fall below +the BPT demarcations when accounting the line flux uncertainties. +Given the factor of two variation in distance and moderate variation +in sensitivity between galaxies, we do not achieve uniform detection +thresholds across the sample. In addition, the various H ii region +morphologies considered by HIIphot do not lead to homogeneous +luminosity thresholds in our source identification. To provide a gen- +eral quantification of our typical H ii region properties per galaxy, +we quote the 10th, 50th and 90th percentiles in both attenuation- +corrected H𝛼 luminosity and H ii region size in Table 3. Here, the +size is taken as the circularized radius that results in an equal area to +the area of the H ii region mask. We note that the vast majority of our +regions are unresolved, as reflected by the close agreement in 10% +and 50% sizes, along with the clear correlation with galaxy distance. +Because of this, we purposefully do not include size measurements +MNRAS 000, 1–27 (2022) + +PHANGS Nebular Catalogue +9 +Figure 3. The distribution of nebulae attenuations, 𝐴𝑉 , derived from the Balmer decrement and the O’Donnell (1994) attenuation curve with an 𝑅𝑉 = 3.1, +compared to the H𝛼 emission line equivalent width determined from the integrated spectra (EW(H𝛼)raw). The central plot shows the 2D histogram of 𝐴𝑉 with +the measured EW(H𝛼)raw, while the outer histograms show the 1D distributions of both properties. The Balmer decrement (H𝛼/H𝛽) is also shown directly on +the top axis. +Table 5. BPT flags included in the catalog +Column name +Value +Meaning +BPT_NII +0 +star formation +−1 +low S/N < 5 +1 +composite +3 +AGN +BPT_SII +0 +star formation +−1 +low S/N < 5 +2 +LI(N)ER +3 +AGN +BPT_OI +0 +star formation +−1 +low S/N < 5 +3 +AGN +in our catalogue. The 10th percentile luminosities provide a general +idea of the completeness limits for each galaxy, and we refer to San- +toro et al. (2022) for more detailed discussion. Future work will aim +to map out of the H ii region selection function more completely and +provide homogenised 150 pc scale catalogues. +4.3 Gas-phase metallicities +To derive the gas-phase metallicity there are a wide range of prescrip- +tions in the literature that can be applied to the nebulae classified as +H ii regions. Systematic differences between these prescriptions are +well known in the literature, and routinely produce absolute measure- +ments that differ by 0.2 dex, and even up to 0.7 dex, in 12+𝑙𝑜𝑔(O/H) +for the same H ii regions (see, e.g., Peimbert et al. 2017; Kewley et al. +2019, for reviews on this problem). While qualitatively the difference +between individual H ii regions is typically maintained (metal-poor +remain poor), the scale in these differences can also be markedly +different, as shown by Kewley & Ellison (2008) in SDSS galaxies. +We demonstrate these differences in Figure 5, where we apply eight +different prescriptions from the literature (Table 6) to our 23,244 H ii +regions and compare the resulting metallicity measurements. We +note that, as the wavelength coverage of our galaxies by MUSE ex- +cludes emission lines below 4800 Å, some of the standard metallicity +prescriptions using lines such as the [O ii]𝜆3727 doublet cannot be +applied here (e.g. Kobulnicky & Kewley 2004; Pilyugin & Thuan +2005; Kewley et al. 2019). In a pair-wise comparison, we compute a +linear conversion between prescriptions, and tabulate the fits (shown +in red in Figure 5) in Table 7. Note that the number of H ii regions in +each panel differs slightly depending upon the detection (𝑆/𝑁 > 5) +of the lines involved. Most values show a positive correlation, though +the strength of the correlation and the scatter between measurements +vary wildly, with offsets of up to 0.2 dex and scatter of up to 0.2 +dex apparent. This should serve as a warning that when comparing +metallicity measurements in the literature, it is important to ensure +consistent prescriptions are applied. The conversions we provide are +MNRAS 000, 1–27 (2022) + +Hα/Hβ +3 +4 +5 +6 +7 +2000 +N +0 +2.5 +2.0 +1.5 +1.0 +0.5 +0 +2 +0.0 +log(N) +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +0 +2000 +Av [mag] +N10 +Groves et al. +Figure 4. The distribution of our nebulae on the three BPT (Baldwin et al. 1981) diagnostic diagrams considered when flagging for ionization mechanism (Table +5). From left to right, we show [O iii]𝜆5007/H𝛽 versus [N ii]𝜆6583/H𝛼, [S ii]𝜆𝜆6716,6730/H𝛼, and [O i]𝜆6300/H𝛼. Overlaid are the Kauffmann et al. (2003) +diagnostic curve (dashed line) and Kewley et al. (2006) diagnostic curves (solid lines). Non-stellar ionizing sources are typically found above these lines, and +objects between the two demarcations are considered composites. Greyscale demonstrates the distribution of nebulae in these diagrams, and 79% of nebulae +are consistent with photoionization. Coloured lines (scaled to the galaxy stellar mass) indicate galaxy trends (binned along the y-axis). [N ii]/H𝛼 and [S ii]/H𝛼 +show clear secondary correlations with stellar mass (corresponding to trends in metallicity). +Table 6. Metallicity prescriptions considered in Figure 5 +Abreviation +Lines used +Reference +Scal-PG16 +H𝛽, [O iii], [N ii], [S ii] +Pilyugin & Grebel (2016) +O3N2-M13 +H𝛼, [O iii], H𝛼, [N ii] +Marino et al. (2013) +O3N2-PP04 +H𝛽, [O iii], H𝛼, [N ii] +Pettini & Pagel (2004) +N2-M13 +H𝛼, [N ii] +Marino et al. (2013) +N2S2-D16 +H𝛼, [N ii], [S ii] +Dopita et al. (2016) +O3S2-C20 +H𝛽, [O iii], H𝛼, [S ii] +Curti et al. (2020) +RS32-C20 +H𝛽, [O iii], H𝛼, [S ii] +Curti et al. (2020) +R3-C20 +H𝛽, [O iii] +Curti et al. (2020) +only applicable over the metallicity range covered by our sample, +defined as the 5–95 percentiles of the distribution. In some cases +(e.g. N2-M13 versus R3-C20) no clear correlation between the pre- +scriptions is observed over the narrow metallicity range covered. +As described in Kreckel et al. (2019), we favour the S calibration +(Scal) prescription defined in Pilyugin & Grebel (2016), hereafter +Scal-PG16, and include these calculated metallicities in our value- +added catalogue. The Scal-PG16 prescription was empirically cali- +brated against a sample of 313 H ii regions where direct auroral line +detections provided measurements of the electron temperature, and +hence more robust determination of 12+log(O/H). As it relies on a +larger number of emission lines than other prescriptions (Table 6), +it is less biased by ionization parameter variations, which can cause +line ratio variations and results in degeneracies in the metallicity +determination when only one or two line ratios are considered for +the prescription (Kewley & Dopita 2002). However, note that for the +range of metallicities encountered in our sample the calibration is +based only on a small fraction of H ii regions. +The Scal-PG16 prescription relies on three standard diagnostic +line ratios: +N2 = ([N ii]𝜆6548 + 𝜆6584)/H𝛽, +S2 = ([S ii]𝜆6717 + 𝜆6731)/H𝛽, +R3 = ([O iii]𝜆4959 + 𝜆5007)/H𝛽, +(1) +where attenuation corrected line fluxes are used (and therefore im- +plicitly includes the ratio of Balmer lines). It is defined separately +over the upper and lower branches in log N2. The upper branch +(log N2 ≥ −0.6) is calculated as +12 + log(O/H) += +8.424 + 0.030 log(R3/S2) + 0.751 log N2 ++ +(−0.349 + 0.182 log(R3/S2) + 0.508 log N2) +× +log 𝑆2 +(2) +and the lower branch (log 𝑁2 < −0.6) is calculated as +12 + log(O/H) += +8.072 + 0.789 log(R3/S2) + 0.726 log N2 ++ +(1.069 − 0.170 log(R3/S2) + 0.022 log N2) +× +log S2 +(3) +The Scal-PG16 prescription is highly correlated with the Dopita +et al. (2016) N2S2-D16 prescription, which is calibrated against +photoionization models but has similarly been designed to minimise +degeneracies with ionization parameter. We calculate the uncertain- +ties associated with the Scal-PG16 metallicity by Monte Carlo error +propagation of the emission line flux errors, with 1000 samples used +to determine the 1𝜎 distribution corresponding to each measured +metallicity. Our H ii regions cover a range of metallicities from 8.1 < +12+log(O/H) < 8.7 with typical statistical uncertainties of 0.01 dex +(Figure 6). +In an independently constructed catalogue, also identified using the +HIIphot package, metallicity measurements from 8,914 H ii regions +in eight of these galaxies have been published previously (Kreckel +et al. 2019). Line flux measurements were made using a different +data reduction pipeline and different analysis approach, such that the +physical extent of the regions may differ and the fitting of the under- +lying stellar continuum has changed. Comparing our new catalogue +with the previous one, we find that ∼7000 H ii regions cross-match +within 1′′, and for these objects the median difference in metallicity +is negligible (0.0003 dex) and the standard deviation is small (< 0.02 +dex). +MNRAS 000, 1–27 (2022) + +0.5 +0.0 +H/[IO] +0 +11 +-0.5 +10 : +-1.0 +6 +1.0 -0.8 -0.6 -0.4 -0.2 +0- '0- 9'0- 8'0- 0'- +0.0 +-2.5 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +log[NIl/Hα +log [Sll]/Hα +log [Ol]/HaPHANGS Nebular Catalogue +11 +Figure 5. A comparison of 12+log(O/H) metallicities across eight different prescriptions (listed in Table 6). The corresponding 1-to-1 line (dashed) is shown in +each panel. A linear fit is given for each (red line), with fit parameters listed in Table 7. Systematic offsets of 0.2 dex and scatter of 0.2 dex is apparent between +prescriptions. Note that the O3N2-PP04 and O3N2-M31 rely on exactly the same line ratios and hence show a perfect correlation. Our favoured prescription is +the Pilyugin & Grebel (2016) Scal, which uses 3 different line ratios (and implicitly the Balmer decrement) to remove degeneracies due to ionization parameter +variations, and shows a strong correlation with N2S2-D16. +4.4 Ionization Parameter +In addition to the actual metal abundance, another parameter of +the ISM that regulates line emissivity is the ionization parameter, +the ratio of the density of ionizing photons to the number den- +sity of hydrogen atoms in the gas (𝑈). Empirically, this is typically +parametrized by line ratios of different ions arising from the same +element. Based on theoretical photoionization models (Kewley & +Dopita 2002), [O iii]/[O ii] is sensitive to changes in 𝑈 but has a +strong secondary dependence on metallicity. [S iii]/[S ii] shows very +little dependence on metallicity but is less commonly used to trace +𝑈 due to the difficulty in observing the red [S iii]𝜆𝜆 9068,9532 lines. +MUSE now gives us coverage of the [S iii]𝜆9068 emission line, en- +abling us to explore ionization parameter variations more directly +using the [S iii]/[S ii] line ratio. +However, while the [S iii]/[S ii] line ratio appears to be a robust +tracer of 𝑈 in theoretical models, comparisons have uncovered large +offsets between model predictions for the ratio and empirical results +(Mingozzi et al. 2020). We apply the prescription of Diaz et al. (1991) +to determine 𝑈 from this diagnostic ratio as +log(𝑈) += +−1.684 log([S iii]𝜆𝜆9068, 9532/[S ii]𝜆𝜆6716, 6730) +− +2.986. +(4) +MNRAS 000, 1–27 (2022) + +6 +2-D1 +9.0 +8.5 +N2. +8.0 +3 +9.0 +-M1 +8.5 +0 +8.0 +O3N2-PP04 +9.0 +8.5 +12+log(0/H) +9.0 +N2-M13 +8.5 +0 +1 +8.0 +log(N) +2 +O3S2-C2 +9.0 +8.5 +RS32-C20 +9.0 +8.5 +9.0 +R +8.0 +8.5 +9.08.0 +8.5 +9.0 +8.0 +8.5 +9.0 +8.5 +9.0 +8.0 +8.5 +9.0 +8.5 +9.0 +8.5 +9.0 +N2S2-D16 +O3N2-PP04 +N2-M13 +O3S2-C20 +Scal-PG16 +O3N2-M13 +RS32-C20 +12+log(O/H)12 +Groves et al. +Table 7. Linear fit parameters for converting metallicities, see red lines in Figure 5. The range (5–95 percentiles) over which this conversion holds is shown in +square brackets. For each conversion we provide in parenthesis the following quantities: intercept, slope, scatter about the conversion. +N2S2-D16 +[8.3, 8.9] +(-7.2, 1.9, 0.06) +O3N2-M13 +[8.4, 8.6] +(5.4, 0.4, 0.05) +(6.8, 0.2, 0.05) +O3N2-PP04 +[8.6, 8.9] +(4.0, 0.6, 0.08) +(6.1, 0.3, 0.08) +(-4.0, 1.5, 0.00) +N2-M13 +[8.4, 8.6] +(5.3, 0.4, 0.03) +(7.2, 0.2, 0.04) +(6.0, 0.3, 0.04) +(6.8, 0.2, 0.04) +O3S2-C20 +[8.6, 8.8] +(8.5, 0.0, 0.06) +(8.6, 0.0, 0.06) +(2.5, 0.7, 0.03) +(4.5, 0.5, 0.03) +(6.9, 0.2, 0.06) +RS32-C20 +[8.5, 8.9] +(5.9, 0.3, 0.06) +(6.8, 0.2, 0.05) +(1.1, 0.9, 0.03) +(3.5, 0.6, 0.03) +(8.3, 0.1, 0.07) +(2.3, 0.7, 0.05) +R3-C20 +[8.7, 8.9] +(6.9, 0.2, 0.06) +(7.5, 0.1, 0.06) +(0.2, 1.0, 0.02) +(2.9, 0.7, 0.02) +(8.2, 0.1, 0.07) +(-0.4, 1.1, 0.04) +(0.4, 1.0, 0.03) +Scal-PG16 +N2S2-D16 +O3N2-M13 +O3N2-PP04 +N2-M13 +O3S2-C20 +RS32-C20 +[8.3, 8.6] +[8.3, 8.9] +[8.4, 8.6] +[8.6, 8.9] +[8.4, 8.6] +[8.6, 8.8] +[8.5, 8.9] +Figure 6. Distribution of errors in 12+log(O/H) as a function of offset from the +radial gradient in each galaxy (Δ(O/H)) based on the Scal-PG16 prescription. +These errors are based on Monte Carlo resampling of the associated line flux +uncertainties, and do not take into account systematic uncertainties related to +the calibration of the metallicity prescription. Typical 1𝜎 uncertainties are +0.01 dex. +Here we measure only the shorter wavelength [S iii]𝜆9068 emis- +sion line but assume a ratio of [S iii]𝜆9532 = 2.5 × [S iii]𝜆9068 fixed +by atomic physics (Osterbrock & Ferland 2006; Tayal et al. 2019). We +require a S/N > 5 in all lines, and are able to compute 𝑈 for 20,781 +objects (66.2% of the sample), nearly all of which (20,083 objects; +97% of objects with measured 𝑈) are classified as H ii regions (see +Section 4.2). +In Figure 7 we see that our H ii regions cover a range of ion- +ization parameters from −3.0 < log𝑈 < −0.5, with no systematic +differences by galaxy stellar mass. Typical uncertainties are 0.04 dex. +Apparent within each galaxy is a positive correlation between 𝑈 and +12+log(O/H), first reported in luminous star-forming galaxies (Do- +pita et al. 2014), and recently identified as a robust trend across H ii +region samples (Kreckel et al. 2019; Grasha et al. 2022). This is con- +trary to theoretical predictions by Massey et al. (2005), and is also not +well reflected in photoionization models (Ji & Yan 2022), indicating +a need for additional model development. No clear correlation of 𝑈 +with stellar mass of the galaxy due to the scatter within each galaxy +and the differing slopes of 𝑈 versus (O/H) found between galaxies. +4.5 Nebulae Environments +The nebulae we identify do not exist in isolation, but rather are +part of the larger scale structure of our galaxies. Therefore we also +include in our catalogue parameters that trace the different galactic +environments in which they occur. +The galaxies in our sample show visible structures (centre, bar, +spiral arm, interarm, disc) that may reflect differences in dynamical +conditions and star formation histories. To define the nebular envi- +ronments, we use the stellar morphological masks that have recently +been identified systematically in Querejeta et al. (2021) based on +Spitzer 3.6 𝜇m images. We locate each of our nebulae with respect +to the simplified environments defined in that paper, as summarised +in Table 8. In Figure 8 we show both the absolute number of H ii +regions within each environment (left) as well as the surface density +of all objects in each environment (right). +In absolute numbers, most of our H ii regions (∼40%) are located +in interarm regions, however this environment also makes up the +largest area in our fields and so correspondingly there is a relatively +low number density of nebulae in this environment. In contrast, +we identify only a small number (∼ 300, ∼2%) of H ii regions in +the galaxy centres, but this corresponds to a high number density. +We find the highest H ii region number density within spiral arm +environments, and the lowest H ii region number density within bar +environments. This reflects the fact that star formation is typically +concentrated into spiral arms, and bar environments (excluding bar +ends) potentially suppress star formation via bar-driven dynamics +(James et al. 2009). Looking at the full object catalogue, including +objects that do not meet our H ii region classification criteria, the +number density of objects in the bar approximately doubles. The +centre environments also contain a high number density of objects +that are not classified as H ii regions. This is expected if AGN and +dynamical shocks are significantly contributing to the gas ionization +in these environments. +5 RESULTS +While the total stellar mass and integrated star formation rate are +global properties that correlate with the integrated properties of the +ISM (Tremonti et al. 2004; Sánchez 2020; Pessa et al. 2021), sig- +nificant secondary correlations are identified within galaxies once +their nebular emission is spatially resolved. The most well known +MNRAS 000, 1–27 (2022) + +0.08 +0.07 +3 +r in 12+log(O/H) +0.06 +0.05 +3g +2 +(N)60l +0.04 +0.03 +error +L +0.02 +0.01 +0.00 +0 +-0.20 +-0.15 +-0.10 +-0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +△ (O/H)PHANGS Nebular Catalogue +13 +Figure 7. Metallicity (12+log(O/H)) as a function of ionization parameter (U) for H ii regions within each galaxy. Galaxies are sorted by stellar mass from low +(top left) to high (bottom right). Linear fits are performed in each galaxy, demonstrating the prevalence of positive correlations across the sample. Apart from +the well-established mass-metallicity relation, no obvious trends are seen as a function of total galaxy stellar mass (red colour scale, bottom right). +Table 8. Our Environmental flags, based on simplified assignments from Querejeta et al. (2021). +Label +Querejeta2021 Environment +Environment +1 +Centre +−→ +Centre +2 +Bar (excluding bar ends) +� +Bar +3 +Bar ends +5 +Spiral arms inside interbar (𝑅gal < 𝑅bar) +� +Arm +6 +Spiral arms (𝑅gal > 𝑅bar) +4 +Interbar (𝑅gal < 𝑅bar) +� +Interarm +7 +Interarm (𝑅gal > 𝑅bar) +8 +Outer disc (𝑅gal > spiral arm ends) +9 +Disc (𝑅gal > 𝑅bar) in galaxies without spiral masks +−→ +Disc +MNRAS 000, 1–27 (2022) + +'NGC5068 +IC5332 +NGC1087 +NGC1385 +8.6 +8.5 +8.4 +8.3 +8.2 +NGC2835 +NGC0628 +NGC4254 +NGC4303 +8.6 +8.5 +8.4 +12+log(0/H) +8.3 +8.2 +NGC4535 +NGC1300 +NGC1672 +NGC432 +8.6 +8.5 +8.4 +8.3 +8.2 +NGC1 +NGC1 +8.6 +8.5 +8.4 +log(M*) +8.3 +10 +11 +8.2 +-3.0-2.5-2.0-1.5-1.0-0.5-3.0-2.5-2.0-1.5-1.0-0.5-3.0-2.5-2.0-1.5-1.0-0.5-3.0-2.5-2.0-1.5-1.0-0.5-3.0-2.5-2.0-1.5-1.0-0.5 +log U14 +Groves et al. +0 +5 +10 +15 +20 +25 +30 +35 +40 +fraction [%] +Centre +Bar +Arm Interarm Disc +0 +2000 +4000 +6000 +8000 +10000 +N +Centre +Bar +Arm +Interarm +Disc +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +N / kpc2 +Figure 8. Distribution of H ii region detections across different galactic environments, both in absolute numbers (left) and in surface density (right). Unfilled +bars (right) indicate the environmental distribution of all nebulae. Interarm regions contain the largest number of H ii regions (∼40% of the sample) but cover +a large area of our fields and as such reflect a relatively low surface density. In contrast, we identify only a small number (∼ 300) of H ii regions in the galaxy +centres, but this corresponds to a high surface density. Nebulae not classified as H ii regions are over-represented in centre and bar environments, where shock +excitation is more likely to contribute to the gas ionization. +of these is the metallicity gradient (for recent reviews see Maiolino +& Mannucci 2019; Kewley et al. 2019). However, gas-phase metal- +licity has also been shown to spatially vary with both stellar mass +surface density (Barrera-Ballesteros et al. 2016) and gas-mass sur- +face density (Barrera-Ballesteros et al. 2018). Once this radial gra- +dient is removed, higher-order variations of the metallicity are seen, +along spiral arms (Sánchez-Menguiano et al. 2016; Ho et al. 2017) +and across discs (Kreckel et al. 2019). We revisit here that work of +Kreckel et al. (2019), an earlier analysis of spatial metallicity varia- +tions in a subset (8 out of 19) of the PHANGS–MUSE galaxies using +data from an earlier version of our reduction and analysis pipeline +(see also Emsellem et al. 2022). +For this analysis, we consider our H ii region sample to be those +objects that are fully contained in the field of view (‘flag_edge’ += 0), are consistent with photoionization (‘BPT_NII’ = ‘BPT_SII’ += 0 and ‘BPT_OI’ <= 0), and where we have high confidence in +our metallicity measurement (‘met_scal_err’ < 0.04 dex; see Figure +6 and Section 4.3). By including a cut on metallicities with large +uncertainties, we exclude only ∼1000 regions that have an average +uncertainty of 0.06 dex. We also exclude six regions with metallicity +values 12+log(O/H) < 8.0 as we believe these are spurious and they +significantly bias our statistics (see Section 6.3). For the following +sections, our catalogue consists of 22,318 H ii regions, with between +477–2556 H ii regions per galaxy. +5.1 Radial Gradients +As described above, it has been clearly established that galaxies in the +local universe systematically have a lower metallicity with increasing +radius (e.g. Moustakas et al. 2010; Pilyugin et al. 2014; Sánchez et al. +2014). These radial trends neglect prominent morphological features +(spiral arms, stellar bars), though do appear to show variations in the +inner and outer parts of galaxies (Sánchez-Menguiano et al. 2018). +The metallicity gradients vary with stellar mass (Sánchez et al. 2014) +and with radius at a given mass (e.g. Boardman et al. 2021), and are +thought to chart the typical inside-out growth of most disc galaxies. +In Figure 9 we show radial trends for a few of the key ISM diagnos- +tics available in our H ii region catalogue, for a representative sample +of four galaxies. These are trends in the H𝛼 luminosity (L(H𝛼)) of in- +dividual H ii regions, attenuation derived from the Balmer decrement +(A𝑉 ), metallicity (12+log(O/H)), and ionization parameter (U). Ra- +dial trends for all 19 galaxies are shown in Appendix C. These four +galaxies in Figure 9 represent a low stellar mass (IC 5332) and high +stellar mass (NGC 1365) galaxy, and systems with a regular spiral +pattern and no bar (NGC 0628) or strong bar and widely separated +arms (NGC 7496). For each galaxy we show the radial trends scaled +to reff, to normalise the sample, and note that in most cases our cov- +erage is limited to the inner parts of each galaxy (<2 reff). We fit +each gradient with a linear relation that neglects the uncertainties, as +variations in these properties are expected to reflect local variations +in the physical conditions and not uncertainties in the measurements. +Representative values at 1 reff are also given in Appendix C. The +individual radial gradients we determine for our PHANGS–MUSE +sample fall largely within the range found from the much larger IFS +surveys such as CALIFA (Espinosa-Ponce et al. 2022) and MaNGA +(Barrera-Ballesteros et al. 2022). Beyond these radial trends, how- +ever, we see a large scatter of these properties within each galaxy due +to the higher spatial resolution of our sample. +In general, galaxies show relatively flat or slightly negative slopes +in L(H𝛼). In many cases central starbursts are also apparent, cov- +ering scales of a few 100 pc and exhibiting high H𝛼 luminosities +(e.g. in NGC 1365), though we note that these measurements could +be biased by our ability to deblend neighbouring H ii regions given +our angular resolution (∼70 pc). There is also a floor imposed by +our region identification methods and sensitivity limits, visible as a +relatively sharp lower bound to our region luminosities, and tracking +the variations in galaxy distances. Most (80%) of the H ii regions +we detect have L(H𝛼) < 1038 erg s−1. About ∼800 (∼3%) of our +H ii regions might be categorized as ‘giant H ii regions’ (L(H𝛼) > +1039 erg s−1). H ii region luminosity functions for each galaxy are +presented in Santoro et al. (2022). +We find flat or slightly negative slopes in A𝑉 , with typical values +ranging from 0–2 mag of extinction, and a median value of 0.75 mag. +We emphasise that these measurements do not represent an unbiased +view of dust in the disc, as most of our nebulae are associated with +star-forming regions that are expected to be dustier (Calzetti et al. +1994; Kreckel et al. 2013). We also potentially miss high attenua- +tion, heavily embedded regions where H𝛽 or even H𝛼 may not be +visible, though the incidence of such obscured nebulae in the lo- +cal Universe is small (Prescott et al. 2007), and likely even rarer in +our low-inclination galaxies. Any such obscured population will be +constrained with our upcoming PHANGS–JWST observations. +MNRAS 000, 1–27 (2022) + +PHANGS Nebular Catalogue +15 +Table 9. Linear fit parameters for radial gradients in 12+log(O/H) using the +Scal prescription, see Figure 10 +Galaxy +intercept +slope [dex/r𝑒 𝑓 𝑓 ] +value at r𝑒 𝑓 𝑓 +𝜎(O/H) +IC5332 +8.475 ± 0.012 +-0.173 ± 0.002 +8.302 +0.066 +NGC0628 +8.533 ± 0.014 +-0.054 ± 0.001 +8.478 +0.048 +NGC1087 +8.479 ± 0.000 +-0.070 ± 0.010 +8.409 +0.032 +NGC1300 +8.617 ± 0.015 +-0.079 ± 0.001 +8.537 +0.042 +NGC1365 +8.666 ± 0.004 +-0.188 ± 0.005 +8.477 +0.040 +NGC1385 +8.459 ± 0.002 +-0.038 ± 0.014 +8.421 +0.033 +NGC1433 +8.569 ± 0.016 +-0.013 ± 0.001 +8.556 +0.051 +NGC1512 +8.581 ± 0.014 +-0.016 ± 0.001 +8.565 +0.042 +NGC1566 +8.613 ± 0.004 +-0.037 ± 0.004 +8.576 +0.037 +NGC1672 +8.566 ± 0.009 +-0.013 ± 0.000 +8.553 +0.033 +NGC2835 +8.555 ± 0.006 +-0.157 ± 0.002 +8.398 +0.040 +NGC3351 +8.579 ± 0.013 +0.007 ± 0.001 +8.587 +0.044 +NGC3627 +8.538 ± 0.003 +0.006 ± 0.004 +8.544 +0.033 +NGC4254 +8.590 ± 0.004 +-0.028 ± 0.003 +8.562 +0.030 +NGC4303 +8.613 ± 0.000 +-0.032 ± 0.006 +8.580 +0.034 +NGC4321 +8.592 ± 0.006 +-0.028 ± 0.004 +8.564 +0.036 +NGC4535 +8.580 ± 0.014 +-0.040 ± 0.003 +8.541 +0.039 +NGC5068 +8.412 ± 0.013 +-0.094 ± 0.001 +8.318 +0.054 +NGC7496 +8.588 ± 0.010 +-0.081 ± 0.003 +8.507 +0.045 +As in Kreckel et al. (2019), we do not see significant radial gradi- +ents in the ionization parameter for any of our galaxies. This reflects +that the localized ionization state of the gas is sensitive mainly to +changes in the ionizing source and local gas density at the cloud +interface. The flat radial gradient also increases our confidence in ra- +dial trends we uncover in metallicity. Ionization parameter variations +can influence diagnostic line ratios and introduce biases depending +on the metallicity prescription used. Our preferred metallicitiy pre- +scription, the Pilyugin & Grebel (2016) S-calibration, is designed to +minimize this bias but it is reassuring that we also do not observe +any radial trends in ionization parameter. +Our most pronounced radial trends are apparent in the H ii region +metallicities (12+log(O/H)), and we show the radial metallicity gra- +dients for all 19 galaxies in Figure 10. A simple linear fit (solid line) +shows very good agreement with the median value in 0.5 reff wide +bins. For the radial fit, we exclude the central 0.5 reff as suggested by +Sánchez-Menguiano et al. (2018). We have also not included the un- +certainties when performing our linear fit, as we allow for an intrinsic +scatter due to physical variations in the gas conditions in excess of +our estimated uncertainties (∼0.01 dex; Figure 6). For each bin we +also track the 1𝜎 scatter (outer lines), and find very good agreement +between these binned radial trends and the linear fits, suggesting +that to first order a linear fit describes the data well. This finding +has been more thoroughly quantified in Williams et al. (2022). Our +galaxies clearly reflect the well-established mass-metallicity relation +(Tremonti et al. 2004), with less massive galaxies exhibiting system- +atically lower metallicities (bottom right panel). +For 10 of the 19 galaxies, measurements of the corotation radius +are available (Williams et al. 2021). This is the location where the gas +rotational dynamics and spiral pattern speed are matched, and can +only be robustly measured through analysis of the stellar kinemat- +ics. Theoretical work has suggested that at this dynamical location, +metallicity variations are predicted to be amplified due to the lower +relative velocity between the gas and spiral pattern overdensity (Spi- +toni et al. 2019). We see no obvious change in the metallicity scatter +at the locations of corotation, or radially with respect to corotation. +Radial gradients for all galaxies are reported in Table 9. Gradients +derived using alternative metallicity prescriptions are also provided +in Appendix D. +5.2 Correlations with Global Properties +Based on the radial trends, we explore correlations of representative +derived properties with global galaxy properties. In particular, we +explore trends with total stellar mass, star formation rate (SFR) and +gas fraction (calculated as the sum of the Hi and H2 gas mass rela- +tive to the total gas and stellar mass). These are all global properties +that are typically associated with the regulation of galaxy evolution +(Genzel et al. 2015). Our galaxies span just over an order of magni- +tude dynamic range in these key properties. While our sample size is +small compared to integral field spectral galaxy surveys like CALIFA +(Sánchez et al. 2012), MaNGA (Bundy et al. 2015), or SAMI (Croom +et al. 2021), our ability to robustly isolate individual H ii regions pro- +vides a novel opportunity to cleanly consider trends relating small +scale properties to global differences. +As both extinction (A𝑉 ) and ionization parameter (U) show no +clear radial trends, we consider the median value measured across +the galaxy disk (Figure 11). A𝑉 shows modest trends for higher +values at higher stellar mass and SFR, consistent with an increased +amount of gas (and hence dust) associated with these systems. We +see no trend with gas fraction. Ionization parameter shows no trends +with global properties, indicating that it is regulated by local physical +conditions in the disk. +Using our radial metallicity gradient fits, we calculate a repre- +sentative metallicity at 1 reff for each galaxy, and consider global +trends in metallicity and metallicity slope (Figure 12). As expected, +we recover the mass-metallicity relation (Tremonti et al. 2004), with +more massive galaxies systematically exhibiting higher metallicities. +Secondary dependencies have been reported with SFR (Mannucci +et al. 2010; Lara-López et al. 2010; de los Reyes et al. 2015), with +galaxies at lower stellar mass exhibiting lower metallicities at fixed +SFR. This effect is broadly seen in our small galaxy sample. From +gas-equilibrium models, this trend has been proposed to derive pri- +marily from a decreasing gas fraction corresponding to high SFRs +(Peeples et al. 2008, 2009; Bothwell et al. 2013), and to some de- +gree this is also reflected in our sample. However, we are generally +in agreement with the larger CALIFA sample of Alvarez-Hurtado +et al. (2022) in that, once the stellar mass metallcity correlation is +removed, the other global properties show no obvious trends. +Trends for steeper metallicity gradients in more massive galaxies +are reported in large galaxy surveys (Belfiore et al. 2017; Poetrodjojo +et al. 2018), but in fact we observe the opposite trend. This could +be due to the radial coverage of our sample being limited, or the +predominance of bar-dominated systems (these have been observed +to exhibit flatter metallicity gradients; Zurita et al. 2021). With those +caveats, we also see trends for flatter slopes at high SFR and low gas +fraction. +5.3 Global metallicity variations +As the linear gradient in metallicity represents the dominant first +order trend, we follow the approach developed in Kreckel et al. (2019) +to fit and subtract this radial gradient and examine the second order +variations in metallicity, Δ(O/H). We further quantify the 1𝜎 scatter +in Δ(O/H) over the entire galaxy as 𝜎(O/H) for each galaxy (Table +9), to understand whether the second order variations in metallicity +are driven by global galaxy properties. +We find 𝜎(O/H) varies across the galaxy sample, with values rang- +ing from 0.03 – 0.06 dex. These values do not change significantly +(<0.005 dex) if we impose a stricter cut on our metallicity uncertain- +ties. In the top panels of Figure 13 we show how 𝜎(O/H) correlates +with different global galaxy properties, including total stellar mass +MNRAS 000, 1–27 (2022) + +16 +Groves et al. +Figure 9. Radial gradients for four representative galaxies in key ISM diagnostics (from top to bottom): H𝛼 luminosity (L(H𝛼)), extinction derived from the +Balmer decrement (A𝑉 ), metallicity (12+log(O/H)), and ionization parameter (U). For each galaxy we show the radial trends normalized to reff. These four +galaxies represent low stellar mass (IC 5332) and high stellar mass (NGC 1365) galaxies, and systems with a regular spiral pattern and no bar (NGC 0628) or +strong bar and widely separated arms (NGC 7496). Most trends are flat or mildly negative, except for the metallicity gradients which show the strongest negative +trends. +(M∗), total star formation rate (SFR), and gas fraction (calculated as +the sum of the Hi and H2 gas mass relative to the gas plus stellar +mass), all properties which might be expected to regulate mixing in +the disc (Krumholz & Ting 2018). We find a weak correlation with +M∗ and SFR, and no correlation with gas fraction. +In Kreckel et al. (2020), the mixing scale for metals, as quantified +via the two point correlation function, was found to display similar +weak correlations with SFR. However, those authors found the most +pronounced correlation with the gas velocity dispersion, indicating +that the homogeneity of the metal distribution in the gas (and cor- +responding mixing scale length) was regulated by gas turbulence. +To test this, we consider 𝜎(O/H) as a global measure of this metal +distribution homogeneity and compare it with three tracers of the +multi-phase gas velocity dispersion. In the bottom panels of Figure +13, we show two constraints from the ionized gas: the median ion- +ized gas velocity dispersion, measured across the entire MUSE map +(𝜎H𝛼,disc) and the median ionized gas velocity dispersion measured +only within the H ii regions (𝜎H𝛼,HII). For both of these we con- +sider only pixels or regions where the H𝛼 emission achieves a S/N > +20, to minimize uncertainties introduced by the low spectral resolu- +tion of MUSE. We also correct for the instrumental dispersion (∼49 +km s−1 at H𝛼, as reported in Bacon et al. 2017). The disc as a whole +shows typically higher dispersions (∼30–35 km s−1) compared to +the H ii regions (∼20–30 km s−1), reflecting elevated gas dispersion +in the diffuse ionized gas (Moiseev & Lozinskaya 2012; Moiseev +et al. 2015; Della Bruna et al. 2020). Both show positive correlations +with 𝜎(O/H). The tightest correlation is seen when considering the +molecular gas velocity dispersion (𝜎CO), measured from the median +value within the ‘strict’ second moment maps (Leroy et al. 2021). +These values are significantly smaller (∼2–4 km s−1), reflecting the +thin mid-plane distribution of this colder and dense ISM component. +These correlations between the global scatter in metallicities (rela- +tive to the radial gradients) and the turbulent state of the ISM (in +both the ionized and molecular material) demonstrate convincingly +MNRAS 000, 1–27 (2022) + +IC5332 +NGC0628 +NGC7496 +NGC1365 +log L(Ha) [erg/s] +42 +41 +40 +39 +38 +31 +36 +3.0 +Av [mag] +2.5 +2.0 +1.5 +1.0 +0.5 +12+log(0/H) +8.6 +8.5 +8.4 +8.3 +8.2 +8:1 +-0.5 +U +601 +-1.0 +.1.5 +-2.0 +2.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 0.0 +0.5 +1.0 +1.5 +2.0 +2.50.0 +0.5 +1.0 +1.5 +2.0 +2.5 0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +R / reffPHANGS Nebular Catalogue +17 +Figure 10. Metallicity 12+log(O/H) radial gradients based on the H ii regions within each of the 19 galaxies. Galaxies are sorted by the stellar mass, from low +(top left) to high (bottom right). For each galaxy we show the radial trends normalized to reff. A simple linear fit (central line) shows very good agreement with +the median value in 0.5 reff wide bins. For each bin we also track the 1𝜎 scatter (outer lines). These radial trends are compared to the locations of co-rotation +(blue vertical lines; Williams et al. 2021), as measured from the stellar kinematics. The bottom right panel shows the linear trends for all galaxies overplotted, +with the colour-scale indicating the total stellar mass. +that the ISM dynamics play a critical role in regulating the mixing +of metals across galaxy discs. +5.4 Local metallicity variations +In Kreckel et al. (2019) we identified a strong correlation between +Δ(O/H) and ionization parameter (as traced by [S iii]/[S ii]). Here, +we revisit secondary correlations between Δ(O/H) and other local +ISM physical conditions. Figure 14 compares Δ(O/H) with the H𝛼 +velocity dispersion measured across the integrated H ii region spec- +tra and with the A𝑉 measured via the Balmer decrement. There is +a weak negative correlation with velocity dispersion echos the result +found globally in Figure 13, and is reflected systematically within in- +dividual galaxies (dashed lines). Given the low instrumental velocity +resolution (∼49 km s−1 at H𝛼) and moderately large integrated scales +(∼70 pc), we expect that our determined ionized gas velocity disper- +sion traces predominantly the larger scale ISM turbulence rather +than local cloud turbulence (likely contributing only ∼10 km s−1; +Relaño et al. 2005; Medina et al. 2014), though we cannot exclude +the possibility that some of these systems experience strong stellar +winds (which can contribute to expansion velocities by as much as +∼60 km s−1; Egorov et al. 2014, 2017). We also identify a positive +correlation between Δ(O/H) and A𝑉 , which is again present within +individual galaxies (dashed lines) though with more variations be- +tween galaxies. However, both of these correlations could also arise +from the correlation of Δ(O/H) with H𝛼 luminosity (see Figure 5 in +Kreckel et al. 2019). +In Figure 15 we qualitatively examine the locations of regions with +particularly high and low metallicity (relative to the radial gradient) +within the galaxy discs. Here, we make a simple cut and highlight +MNRAS 000, 1–27 (2022) + +8.7 +NGC5068 +IC5332 +NGC1087 +NGC1385 +NGC7496 +8.6 +8.5 +8.4 +8.3 +8.2 +8.1 +NGC2835 +NGC0628 +NGC4254 +NGC4303 +8.6 +8.5 +8.4 +2 +log(O/H) +8.3 +8.2 +8.1 +NGC4535 +NGC1300 +NGC1512 +NGC1672 +NGC4321 +8.6 +2 +8.5 +8.4 +8.3 +8.2 +8.1 +NGC1566 +NGC3627 +NGT1433 +·NGC1365 +8.6 +8.5 +8.4 +8.3 +log(M) +8.2 +8.1 +9 +10, +11 +3 +0 +3 +4 +0 +1 +2 +3 +4 +0 +4 +2 +3 +4 +R / reff18 +Groves et al. +109 +1010 +1011 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +median AV [mag] +r = 0.43 +100 +101 +r = 0.28 +0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 +r = -0.01 +109 +1010 +1011 +M ∗ [M ⊙ ] +−2.1 +−2.0 +−1.9 +−1.8 +−1.7 +−1.6 +−1.5 +−1.4 +median log(U) +r = 0.26 +100 +101 +SFR [M ⊙ / year] +r = 0.26 +0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 +gas fraction +r = 0.24 +Figure 11. Comparison of median extinction (A𝑉 ; top) and median ionization parameter (𝑈; bottom) as a function of global galaxy properties (stellar mass, star +formation rate, gas fraction). In the top left of each plot we show the Pearson correlation coefficient of those quantities, 𝑟. A𝑉 shows modest correlations with +stellar mass and SFR, reflecting an increased gas content (and hence dust content) in these systems. 𝑈 shows no correlations, reflecting that it is set primarily +by local ISM conditions. +Figure 12. Comparison of parameters derived from the radial metallicity gradients with global galaxy properties (stellar mass, star formation rate, gas fraction). +These include the metallicity measured at 1 reff(top), and the metallicity slope (bottom). Points are colour coded by the total stellar mass. +MNRAS 000, 1–27 (2022) + +8.65 +8.60 +11.00 +8.55 +1 +10.75 +8.50 +(H/0)60I+2 +8.45 +10.50 +8.40 +8.35 +2 +10.25 +8.30 +[m +8.25 +109 +1010 +1011 +100 +101 +0.000.050.100.150.200.250.300.35 +10.00 +(M +0.05 +log10( +9.75 +0.00 + slope +-0.05 +9.50 +etallicity +-0.10 +9.25 +-0.15 +met +9.00 +-0.20 +109 +1010 +1011 +100 +101 +0.000.050.100.150.200.250.300.35 +M*[M。] +SFR [M。 / year] +gas fractionPHANGS Nebular Catalogue +19 +109 +1010 +1011 +M ∗ [M ⊙ ] +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +σ (O/H) +r = -0.22 +100 +101 +SFR [M ⊙ / year] +r = -0.31 +0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 +gas fraction +r = -0.23 +20 +25 +30 +35 +40 +σHα, disc [km/s] +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +σ (O/H) +r = -0.63 +20 +25 +30 +35 +40 +σHα, HII [km/s] +r = -0.65 +0 +1 +2 +3 +4 +5 +6 +7 +8 +σCO [km/s] +r = -0.67 +Figure 13. Correlations between the global scatter in metallicity (𝜎(O/H)), after removing the first order radial gradient, as a function of galaxy integrated +properties. In the top right of each plot we show the Pearson correlation coefficient of those quantities, 𝑟. The total stellar mass (M∗), total star formation rate +(SFR), and gas fraction show weak or absent trends. More pronounced correlations are seen with the ionized gas velocity dispersion, as measured across the full +disc (𝜎H𝛼,disc) or just the H ii regions (𝜎H𝛼,HII), as well as with the molecular gas velocity dispersion (𝜎CO). +H ii regions with Δ(O/H) > 0.05 in red, and H ii regions with Δ(O/H) +< −0.05 in blue. While in some cases the enriched regions appear +concentrated along spiral arms or at bar-ends, they can also be found +distributed across the galaxy discs. Similarly, the regions with de- +creased abundances show some clustering (reflecting the homogene- +ity on kpc scales quantified in Kreckel et al. 2020), but no obvious +patterns with galaxy environments. There is no clear difference in +Δ(O/H) between the different environmental masks of Querejeta et al. +(2021), as was also shown by Williams et al. (2022). This reflects +a complicated relation between enrichment patterns and individual +galaxy dynamics. A more detailed analysis of a larger sample of H ii +regions with a larger dynamic range in these quantities will be need +to disentangle what drives these second-order metallicity variations +in galaxies. +6 DISCUSSION +6.1 Metallicity variations +As reported in Williams et al. (2022), our metallicity gradients are +dominated by linear radial trends (Figure 10) following earlier works +on larger samples (Sánchez et al. 2014; Espinosa-Ponce et al. 2022; +Barrera-Ballesteros et al. 2022, e.g.). Following Sánchez-Menguiano +et al. (2018) we do not measure the gradient within 0.5reff, however +unlike that work we see no consistent indication for a flattening within +our galaxy sample at larger radii. Our smaller sample as compared to +the 109 galaxies in Sánchez-Menguiano et al. (2018) mean its harder +to draw conclusions why this is, but the higher physical resolution +of the PHANGS–MUSE sample (∼ 70 pc) as compared to theirs +(∼ 460 pc) could be one possible reason. Second-order variations +are small, typically below ∼0.05 dex, and reflect a remarkable level +of homogeneity in the metallicity distribution across galaxies. They +do not show any correlation with co-rotation radius, which has been +predicted by simulations to influence the efficiency of mixing within +the disc (Spitoni et al. 2019). +We establish that there is a weak global correlation between the +magnitude of metallicity variations (𝜎(O/H)) and star formation rate +(Figure 13, top), but tighter correlations are observed with measures +of the global median gas velocity dispersion (Figure 13, bottom). This +holds when considering both ionized and molecular gas phases. This +is consistent with more turbulent ISM conditions leading to mixing +on larger scales, resulting in overall more homogeneity in the metal +distribution. We do not aim to investigate the source of turbulence +in this paper, and careful work will be needed to disentangle the +effects of star formation, secular dynamical processes and external +gas accretion processes. +We identify H ii regions with metallicities significantly different +from the linear radial gradients, and observe that on local scales +this correlation holds, with enriched H ii regions correlating with +lower local ionized gas velocity dispersions (Figure 14). These also +correlate with dustier local environments, as traced by the Balmer +decrement. Assuming gas and dust are well mixed, this would suggest +that more enriched regions are associated with higher density gas. +Together with the trend established with H𝛼 velocity dispersion, this +leads to a picture where a denser, calmer ISM facilitates localized +pockets of enrichment. At the other end of the spectrum, relatively +more metal poor gas is associated with lower gas densities and in- +creased turbulence. The local metallicity variations we report are +MNRAS 000, 1–27 (2022) + +20 +Groves et al. +Figure 14. Metallicity variations measured after removing the radial gradient (Δ(O/H)) show a negative correlation with the H𝛼 velocity dispersion (𝜎𝐻 𝛼) +measured across the H ii region (left), and a positive correlation with the A𝑉 measured from the Balmer decrement (right). The greyscale indicates the distribution +density, and linear fits performed for individually galaxies are shown as dashed lines. Relative to a perfectly linear metallicity gradient (blue line), enriched +regions are found at low velocity dispersions and dusty environments. +consistent with the picture developed in Kreckel et al. (2020), which +quantified the mixing scales within eight of these 19 galaxies and +determined that metallicity variations are likely reflecting dilution +rather than pollution of the ISM. +Another key result from Kreckel et al. (2019) in their initial study +of eight of the PHANGS–MUSE targets is the identification of sys- +tematic azimuthal variations in the metallicity distribution. This has +been confirmed for the full sample of 19 galaxies in Williams et al. +(2022), who analyzed interpolated metallicity maps. What remains +less clear is how these variations may or may not correlate with +galaxy environments (centre, bar, spiral arm, interarm). Looking at +integrated environments across the sample, Williams et al. (2022) +were unable to recover any systematic trends aside from systematic +enrichment of galaxy centres. This is somewhat in conflict with pre- +vious results on individual galaxies (Ho et al. 2017; Vogt et al. 2017; +Ho et al. 2018), and claims based on growing samples (Sánchez- +Menguiano et al. 2019) for correlations in metallicity variations with +spiral arms. In the initial sample of eight galaxies of Kreckel et al. +(2019), half were found to have variations correlating with spiral +structure but often in only a single spiral arm. +We do not revisit this interesting topic as we believe it requires +careful dynamical considerations, tailored to each galaxy, but it +demonstrates the complicated abundance patterns in relation to the +galaxy environments (Figure 15). While in some galaxies the en- +riched regions (in red) appear to strongly trace the spiral pattern +(e.g. NGC 1365, NGC 1566, NGC 1672), they can also generally +be found throughout the entire disk. Qualitatively, the regions with +reduced metallicity (in blue) often appear somewhat clustered and +located at bar-ends. These maps highlight the challenges in estab- +lishing the role of galaxy environment and the role of gas flows in +regulating the enrichment patterns in galaxy discs. +6.2 Missing nebulae and objects failing the BPT cuts +Our catalogue consists of objects that are selected to be bright in H𝛼, +but as is apparent in the BPT diagrams (Figure 4) these are not all +nebulae where the ionization is dominated by photoionization from +young massive stars. Based on our consideration of diagnostic line +ratios (Section 4.2), this results in a sample of 23,244 nebulae that +we classify as H ii regions. However, it leaves 7,546 objects in our +catalogue for which we provide no definitive classification. These +could be H ii regions blended with strong DIG or AGN emission, +SNRs or PNe. Note that 609 objects are already excluded entirely +from this analysis as they fall at the field edge. +Of the unclassified objects, we find that 4,688 are labelled as +‘composite’ based on the [O iii]/H𝛽 vs. [N ii]/H𝛼 BPT diagnostic +(BPT_NII = 1), and are quite likely H ii regions. The commonly +used BPT demarcation empirically established by Kauffmann et al. +(2003) was developed for classification of central kpc-scale and in- +tegrated galaxy spectra. Recent work has begun to explore whether +this parameter space is sufficiently represented once outer disc envi- +ronments are considered, and wider parameter space including kine- +matic diagnostics are included (Law et al. 2021). With our work, we +consider even smaller physical scales (<100 pc), and indeed recent +modelling has shown that individual H ii regions throughout their +evolution may populate the ‘composite’ regions of the BPT diagram +(falling between the Kauffmann et al. 2003 and Kewley et al. 2001 +demarcations) for short periods during their earliest phase of evolu- +tion (Pellegrini et al. 2020). One of the main long-term science goals +for producing this catalogue is to provide the necessary database of +high-quality emission-line fluxes necessary to continue such detailed +comparisons with cutting-edge models. Within PHANGS, ongoing +work applies a bayesian framework to match line ratios in emission +line objects with different model grids, with the goal of establishing +new classification methods (Congiu et al. in prep). +MNRAS 000, 1–27 (2022) + +0.20 +3 +0.15 +0.10 +0.05 +2 +/ / II +(N)60) +0.00 +-0.05 +1 +-0.10 +-0.15 +0.20 +0 +0 +10 +20 +30 +40 +0.0 +0.5 +1.0 +1.5 +2.0 +[km/s] +Ay [mag] +OHαPHANGS Nebular Catalogue +21 +Figure 15. Maps of the H𝛼 emission overlaid with locations of H ii regions with particularly high (red) or low (blue) metallicities compared to the radial +gradient. Here we have required low statistical uncertainties (<0.01 dex) and be offset from the radial trend by more than 0.05 dex. While patterns appear in some +galaxies relative to the spiral pattern (e.g. NGC 1672, NGC 1365), and spatial clustering is apparent in many systems (e.g. NGC 2835, NGC 3627), variations +in the metallicity patterns are not systematically observed in all galaxies. +Active Galactic Nuclei (AGN) provide another potential source of +gas excitation, and are present in 7 (37%) of our galaxies (as labelled +in Table 1), with four of these AGN hosting molecular gas outflows +(three galaxies without AGN also host molecular gas outflows; Stuber +et al. 2021). In some cases (e.g. NGC 1365; Venturi et al. 2018) they +represent a remarkable dominant source of ionization, with [O iii] +bright ionization cones visible across the central kpc of the galaxy +and extending over nearly the full MUSE field of view. In cases +of lower luminosity AGN (NGC 1433, NGC 4303, NGC 7496), it +can be difficult to spatially isolate any AGN contributions, as they +appear as extended ionized structures associated with (presumably) +outflowing material, and seen in projection with H ii regions will +bias the emission line diagnostics. Visualization of the emission line +maps for all galaxies is available in Emsellem et al. (2022). +MNRAS 000, 1–27 (2022) + +ngc1087 +ngc0628 +ngc1300 +ngct +kpc +. kpc +1 kpc +ngc1385 +ngc1566 +ngc1512 +ngc1433 +167 +kpo +1kpc +1kpc +ngc3627 +ngc2835 +ngc4303 +1 kpc +kpc +ngc4535 +ngc7496 +ngc5068 +ngc432 +"kpo22 +Groves et al. +Supernova remnants (SNRs) are the network of shocks caused +by supernova explosions as they expand into the surrounding ISM. +They are often identified via their strong [S ii] and H𝛼 emission (e.g. +Long et al. 2022). As predicted by shock models (Allen et al. 2008), +their diagnostic line ratios populate regions of the BPT diagrams +that partially overlap with photoionization. With typical sizes of less +than 100 pc, most of these objects are expected to be unresolved in +our data. Ongoing projects within the PHANGS collaboration are +building catalogues of SNRs, focusing on detection of these objects +via their distinctive line ratios and broadened line kinematics (Li +et al. in prep, Congiu et al. in prep). Preliminary results suggest +between 1000–5000 SNRs are present in our data, and may make up +a significant fraction of the unclassified objects. +Planetary nebulae (PNe) are shells of gas expelled by intermediate +mass stars (1-8 M⊙; Parker 2022) and ionized by the central source +in the end phase of their life. They are particular bright in [O iii] line +emission (Parker 2022), but can also emit strongly in H𝛼, and are un- +resolved at the distances of our target galaxies. Given that the central +stars exhibit a harder ionizing spectrum than typical H ii regions, they +could be expected to fall in regions of the BPT diagrams tradition- +ally populated by AGN. Recently, Scheuermann et al. (2022) used +the PHANGS–MUSE observations to identify PNe, with selection +based on [O iii] emission and source classification confirmed based +on diagnostic line ratios. They find a total of 899 PNe across these 19 +galaxies, 193 of which are within 1′′ of sources in our nebular cata- +logue. 158 of these (82%) are unclassified, failing our BPT cuts, and +making up a very small fraction of the unclassified objects. Overall, +these make up a very small fraction of our nebular catalogue, which +is unsurprising as many of the PNe are faint or undetected in H𝛼. +6.3 Odd/misclassfied nebulae +Because of our automated approach to object identification, line +fitting, and object classification, our catalogue contains objects that +appear as unusual outliers in critical diagnostics. We examine the +spectra of a subset of these objects to understand if there are any +systematic problems and provide some explanation for these outliers. +One category of unusual object are H ii regions identified as having +particular large H𝛼 velocity dispersions, and we look at the 12 objects +𝜎𝐻 𝛼 > 200 km s−1. One object (NGC1385, #12) clearly corresponds +to a foreground star missed by our flagging. We find that the central +AGN in both NGC 1365 and NGC 1566 end up classified as H ii +regions because we perform only single Gaussian fits to our emission +lines, and the underlying broad line component results in a biased +fit. Two other objects (#589 in NGC 1300 and #35 NGC 3627) +appear to be a result of poorly constrained continuum fits and the +misclassification of noise as spectral lines within the spectra. The +remaining seven objects clearly contain a broader secondary line +component that is coincident with the H𝛼 line emission. Whether +this is a signature of massive stars driving strong winds or due to +a background galaxy seen in projection is difficult to distinguish +with these spectra alone. PHANGS–HST imaging reveals unresolved +bright sources coincident with two of the nebulae, but no obvious +background galaxy or bright stellar source at the other positions. +Six objects that we identify as H ii regions result in inferred metal- +licity measurements that are unusually low, 12+log(O/H) < 8.0, and +in fact all appear to be spurious sources. On closer inspection, two are +faint foreground stars that had not been identified, one corresponds +to an object where the stellar continuum is poorly constrained and +noise peaks are fit as emission lines, and the remaining three show +secondary broadened Gaussian profiles that bias the H𝛼 fit and skew +the resulting line ratio diagnostics. Many of these exhibit S/N∼10 in +Figure 16. A comparison of MUSE and HST H𝛼 imaging in a star-forming +complex of NGC 1672 (Barnes et al. 2022). While it is clear that MUSE +does separate distinct complexes, the high ∼8 pc HST resolution reveals +complicated H𝛼 morphologies including clustered compact regions which +cannot be distinguished from neighbouring brighter regions at the 70 pc +MUSE resolution. +fainter lines ([S ii], [O iii]), suggesting more stringent S/N cuts would +be effective in excluding spurious sources. +Finally, our observation of NGC 1672 happened to occur only +shortly after a supernova event in the galaxy, AT2017gax, and as a +result is poorly fit in our catalogue and has ID #429. +6.4 Impact of Resolution +Given that star formation is often observed to be clustered (that is, +star formation occurs in high gas density regions like spiral arms), +one limiting factor in constructing our nebular catalogue is the phys- +ical resolution of our observations. In Table 1 we list the physical +resolution within each galaxy, which reflects both the seeing condi- +tions during the observation and the distance to the source. Values +range from 23–104 pc, sufficient to identify star forming regions that +are isolated but likely insufficient to separate clustered star-forming +complexes. In comparison with HST H𝛼 narrowband imaging (Fig- +ure 16), it is apparent that objects identified as a single source in +MUSE can be comprised of several neighbouring complexes. We are +not able to fully account for this effect without additional HST H𝛼 +imaging, which is currently only available for a handful of our targets +(NGC 628, NGC 1672, NGC 3351) but will be available for the full +sample with an upcoming PHANGS HST narrow band survey. +The effect of blending of objects within this catalogue is discussed +in Santoro et al. (2022) in relation to its impact on the H ii region +luminosity function. Those authors estimate the mean separation +between H ii regions per galaxy and find that it shows only a modest +correlation with changes in the H ii region luminosity function slope +and cannot be responsible for differences in luminosity function slope +that are observed between galaxies. +Another location where blending becomes a clear issue is in the +centres of galaxies hosting starbursting rings, where the extreme spa- +tial concentration of bright H ii regions make decomposition chal- +lenging. Region identification is further complicated by the increased +levels of diffuse ionized gas, associated with the high stellar den- +sity contributing an additional ionization component (see Section +MNRAS 000, 1–27 (2022) + +MUSE +Nebulae +MUSE +HST +500pc +Ha +Ha +500pcPHANGS Nebular Catalogue +23 +6.5; Belfiore et al. 2022). Three galaxies (Figure 17) host particu- +larly bright nuclear star-forming rings (NGC 1672, NGC 3351, NGC +4321), and we caution against over-interpretation of these integrated +nebular fluxes without more careful spatial decomposition or de- +blending of objects. +While the HST comparisons and H ii region luminosity functions +suggest blending of nebulae is not a huge issue (excluding starburst- +ing rings), even if it is occurring, the resolutions we achieve with +MUSE are generally insufficient for accurate determination of H ii +region sizes. This has lead us to exclude any size determination +from our catalogue. This issue was been highlighted in comparisons +of MUSE H ii regions with HST narrow band H𝛼 imaging (Barnes +et al. 2022), where it is apparent that at ∼70 pc scales MUSE is unable +to provide robust quantification of sizes or morphologies (Hannon +et al. 2022). In our catalogue, we therefore provide the number of +0.2′′× 0.2′′pixels associated with the spatial mask of each H ii re- +gion, as a way of distinguishing the larger objects and derive surface +brightnesses. +6.5 Diffuse Ionized Gas +A prominent feature in our deep emission line maps is the pervasive +diffuse ionized gas (DIG) component. As was shown in Zurita et al. +(2000) and more recently revisited for the PHANGS–MUSE galaxies +(Belfiore et al. 2022), the bulk of this emission is spatially correlated +with H ii region locations, and can be explained well by a model +where ionizing photons leak from their H ii regions and propagate out +to ∼kpc scales in the disc. However, with an additional consideration +of the emission line ratios in the gas, it is clear that an additional +contribution from hot low-mass evolved stars (HOLMES) is required +(Belfiore et al. 2022). +Systematic differences in line ratios observed in H ii regions com- +pared to the DIG were first established in Milky Way observations +(Haffner et al. 2009), and have the potential to significantly bias line +flux measurements particularly for [S ii] and [N ii] which are emitted +strongly in the DIG. However, the irregular spatial distribution and +low-surface brightness of the DIG makes it difficult to model and sub- +tract this component. In our construction of the nebular catalogues, +we have not applied corrections for the DIG along the line of sight. +As most of our objects are bright, exhibiting median H𝛼 luminosities +of 2×1037 ergs s−1 and H𝛼 surface brightnesses of 2×1039 ergs s−1 +kpc−2, we expect that the DIG should not strongly impact our line +ratios or derived quantities (e.g. dust attenuation, metallicity, ion- +ization parameter). However, we still caution that physical quantities +interpreted from integrated line fluxes may still be impacted, espe- +cially for H ii regions with low surface brightnesses or in crowded +environments. +7 CONCLUSIONS +We present a new catalogue of 30,790 nebulae, selected morpholog- +ically in H𝛼 images, that are fully contained within the PHANGS– +MUSE coverage of 19 nearby star-forming spiral galaxies. For all +nebulae, we construct an integrated spectrum and measure integrated +line fluxes and line kinematics for strong lines across the optical +(4800–9300Å) spectrum (Table 2). We calculate derived properties, +including dust attenuation via the Balmer decrement, characterise +their galactic environments (centre, bar, spiral arm, interarm, disk), +and classify objects based on their emission line (BPT) diagnostics. +For the 23,244 nebulae that we classify as H ii regions (Section 4.2), +we calculate the gas-phase metallicity and ionization parameter. A +full list of properties characterised in our catalogue is included in +Table 4. +We examine the radial gradients of the nebula physical properties +within galaxies, finding that the metallicity gradients demonstrate +the most pronounced trends (Table 9). We also look in more detail at +the residual variation in metallicity (Δ(O/H)), after fitting and sub- +tracting the linear radial trends in log(O/H), and quantify the global +scatter in this residual metallicity within each galaxy (𝜎(O/H)). As +was previously shown using a sub-sample of our galaxies (Kreckel +et al. 2019), we recover small scatters with 𝜎(𝑂/𝐻) =0.03 – 0.07 +dex, and show that this scatter correlates weakly with the global star +formation rate and correlates strongly with both ionized and molecu- +lar gas velocity dispersions. Local metallicity variations show further +support for these trends with the turbulent condition of the ISM, and +we find that enriched regions preferentially show lower H𝛼 velocity +dispersion and higher dust attenuation, reflecting a correspondence +with calmer pockets of the ISM. These correlations between the +metallicity variations (relative to the radial gradients) and the turbu- +lent state of the ISM (in both the ionized and molecular material) +demonstrate convincingly that the ISM dynamics play a critical role +in regulating the mixing of metals across galaxy discs. Correlations +with galaxy environment are qualitatively apparent in some galax- +ies, but systematic trends are less clear and more detailed dynamical +modelling is required. +With this catalogue, we do not yet include a full analysis of the +temperature sensitive auroral emission lines ([N ii]5755, [S iii]6313, +[O ii]7320,7330), which are contained within our wavelength range +and detected in about a thousand H ii regions. Careful fitting of these +faint lines (∼1% of the intensity of H𝛼) is required to determine robust +line fluxes, and will be the subject of future work. In combination +with the collisionally excited strong-lines, we aim to derive electron +temperatures and simultaneously (using the [S ii] density diagnostic) +determine electron densities for a sub-sample of our catalogue. +This catalogue of the young, ionized nebulae represents a key pa- +rameter when developing a model of the baryon cycle within galaxies +on resolved (<100 pc) scales. Within the PHANGS collaboration, +we aim to quantify key stages in this process by characterising with +PHANGS–ALMA the molecular gas (Sun et al. 2020) and individ- +ual giant molecular clouds (GMCs; Rosolowsky et al. 2021; Hughes +et al. in prep), and with PHANGS–HST the individual star clusters +and stellar associations (Turner et al. 2021; Larson et al. in prep). +Comparison of the H ii region and GMC distributions reveal rela- +tively little spatial overlap (Kreckel et al. 2018), indicative of short +cloud disruption timescales (Kim et al. 2021; Chevance et al. 2022). +Ongoing work links individual H ii regions with their parent GMC, +to determine if stellar feedback has a measurable impact on molec- +ular cloud properties (Zakardjian et al. in prep), and searches for +super-bubble morphologies in the molecular gas, to quantify feed- +back energetics (Watkins et al. in prep). Cross-matching our H ii re- +gion catalogue with young stellar associations provides quantitative +constraints on the mass and age of the stars powering these ionized +nebulae, uniquely enabling us to constrain the evolutionary sequence +and link the ionizing photon budget to the ionization state of the gas +(Barnes et al. 2022, Scheuermann et al. submitted, Egorov et al. in +prep). In the near future, these results will be further complemented +by PHANGS–JWST maps of the earliest embedded phases of star +formation and constraints on the dust chemistry, further completing +our view of the complete star-formation cycle. +MNRAS 000, 1–27 (2022) + +24 +Groves et al. +Figure 17. Galaxy centres from NGC 1672, NGC 3351 and NGC 4321, which host nuclear star-forming rings. H𝛼 emission (greyscale) shows clustered and +clumpy star-forming regions, which are only moderately well separated into individual H ii regions (red contours). Region sizes in this particular environment +are also significantly overestimated. +ACKNOWLEDGEMENTS +This work was carried out as part of the PHANGS collaboration. +Based on observations collected at the European Southern Ob- +servatory under ESO programmes 1100.B-0651, 095.C-0473, and +094.C-0623 (PHANGS–MUSE; PI Schinnerer), as well as 094.B- +0321 (MAGNUM; PI Marconi), 099.B-0242, 0100.B-0116, 098.B- +0551 (MAD; PI Carollo) and 097.B-0640 (TIMER; PI Gadotti). +KK, FS, and OE gratefully acknowledge funding from the Ger- +man Research Foundation (DFG) in the form of an Emmy Noether +Research Group (grant number KR4598/2-1, PI Kreckel). KK, EJW +and SCOG acknowledge support from the Deutsche Forschungs- +gemeinschaft (DFG, German Research Foundation) – Project-ID +138713538 – SFB 881 (“The Milky Way System”, subprojects B1, +B2, B8 and P1). SCOG also acknowledges funding from the European +Research Council via the ERC Synergy Grant “ECOGAL – Under- +standing our Galactic ecosystem: From the disk of the Milky Way +to the formation sites of stars and planets” (project ID 855130) and +from the Heidelberg Cluster of Excellence (EXC 2181 - 390900948) +“STRUCTURES: A unifying approach to emergent phenomena in +the physical world, mathematics, and complex data”, funded by the +German Excellence Strategy. ATB and FB would like to acknowl- +edge funding from the European Research Council (ERC) under +the European Union’s Horizon 2020 research and innovation pro- +gramme (grant agreement No.726384/Empire). FS, ES, and TGW +acknowledge funding from the European Research Council (ERC) +under the European Union’s Horizon 2020 research and innovation +programme (grant agreement No. 694343). E.C. acknowledge sup- +port from ANID Basal projects ACE210002 and FB210003. PSB +acknowledge support from the project project PID2019-107427-GB- +31 funded by the MCIN/AEI/10.13039/50110001103. This research +has made use of the NASA/IPAC Extragalactic Database (NED) +which is operated by the Jet Propulsion Laboratory, California In- +stitute of Technology, under contract with NASA. It also made use +of a number of python packages, namely the main astropy package +(Astropy Collaboration et al. 2013, 2018), numpy (Harris et al. 2020) +and matplotlib (Hunter 2007). +DATA AVAILABILITY +The MUSE data underlying this work are presented in Emsellem et al. +(2022) and are available at the ESO archive 5 and CADC archive 6. +The catalogue of all identified nebulae, along with fits files containing +masks of the individual nebulae locations, are published along with +this paper and available in the online supplementary material of the +journal as well as at the CADC archive. +REFERENCES +Abazajian K. N., et al., 2009, ApJS, 182, 543 +Allen M. G., Groves B. A., Dopita M. A., Sutherland R. S., Kewley L. J., +2008, ApJS, 178, 20 +Alvarez-Hurtado P., Barrera-Ballesteros J. K., Sánchez S. F., Colombo D., +López-Sánchez A. R., Aquino-Ortíz E., 2022, ApJ, 929, 47 +Anand G. S., et al., 2021, MNRAS, 501, 3621 +Astropy Collaboration et al., 2013, A&A, 558, A33 +Astropy Collaboration et al., 2018, AJ, 156, 123 +Bacon R., et al., 2010, in Ground-based and Airborne Instrumentation for +Astronomy III. p. 773508, doi:10.1117/12.856027 +Bacon R., et al., 2017, A&A, 608, A1 +Baldwin J. A., Phillips M. M., Terlevich R., 1981, PASP, 93, 5 +Barnes A. T., et al., 2021, MNRAS, 508, 5362 +Barnes A. T., et al., 2022, arXiv e-prints, p. arXiv:2205.05679 +Barrera-Ballesteros J. K., et al., 2016, MNRAS, 463, 2513 +Barrera-Ballesteros J. K., et al., 2018, ApJ, 852, 74 +Barrera-Ballesteros J. K., et al., 2022, arXiv e-prints, p. arXiv:2206.07058 +Belfiore F., et al., 2017, MNRAS, 469, 151 +Belfiore F., et al., 2022, A&A, 659, A26 +Berg D. A., Pogge R. W., Skillman E. D., Croxall K. V., Moustakas J., Rogers +N. S. J., Sun J., 2020, ApJ, 893, 96 +Blanc G. A., et al., 2013, AJ, 145, 138 +Boardman N. F., Zasowski G., Newman J. A., Sanchez S. F., Schaefer A., +Lian J., Bizyaev D., Drory N., 2021, MNRAS, 501, 948 +Bothwell M. S., Maiolino R., Kennicutt R., Cresci G., Mannucci F., Marconi +A., Cicone C., 2013, MNRAS, 433, 1425 +Bundy K., et al., 2015, ApJ, 798, 7 +Calzetti D., 2001, PASP, 113, 1449 +Calzetti D., Kinney A. L., Storchi-Bergmann T., 1994, ApJ, 429, 582 +5 https://archive.eso.org/scienceportal/home?data_collection=PHANGS +6 https://www.canfar.net/storage/vault/list/phangs/RELEASES/PHANGS- +MUSE +MNRAS 000, 1–27 (2022) + +NGC1672 +NGC3351 +NGC4321 +15°49'35" +00 +11°42'25" +-59°14'40" +30' +20" +25" +(ICRS) +15" +(ICRS) +50 +Dec ( +20" +Dec +C +10" +C +55" +15" , +8 +05 +15'00" +10″ +00" +1 kpc +05" +h45m44 +43s +42s +4is +10h43m58.5s +58.0s +57.55 +57.05 +2h22m55.5 +55.05 +54.5 +54.09 +RA (ICRS) +RA (ICRS) +RA (ICRS)PHANGS Nebular Catalogue +25 +Cappellari M., 2017, MNRAS, 466, 798 +Cappellari M., Copin Y., 2003, MNRAS, 342, 345 +Cardelli J. A., Clayton G. C., Mathis J. S., 1989, ApJ, 345, 245 +Chabrier G., 2003, PASP, 115, 763 +Chevance M., et al., 2022, MNRAS, 509, 272 +Ciardullo R., Feldmeier J. J., Jacoby G. H., Kuzio de Naray R., Laychak +M. B., Durrell P. R., 2002, ApJ, 577, 31 +Croom S. M., et al., 2021, MNRAS, 505, 991 +Croxall K. V., van Zee L., Lee H., Skillman E. D., Lee J. C., Côté S., Kennicutt +Robert C. J., Miller B. W., 2009, ApJ, 705, 723 +Curti M., Mannucci F., Cresci G., Maiolino R., 2020, MNRAS, 491, 944 +De Marchi G., et al., 2016, MNRAS, 455, 4373 +Della Bruna L., et al., 2020, A&A, 635, A134 +Diaz A. I., Terlevich E., Vilchez J. M., Pagel B. E. J., Edmunds M. G., 1991, +MNRAS, 253, 245 +Dopita M. A., Rich J., Vogt F. P. A., Kewley L. J., Ho I. T., Basurah H. M., +Ali A., Amer M. A., 2014, Ap&SS, 350, 741 +Dopita M. A., Kewley L. J., Sutherland R. S., Nicholls D. C., 2016, Ap&SS, +361, 61 +Draine B. T., 2011, Physics of the Interstellar and Intergalactic Medium. +Princeton University Press +Driver S. P., et al., 2009, Astronomy and Geophysics, 50, 5.12 +Egorov O. V., Lozinskaya T. A., Moiseev A. V., Smirnov-Pinchukov G. V., +2014, MNRAS, 444, 376 +Egorov O. V., Lozinskaya T. A., Moiseev A. V., Shchekinov Y. A., 2017, +MNRAS, 464, 1833 +Emsellem E., et al., 2022, A&A, 659, A191 +Epinat B., Amram P., Marcelin M., 2008, MNRAS, 390, 466 +Erroz-Ferrer S., et al., 2019, MNRAS, 484, 5009 +Espinosa-Ponce C., Sánchez S. F., Morisset C., Barrera-Ballesteros J. K., +Galbany L., García-Benito R., Lacerda E. A. D., Mast D., 2020, MNRAS, +494, 1622 +Espinosa-Ponce C., Sánchez S. F., Morisset C., Barrera-Ballesteros J. K., +Galbany L., García-Benito R., Lacerda E. A. D., Mast D., 2022, MNRAS, +512, 3436 +Gadotti D. A., et al., 2019, MNRAS, 482, 506 +Gaia Collaboration et al., 2018, A&A, 616, A1 +Genzel R., et al., 2015, ApJ, 800, 20 +Grasha K., et al., 2022, ApJ, 929, 118 +Haffner L. M., et al., 2009, Reviews of Modern Physics, 81, 969 +Hannon S., et al., 2022, MNRAS, 512, 1294 +Harris C. R., et al., 2020, Nature, 585, 357 +Ho I. T., et al., 2017, ApJ, 846, 39 +Ho I. T., et al., 2018, A&A, 618, A64 +Hunter J. D., 2007, Computing in Science & Engineering, 9, 90 +James P. A., Bretherton C. F., Knapen J. H., 2009, A&A, 501, 207 +Ji X., Yan R., 2022, A&A, 659, A112 +Kauffmann G., et al., 2003, MNRAS, 346, 1055 +Kennicutt Jr. R. C., 1989, ApJ, 344, 685 +Kewley L. J., Dopita M. A., 2002, ApJS, 142, 35 +Kewley L. J., Ellison S. L., 2008, ApJ, 681, 1183 +Kewley L. J., Heisler C. A., Dopita M. A., Lumsden S., 2001, ApJS, 132, 37 +Kewley L. J., Groves B., Kauffmann G., Heckman T., 2006, MNRAS, 372, +961 +Kewley L. J., Nicholls D. C., Sutherland R. S., 2019, ARA&A, 57, 511 +Kim J., et al., 2021, MNRAS, 504, 487 +Kobulnicky H. A., Kewley L. J., 2004, ApJ, 617, 240 +Kreckel K., et al., 2013, ApJ, 771, 62 +Kreckel K., Blanc G. A., Schinnerer E., Groves B., Adamo A., Hughes A., +Meidt S., 2016, ApJ, 827, 103 +Kreckel K., et al., 2018, ApJ, 863, L21 +Kreckel K., et al., 2019, ApJ, 887, 80 +Kreckel K., et al., 2020, MNRAS, 499, 193 +Krumholz M. R., Ting Y.-S., 2018, MNRAS, 475, 2236 +Lang P., et al., 2020, ApJ, 897, 122 +Lara-López M. A., et al., 2010, A&A, 521, L53 +Law D. R., et al., 2021, ApJ, 915, 35 +Le Fevre O., et al., 2003, in Guhathakurta P., ed., Society of Photo-Optical +Instrumentation Engineers (SPIE) Conference Series Vol. 4834, Discov- +eries and Research Prospects from 6- to 10-Meter-Class Telescopes II. +pp 173–182, doi:10.1117/12.457547 +Lee J. C., et al., 2022, ApJS, 258, 10 +Leroy A. K., et al., 2019, ApJS, 244, 24 +Leroy A. K., et al., 2021, ApJS, 257, 43 +Long K. S., et al., 2010, ApJS, 187, 495 +Long K. S., Blair W. P., Winkler P. F., Della Bruna L., Adamo A., McLeod +A. F., Amram P., 2022, ApJ, 929, 144 +Lugo-Aranda A. Z., Sánchez S. F., Espinosa-Ponce C., López-Cobá C., Gal- +bany L., Barrera-Ballesteros J. K., Sánchez-Menguiano L., Anderson +J. P., 2022, RAS Techniques and Instruments, 1, 3 +Maiolino R., Mannucci F., 2019, A&ARv, 27, 3 +Makarov D., Prugniel P., Terekhova N., Courtois H., Vauglin I., 2014, A&A, +570, A13 +Mannucci F., Cresci G., Maiolino R., Marconi A., Gnerucci A., 2010, MN- +RAS, 408, 2115 +Marino R. A., et al., 2013, A&A, 559, A114 +Massey P., Puls J., Pauldrach A. W. A., Bresolin F., Kudritzki R. P., Simon +T., 2005, ApJ, 627, 477 +Medina S. N. X., Arthur S. J., Henney W. J., Mellema G., Gazol A., 2014, +MNRAS, 445, 1797 +Metha B., Trenti M., Chu T., 2021, MNRAS, 508, 489 +Mingozzi M., et al., 2020, A&A, 636, A42 +Moiseev A. V., Lozinskaya T. A., 2012, MNRAS, 423, 1831 +Moiseev A. V., Tikhonov A. V., Klypin A., 2015, MNRAS, 449, 3568 +Moustakas J., Kennicutt Jr. R. C., Tremonti C. A., Dale D. A., Smith J.-D. T., +Calzetti D., 2010, ApJS, 190, 233 +O’Donnell J. E., 1994, ApJ, 422, 158 +Oey M. S., et al., 2007, ApJ, 661, 801 +Osterbrock D. E., Ferland G. J., 2006, Astrophysics of gaseous nebulae and +active galactic nuclei. University Science Books +Parker Q. A., 2022, Frontiers in Astronomy and Space Sciences, 9, 895287 +Peeples M. S., Pogge R. W., Stanek K. Z., 2008, ApJ, 685, 904 +Peeples M. S., Pogge R. W., Stanek K. Z., 2009, ApJ, 695, 259 +Peimbert M., Peimbert A., Delgado-Inglada G., 2017, PASP, 129, 082001 +Pellegrini E. W., Rahner D., Reissl S., Glover S. C. O., Klessen R. S., +Rousseau-Nepton L., Herrera-Camus R., 2020, MNRAS, 496, 339 +Pessa I., et al., 2021, A&A, 650, A134 +Pettini M., Pagel B. E. J., 2004, MNRAS, 348, L59 +Pietrinferni A., Cassisi S., Salaris M., Castelli F., 2004, ApJ, 612, 168 +Pilyugin L. S., Grebel E. K., 2016, MNRAS, 457, 3678 +Pilyugin L. S., Thuan T. X., 2005, ApJ, 631, 231 +Pilyugin L. S., Grebel E. K., Kniazev A. Y., 2014, AJ, 147, 131 +Poetrodjojo H., et al., 2018, MNRAS, 479, 5235 +Poetrodjojo H., D’Agostino J. J., Groves B., Kewley L., Ho I. T., Rich J., +Madore B. F., Seibert M., 2019, MNRAS, 487, 79 +Prescott M. K. M., et al., 2007, ApJ, 668, 182 +Querejeta M., et al., 2021, A&A, 656, A133 +Relaño M., Beckman J. E., Zurita A., Rozas M., Giammanco C., 2005, A&A, +431, 235 +Reynolds R. J., 1990, ApJ, 349, L17 +Rosolowsky E., et al., 2021, MNRAS, 502, 1218 +Rousseau-Nepton L., et al., 2019, MNRAS, 489, 5530 +Salo H., et al., 2015, ApJS, 219, 4 +Sánchez S. F., 2020, ARA&A, 58, 99 +Sánchez-Menguiano L., et al., 2016, ApJ, 830, L40 +Sánchez-Menguiano L., et al., 2018, A&A, 609, A119 +Sánchez-Menguiano L., Sánchez Almeida J., Muñoz-Tuñón C., Sánchez +S. F., Filho M., Hwang H.-C., Drory N., 2019, arXiv e-prints, p. +arXiv:1904.03930 +Sánchez-Menguiano L., Sánchez S. F., Pérez I., Ruiz-Lara T., Galbany L., +Anderson J. P., Kuncarayakti H., 2020, MNRAS, 492, 4149 +Sánchez S. F., et al., 2012, A&A, 538, A8 +Sánchez S. F., et al., 2014, A&A, 563, A49 +Santoro F., et al., 2022, A&A, 658, A188 +Scheuermann F., et al., 2022, MNRAS, 511, 6087 +Schlafly E. F., Finkbeiner D. P., 2011, ApJ, 737, 103 +MNRAS 000, 1–27 (2022) + +26 +Groves et al. +Sil’chenko O. K., Moiseev A. V., Egorov O. V., 2019, ApJS, 244, 6 +Smith R. C., Points S. D., Chu Y. H., Winkler P. F., Aguilera C., Leiton +R., MCELS Team 2005, in American Astronomical Society Meeting +Abstracts. p. 25.07 +Spitoni E., Cescutti G., Minchev I., Matteucci F., Silva Aguirre V., Martig +M., Bono G., Chiappini C., 2019, A&A, 628, A38 +Stuber S. K., et al., 2021, A&A, 653, A172 +Sun J., et al., 2020, ApJ, 892, 148 +Tayal S. S., Zatsarinny O., Sossah A. M., 2019, ApJS, 242, 9 +Thilker D. A., Braun R., Walterbos R. A. M., 2000, AJ, 120, 3070 +Tremonti C. A., et al., 2004, ApJ, 613, 898 +Turner J. A., et al., 2021, MNRAS, 502, 1366 +Vacca W. D., 1994, ApJ, 421, 140 +Vazdekis A., Koleva M., Ricciardelli E., Röck B., Falcón-Barroso J., 2016, +MNRAS, 463, 3409 +Veilleux S., 2002, in Rosada M., Binette L., Arias L., eds, Astronomical +Society of the Pacific Conference Series Vol. 282, Galaxies: the Third +Dimension. p. 293 +Venturi G., et al., 2018, A&A, 619, A74 +Véron-Cetty M. P., Véron P., 2010, A&A, 518, A10 +Vogt F. P. A., Pérez E., Dopita M. A., Verdes-Montenegro L., Borthakur S., +2017, A&A, 601, A61 +Westfall K. B., et al., 2019, AJ, 158, 231 +Williams J. P., de Geus E. J., Blitz L., 1994, ApJ, 428, 693 +Williams T. G., et al., 2021, AJ, 161, 185 +Williams T. G., et al., 2022, MNRAS, 509, 1303 +Zhang K., et al., 2017, MNRAS, 466, 3217 +Zurita A., Rozas M., Beckman J. E., 2000, A&A, 363, 9 +Zurita A., Florido E., Bresolin F., Pérez I., Pérez-Montero E., 2021, MNRAS, +500, 2380 +de Vaucouleurs G., de Vaucouleurs A., Corwin Herold G. J., Buta R. J., Paturel +G., Fouque P., 1991, Third Reference Catalogue of Bright Galaxies. +Springer Science+Business Media +de los Reyes M. A., et al., 2015, AJ, 149, 79 +MNRAS 000, 1–27 (2022) + +PHANGS Nebular Catalogue +27 +Table C1. Representative values of key physical parameters at 1 reff +Galaxy +A𝑉 +log EW(H𝛼) +log L(H𝛼) +log U +[mag] +[Å] +[erg s−1] +NGC0628 +0.72 +1.3 +36.9 +-1.9 +NGC1087 +0.80 +1.5 +37.6 +-1.6 +NGC1300 +0.97 +1.3 +37.3 +-1.5 +NGC1365 +0.69 +1.6 +37.6 +-1.5 +NGC1385 +0.86 +1.6 +37.7 +-1.6 +NGC1433 +0.69 +1.0 +37.2 +-1.6 +NGC1512 +0.71 +1.2 +37.7 +-1.7 +NGC1566 +0.86 +1.3 +37.8 +-1.5 +NGC1672 +0.95 +1.4 +38.0 +-1.7 +NGC2835 +0.45 +1.4 +37.2 +-1.7 +NGC3351 +0.85 +1.1 +37.0 +-1.7 +NGC3627 +1.18 +1.3 +37.9 +-2.1 +NGC4254 +1.25 +1.4 +37.9 +-1.7 +NGC4303 +0.96 +1.5 +38.0 +-1.7 +NGC4321 +1.14 +1.3 +37.7 +-1.6 +NGC4535 +0.87 +1.5 +37.1 +-1.7 +NGC5068 +0.36 +1.4 +36.5 +-1.9 +NGC7496 +0.74 +1.4 +37.4 +-1.6 +IC5332 +0.25 +1.6 +36.4 +-1.6 +APPENDIX A: IMAGE ATLAS +To provide a visual impression of the size and distribution of our +identified nebulae, we provide images (Figures A1 - A16) of the +H𝛼 emission in all 19 galaxies with the nebular catalogue masks +overplotted with the colours indicating the intrinsic (dust corrected) +H𝛼 luminosity of each nebulae. +APPENDIX B: BPT DIAGRAMS FOR ALL GALAXIES +BPT (Baldwin et al. 1981) diagnostics are commonly used to distin- +guish the ionization source both in individual ionized nebulae and +across integrated galaxy spectra. We demonstrate the range of line +ratios observed within our complete nebular catalogue in Section +4.2 and Figure 4, and include here the breakdown for each individ- +ual galaxy (Figures B1-B3). In all diagrams, regions consistent with +photoionization across all three diagnostics are marked in blue. The +complete nebular catalogue is shown in grey. Overlaid are the Kauff- +mann et al. (2003) diagnostic curve (dashed line) and Kewley et al. +(2006) diagnostic curves (solid lines). +APPENDIX C: RADIAL GRADIENTS FOR ALL GALAXIES +To give a more comprehensive look at the radial variations in nebu- +lar properties, we show the dust extinction derived from the Balmer +decrement (A𝑉 ; Figure C1), the extinction corrected H𝛼 luminosity +(Figure C2), the ionization parameter (U; Figure C3), and the equiv- +alent width of H𝛼 (EW H𝛼; Figure C4) as a function of effective +radius (reff) for all galaxies in our sample. In Table C1 we tabulate a +representative value for each based on linear radial fits at 1 reff. +APPENDIX D: RADIAL METALLICITY GRADIENTS FOR +ALTERNATE PRESCRIPTIONS +A zoomed in view of the radial metallicity gradients for each galaxy is +shown for metallicities calculated using the Scal-PG16 (Figure D1), +O3N2-M13 (Figure D2), and N2S2-D16 (Figure D3) prescriptions. +Table D1. Linear fit parameters for radial gradients in 12+log(O/H) using the +O3N2 (Marino et al. 2013) prescription, see Figure D3 +Galaxy +intercept +slope [dex/r𝑒 𝑓 𝑓 ] +value at r𝑒 𝑓 𝑓 +𝜎(O/H) +IC5332 +8.528 ± 0.010 +-0.043 ± 0.000 +8.485 +0.063 +NGC0628 +8.605 ± 0.010 +-0.041 ± 0.002 +8.564 +0.046 +NGC1087 +8.525 ± 0.002 +-0.032 ± 0.007 +8.494 +0.041 +NGC1300 +8.649 ± 0.011 +-0.071 ± 0.000 +8.579 +0.049 +NGC1365 +8.671 ± 0.003 +-0.186 ± 0.003 +8.486 +0.054 +NGC1385 +8.495 ± 0.002 +-0.001 ± 0.009 +8.494 +0.039 +NGC1433 +8.605 ± 0.013 +-0.005 ± 0.001 +8.600 +0.051 +NGC1512 +8.584 ± 0.012 +0.006 ± 0.002 +8.590 +0.042 +NGC1566 +8.607 ± 0.003 +-0.023 ± 0.003 +8.583 +0.040 +NGC1672 +8.568 ± 0.005 +-0.009 ± 0.000 +8.559 +0.045 +NGC2835 +8.581 ± 0.004 +-0.093 ± 0.001 +8.488 +0.063 +NGC3351 +8.563 ± 0.012 +0.038 ± 0.001 +8.601 +0.055 +NGC3627 +8.581 ± 0.002 +0.015 ± 0.003 +8.596 +0.036 +NGC4254 +8.623 ± 0.003 +-0.030 ± 0.002 +8.594 +0.039 +NGC4303 +8.623 ± 0.000 +-0.029 ± 0.004 +8.594 +0.039 +NGC4321 +8.580 ± 0.005 +0.007 ± 0.002 +8.587 +0.040 +NGC4535 +8.611 ± 0.014 +-0.011 ± 0.003 +8.599 +0.050 +NGC5068 +8.547 ± 0.007 +-0.066 ± 0.001 +8.481 +0.062 +NGC7496 +8.624 ± 0.005 +-0.081 ± 0.002 +8.543 +0.050 +Table D2. Linear fit parameters for radial gradients in 12+log(O/H) using the +N2S2 (Dopita et al. 2016) prescription, see Figure D3 +Galaxy +intercept +slope [dex/r𝑒 𝑓 𝑓 ] +value at r𝑒 𝑓 𝑓 +𝜎(O/H) +IC5332 +8.575 ± 0.018 +-0.263 ± 0.003 +8.312 +0.120 +NGC0628 +8.740 ± 0.020 +-0.119 ± 0.002 +8.622 +0.103 +NGC1087 +8.574 ± 0.004 +-0.110 ± 0.012 +8.464 +0.072 +NGC1300 +8.919 ± 0.017 +-0.175 ± 0.003 +8.744 +0.093 +NGC1365 +8.947 ± 0.008 +-0.348 ± 0.005 +8.599 +0.091 +NGC1385 +8.552 ± 0.000 +-0.066 ± 0.013 +8.486 +0.068 +NGC1433 +8.831 ± 0.020 +-0.034 ± 0.001 +8.797 +0.111 +NGC1512 +8.817 ± 0.022 +-0.032 ± 0.003 +8.784 +0.097 +NGC1566 +8.877 ± 0.008 +-0.076 ± 0.004 +8.802 +0.088 +NGC1672 +8.745 ± 0.014 +-0.019 ± 0.000 +8.727 +0.086 +NGC2835 +8.712 ± 0.012 +-0.265 ± 0.003 +8.447 +0.083 +NGC3351 +8.867 ± 0.024 +0.010 ± 0.002 +8.878 +0.099 +NGC3627 +8.737 ± 0.006 +0.010 ± 0.005 +8.747 +0.067 +NGC4254 +8.872 ± 0.008 +-0.068 ± 0.003 +8.803 +0.071 +NGC4303 +8.916 ± 0.004 +-0.093 ± 0.005 +8.823 +0.087 +NGC4321 +8.897 ± 0.014 +-0.080 ± 0.001 +8.817 +0.081 +NGC4535 +8.868 ± 0.023 +-0.105 ± 0.004 +8.763 +0.084 +NGC5068 +8.466 ± 0.015 +-0.153 ± 0.000 +8.313 +0.092 +NGC7496 +8.825 ± 0.016 +-0.155 ± 0.002 +8.669 +0.101 +Corresponding linear radial fits are provided for the Scal-PG16 (Table +9), O3N2-M13 (Table D1), and N2S2-D16 (Table D2) prescriptions. +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–27 (2022) + +28 +Groves et al. +Figure A1. The spatial distribution of nebulae in the galaxy IC5332. The background greyscale image shows the H𝛼 emission in log scale, and the colour of the +nebulae indicates their intrinsic (dust corrected) H𝛼 luminosity. +MNRAS 000, 1–27 (2022) + +38.5 +-36°05'00" +38.0 ++ +37.5 +30" +37.0 +Dec +..00.90 +36.5 +30" +36.0 +07'00" +1 kpc +35.5 +30" +35.0 +23h34m32s +30s +28s +26s +24s +R.A.PHANGS Nebular Catalogue +29 +Figure A2. As in figure A1 but for NGC 628. +MNRAS 000, 1–27 (2022) + +15°49' +39 +48' +38 +47' +Dec +37 +46' +36 +1 kpc +45' +35 +1h36m50s +45s +40s +35s +R.A.30 +Groves et al. +Figure A3. As in figure A1 but for NGC 1087 +MNRAS 000, 1–27 (2022) + +40 +-0°28'30" +29'00" +39 +30" +log1o(L(Hα) [erg s-1] +38 +Dec +30'00" +37 +30" +36 +31'00"- +I kpc. +35 +2h46m28s +26s +24s +22s +R.A.PHANGS Nebular Catalogue +31 +Figure A4. As in figure A1 but for NGC 1300 +MNRAS 000, 1–27 (2022) + +39 +-19°23'30" +24'00" +30" +Dec +25'00" +37 +30"- +1 kpc +26'00" +36 +3h19m50s +45s +40s +35s +R.A. +3532 +Groves et al. +Figure A5. As in figure A1 but for NGC 1365 +MNRAS 000, 1–27 (2022) + +41 +-36°07'00" +40 +30" +log1o(L(Hα) [erg s-1] +08'00" +39 +Dec +30" +38 +..00.60 +37 +30″- +1 kpc +36 +3h33m48s +42s +36s +30s +R.A. +35PHANGS Nebular Catalogue +33 +Figure A6. As in figure A1 but for NGC 1385 +MNRAS 000, 1–27 (2022) + +40.0 +39.5 +-24°29'00". +39.0 +30" +38.5 +Dec +30'00" +38.0 +37.5 +30" +37.0 +31'00"- +36.5 +1 kpc +36.0 +30" +3h37m32s +30s +28s +26s +R.A.34 +Groves et al. +Figure A7. As in figure A1 but for NGC 1512 +MNRAS 000, 1–27 (2022) + +-43°19'30" +.+. +39 +20'00" +30" +38 +Dec +21'00" +37 +30" +-36 +22'00" +1 kpc +35 +4h04m00s +03m56s +52s +48s +R.A.PHANGS Nebular Catalogue +35 +Figure A8. As in figure A1 but for NGC 1566 +MNRAS 000, 1–27 (2022) + +-54°54' +40 +39 +55' +38 +Dec +56' +37 +57' +36 +1 kpc +35 +4h20m10s +05s +00s +19m55s +R.A.36 +Groves et al. +Figure A9. As in figure A1 but for NGC 1672 +MNRAS 000, 1–27 (2022) + +40 +-59°14'00"- +39 +30" +38 +Dec +15'00" +37 +30" +1 kpc +36 +4h45m54s +48s +42s +36s +30s +R.A. +35PHANGS Nebular Catalogue +37 +Figure A10. As in figure A1 but for NGC 2835 +MNRAS 000, 1–27 (2022) + +-22°20'00" +39 +30" +38 +log1o(L(Hα)) [erg s-1] +21'00" +Dec +30" +37 +22'00" +36 +1 kpc +30" +35 +gh17m57s +54s +51s +48s +R.A.38 +Groves et al. +Figure A11. As in figure A1 but for NGC 3351 +MNRAS 000, 1–27 (2022) + +40 +11°43'30" +39 +1 +00" +38 +42'30" +Dec +00" +": +37 +? +41'30" ++ +36 +1 kpc. +1. +00" +35 +10h44m03s +0os +43m57s +54s +R.A.PHANGS Nebular Catalogue +39 +Figure A12. As in figure A1 but for NGC 3627 +MNRAS 000, 1–27 (2022) + +40 +13°00' +39 +log1o(L(Hα) [erg s-1] +38 +12°59' +Dec +37 +58' +36 +1 kpc +57' +35 +11h20m18s 16s +14s +12s +R.A.40 +Groves et al. +Figure A13. As in figure A1 but for NGC 4254 +MNRAS 000, 1–27 (2022) + +14°27' +39.5 +39.0 +38.5 +26' +38.0 +Dec +37.5 +25' +37.0 +36.5 +24' +kp +36.0 +12h18m57s +54s +51s +48s +45s +R.A.PHANGS Nebular Catalogue +41 +Figure A14. As in figure A1 but for NGC 4535 +MNRAS 000, 1–27 (2022) + +8°13'00" +40 +12'30" +39 +00" +log1o(L(Hα) [erg s-1] ++++ +38 +Dec +11'30" +37 +00" +36 +10'30" +1 kpc +35 +12h34m24s +22s +20s +18s +R.A.42 +Groves et al. +Figure A15. As in figure A1 but for NGC 5068 +MNRAS 000, 1–27 (2022) + +39.0 +38.5 +-21°01' +38.0 ++. +37.5 +02' +Dec +37.0 +36.5 +03' +36.0 +I kpc +35.5 +04' +13h19m00s +18m56s +52s +48s +R.A.PHANGS Nebular Catalogue +43 +Figure A16. As in figure A1 but for NGC 7496 +MNRAS 000, 1–27 (2022) + +40.0 +-43°24'30" +39.5 +: +25'00" +39.0 +11. +log1o(L(Hα)) [erg s-1] +38.5 +30" +Dec +38.0 +26'00" +37.5 +30" +37.0 +. +kpc +36.5 +27'00" +36.0 +23h09m52s 50s +48s +46s +44s +R.A.44 +Groves et al. +Figure B1. [O iii]/H𝛽 as a function of [N ii]/H𝛼 for each galaxy individually. Galaxies are ordered by stellar mass from low (top left) to high (bottom right). +The full nebular catalogue is shown in grey, H ii regions identified as consistent with photoionization across all three BPT diagnostics (see Section 4.2 and Table +5) are marked in blue. Overlaid are the Kauffmann et al. (2003) diagnostic curve (dashed line) and Kewley et al. (2006) diagnostic curves (solid line). +MNRAS 000, 1–27 (2022) + +0.5 +0.0 +0.5 +1.0 +NGC5068 +NGC1087 +NGC1385 +NGC7496 +0.5 +0.0 +-0.5 +qH/[IO] f +1.0 +NGC2835 +NGC0628 +NGC +NGC4254 +NGC4303 +0.5 +601 +0.0 +0.5 +1.0 +NGC1300 +NGC1512 +NGC1672 +NGC4321 +0.5 +.0-0.8-0.6-0.4-0.20.0 0.2 +0.0 +0.5 +1.0 +NGC1566 +NGC3627 +NGC1365 +1.0-0.8-0.6-0.4-0.20.0-0120-0.80.6-0.4-0.2 0.0 -0120-0.8-0.6-0.4-0.2 0.0 -0120-0.8-0.6-0.4-0.20.0 0.2 +log [NII]/HaPHANGS Nebular Catalogue +45 +Figure B2. [O iii]/H𝛽 as a function of [S ii]/H𝛼 for each galaxy individually. Galaxies are ordered by stellar mass from low (top left) to high (bottom right). The +full nebular catalogue is shown in grey, H ii regions identified as consistent with photoionization across all three BPT diagnostics (see Section 4.2 and Table 5) +are marked in blue. Overlaid is the Kewley et al. (2006) diagnostic curve (solid line). +MNRAS 000, 1–27 (2022) + +0.5 +0.0 +-0.5 +-1.0 +NGC5068 +IC5332 +NGC1087 +NGC1385 +NGC7496 +0.5 +0.0 +-0.5 +qH/[IIO] f +1.0 +NGC2835 +NGC0628 +NGC4254 +NGC4303 +0.5 +601 +0.0 +0.5 +1.0 +NGC1300 +NGC1512 +NGC1672 +NGC4321 +0.5 +.0-0.8-0.6-0.4-0.20.0 0.2 +0.0 +-0.5 +1.0 +NGC1566 +NGC3627 +NGC1365 +1.0-0.8-0.6-0.4-0.20.0-0120-0.80.6-0.4-0.2 0.0 -0120-0.8-0.6-0.4-0.2 0.0 -0120-0.8-0.6-0.4-0.20.0 0.2 +log [SI]/Ha46 +Groves et al. +Figure B3. [O iii]/H𝛽 as a function of [O i]/H𝛼 for each galaxy individually. Galaxies are ordered by stellar mass from low (top left) to high (bottom right). The +full nebular catalogue is shown in grey, H ii regions identified as consistent with photoionization across all three BPT diagnostics (see Section 4.2 and Table 5) +are marked in blue. Overlaid is the Kewley et al. (2006) diagnostic curve (solid line). +MNRAS 000, 1–27 (2022) + +0.5 +0.0 +-0.5 +1.0 +NGC5068 +IC5332 +NGC1087 +NGC1385 +NGC7496 +0.5 +0.0 +-0.5 +qH/[IIO] f +1.0 +NGC2835 +NGC4254 +NGC4303 +0.5 +601 +0.0 +0.5 +1.0 +NG +NGC1300 +NGC1512 +NGC1672 +NGC4321 +0.5 +2.5-2.0-1.5-1.0-0.5 +0.0 +-0.5 +1.0 +NGC1566 +NGC36 +NGC1365 +log [O1]/HaPHANGS Nebular Catalogue +47 +Figure C1. A𝑉 radial gradients. +MNRAS 000, 1–27 (2022) + +NGC5068 +IC5332 +NGC1087 +NGC1385 +NGC7496 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +NGC2835 +NGC0628 +NGC3351 +NGE +2.5 +2.0 +1.5 +1.0 +[mag] +0.5 +0.0 +Av +NGC1300 +NGC1512 +NGC1672 +NGC4321 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +NGC1566 +NGC1433 +NGC1365 +log(M) +2.5 +2.0 +11 +1.5 +1.0 +0.5 +0.0 +0 +4 +4 +0 +R / reff48 +Groves et al. +Figure C2. L(H𝛼) radial gradients. +MNRAS 000, 1–27 (2022) + +41 +NGC5068 +IC5332 +NGC1087 +NGC1385 +NGC7496 +40 +39 +38 +37 +36 +35 +41 +NGC2835 +NGC0628 +NGC3351 +NGC4254 +NGC4303 +40 +39 +[erg/s] +38 +37 +36 +L(Hα) +35 +41 +NGC4535 +NGC1300 +NGC1512 +NGC1672 +NGC4321 +60l +40 +39 +38 +37 +36 +35 +41 +NGC1566 +NGC3627 +NGC1433 +NGC1365 +log(M) +40 +9 +10 +11 +39 +38 +37 +36 +35 +2 +3 +4 +4 +0 +4 +R / reffPHANGS Nebular Catalogue +49 +Figure C3. Ionization parameter (U) radial gradients. +MNRAS 000, 1–27 (2022) + +0.0 +NGC5068 +NGC08 +NGC1385 +NGC7496 +0.5 +1.0 +1.5 +2.0 +2.5 +-3.0 +0.0 +-0.5 +1.0 +1.5 +2.0 +-2.5 +U +g +3.0 +0.0 +0.5 +1.0 +1.5 +-2.0 +2.5 +-3.0 +0.0 +NGC1566 +log(M) +0.5 +10 +11 +1.0 +1.5 +-2.0 +2.5 +-3.0 +4 +0 +R / reff50 +Groves et al. +Figure C4. EW(Ha) radial gradients. +MNRAS 000, 1–27 (2022) + +3.0 +NGC5068 +IC5332 +NGC1087 +NGC1385 +NGC7496 +2.5 +2.0 +1.5 +1.0 +0.5 +3.0 +NGC2835 +NGC0628 +NGC3351 +NGC4254 +NGC4303 +2.5 +2.0 +1.5 +EW(Hα)raw +1.0 +0.5 +3.0 +NGC4535 +NGC1300 +NGC1512 +NGC1 +NGC4321 +2.5 +g +2.0 +1.5 +1.0 +0.5 +3.0 +NGC1566 +NGC3627 +NGC1433 +NGC1365 +log(M) +2.5 +2.0 +LU +1.5 +1.0 +0.5 +4 +0 +R / reffPHANGS Nebular Catalogue +51 +Figure D1. Metallicity radial gradients, using the Scal-PG16 prescription. All galaxies show the same dynamic range in 12+log(O/H). +MNRAS 000, 1–27 (2022) + +IC5332 +8.7 +NGC0628 +8.6 +NGC1087 +8.7 +NGC1300 +NGC1365 +8.5 +8.7 +8.6 +8.5 +8.6 +8.4 +8.6 +8.5 +8.4 +8.5 +8.3 +d8.5 +8.4 +8.3 +8.4 +8.2 +8.4 +8.3 +8.2 +8.3 +8.1 +8.3 +0.00.20.40.60.81.01.21.4 +0.0 +0.5 +1.0 +1.5 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.01.5 +2.0 +0.0 0.2 0.4 0.6 0.8 1.0 1.2 +NGC1385 +8.8 +NGC1433 +NGC1512 +NGC1566 +NGC1672 +8.6 +8.7 +8.7 +8.7 +8.7 +8.5 +8.6 +8.6 +8.6 +8.6 +cal +8.4 +8.5 +8.5 +8.5 +8.5 +8.3 +8.4 +8.4 +8.4 +8.4 +(H/0)60|+ +8.2 +8.3 +8.3 +18.3 +0.00.5 +1.0 +1.5 +2.0 +0.00.51.01.52.02.53.03.5 +0.00.51.01.5 2.0 2.5 +0.00.51.01.52.02.53.03.54.0 +0 +1 +2 +3 +4 +8.6 +NGC2835 +8.8 +NGC3351 +NGC3627 +8.8 +NGC4254 +NGC4303 +8.7 +8.7 +8.5 +8.7 +8.7 +2 +8.6 +8.6 +8.4 +8.6 +8.6 +8.5 +8.5 +8.3 +8.5 +8.5 +8.4 +8.4 +8.2 +8.4 +8.4 +8.3 +8.3 +0.0 0.5 +1.01.52.0 +0.0 0.5 1.0 1.5 2.0 +0.0 0.5 1.0 1.5 2.0 2.5 +0 +1 +2 +3 +4 +0.00.51.01.52.02.53.0 +NGC4321 +NGC4535 +NGC5068 +8.5 +8.7 +NGC7496 +8.7 +8.7 +8.4 +8.6 +8.6 +8.6 +8.3 +8.5 +8.5 +8.5 +8.2 +8.4 +8.4 +8.4 +8.1 +8.3 +8.3. +8.3 +0.00.5 +1.0 +1.5 +2.0 +0.00.20.40.60.81.01.21.41.6 +0.00.5 +1.01.5 2.02.5 +0.0 +0.5 +1.0 +1.5 2.0 +R / reff52 +Groves et al. +Figure D2. Metallicity radial gradients, using the O3N2-M13 prescription. All galaxies show the same dynamic range in 12+log(O/H). +MNRAS 000, 1–27 (2022) + +IC5332 +NGC0628 +8.7 +NGC1087 +8.8 +8.7 +NGC1300 +NGC1365 +8.7 +8.7 +8.6 +8.7 +8.6 +8.6 +8.6 +8.5 +8.6 +8.5 +8.5 +8.5 +8.4 +8.5 +8.4 +8.4 +8.4 +8.3 +8.4 +8.3 +8.3 +8.3 +0.00.20.40.60.81.01.21.4 +0.0 +0.5 +1.0 +1.5 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 0.20.4 0.6 0.8 1.0 1.2 +18.8 +8.8 +8.8 +8.7 +NGC1385 +NGC1433 +8.8 +NGC1512 +NGC1566 +NGC1672 +8.7 +8.7 +8.7 +8.6 +8.7 +8.6 +8.6 +8.6 +03N2 +8.5 +8.6 +8.5 +8.5 +8.5 +8.4 +0 +8.5 +8.4 +8.4 +8.4 +8.3 +8.4 +(H/0)60|+ +0.00.5 +1.0 +1.5 +2.0 +0.00.51.01.52.02.53.03.5 +0.0 0.51.01.5 2.0 2.5 +0.00.51.01.52.02.53.03.54.0 +0 +1 +2 +3 +4 +8.7 +NGC2835 +NGC3351 +NGC3627 +8.8 +8.8 +NGC4254 +8.8 +8.8 +NGC4303 +8.6 +8.7 +8.7 +8.7 +8.7 +2 +8.5 +8.6 +8.6 +8.6 +8.6 +8.4 +8.5 +8.5 +8.5 +8.5 +8.3 +8.4 +8.4 +8.4 +8.4 +0.0 0.5 +1.01.5 2.0 +0.0 0.5 1.0 1.5 2.0 +0.0 0.5 1.01.52.02.5 +0 +1 +2 +3 +4 +0.0 0.51.0 1.5 2.0 2.5 3.0 +8.8 +8.8 +NGC4321 +NGC4535 +8.7 +8.8 +NGC5068 +NGC7496 +8.7 +8.7 +8.6 +8.7 +8.6 +8.6 +8.5 +8.6 +18.5 +8.5 +8.4 +8.5 +8.4 +8.4 +8.3 +8.4 +0.00.5 +1.0 +1.5 +2.0 +0.00.20.40.60.81.01.21.41.6 +0.00.5 +1.01.5 2.02.5 +0.0 +0.5 +1.0 +1.5 +2.0 +R / reffPHANGS Nebular Catalogue +53 +Figure D3. Metallicity radial gradients, using the N2S2-D16 prescription. All galaxies show the same dynamic range in 12+log(O/H). +MNRAS 000, 1–27 (2022) + +8.7 +NGC0628 +8.8 +8.9 +NGC1087 +9.0 +NGC1300 +9.0 +NGC1365 +8.6 +8.7 +8.9 +8.9 +8.8 +8.5 +8.6 +8.7 +8.8 +8.8 +8.4 +8.5 +8.6 +8.7 +8.7 +8.3 +8.4 +8.5 +8.6 +8.6 +8.2 +8.3 +8.4 +8.5 +8.5 +8.1 +8.2 +8.3 +8.4 +8.4 +0.00.20.40.60.81.01.21.4 +0.0 +0.5 +1.0 +1.5 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.01.5 +2.0 +0.0 0.2 0.4 0.6 0.8 1.0 1.2 +8.8 +NGC1385 +9.0 +9.1 +NGC143 +NGC1512 +NGC1566 +9.0 +NGC1672 +8.7 +9.0 +9.0 +8.9 +8.9 +8.6 +8.9 +8.9 +8.8 +8.8 +8.8 +2S2 +8.5 +8.8 +8.7 +8.7 +8.4 +8.7 +8.7 +N2: +8.6 +8.6 +8.6 +8.3 +8.6 +8.5 +8.5 +8.5 +8.2 +8.4 +8.5 +(H/O) +8.4 +8.4 +0.0 +0.5 +1.0 +1.5 +0.00.51.01.52.02.53.03.5 +0.00.51.01.52.0 2.5 +0.00.51.01.52.02.53.03.54.0 +0 +1 +2 +3 +4 +8.7 +NGC2835 +NGC3351 +NGC3627 +NGC4254 +NGC4303 +9.1 +9.0 +9.0 +8.6 +9.0 +8.9 +8.9 +9.0 +8.5 +8.9 +8.9 +8.8 +8.8 +2 +8.4 +8.8 +8.7 +8.7 +8.8 +8.3 +8.7 +8.6 +8.6 +8.7 +8.2 +8.6 +8.6 +8.5 +8.5 +8.1 +8.5 +8.5 +8.4 +8.4 +0.0 0.5 +1.0 +1.5 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Dale10, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Glover11, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Pessa3, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Sanchez-Blazquez12,13, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Williams3 1International Centre for Radio Astronomy Research, University of Western Australia, 7 Fairway, Crawley, 6009 WA, Australia 2Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Mönchhofstraße 12-14, 69120 Heidelberg, Germany 3Max Planck Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany 4INAF – Osservatorio Astrofisico di Arcetri, Largo E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Fermi 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' I-50157 Firenze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Italy 5 Departamento de Astronomía,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Universidad de Chile,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Camino del Observatorio 1515,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Las Condes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Santiago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Chile 6European Southern Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Karl-Schwarzschild Straße 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' D-85748 Garching bei München,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Germany 7Research School of Astronomy and Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Australian National University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Canberra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' ACT 2611,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Australia 8Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 140 West 18th Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Columbus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Ohio 43210,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' USA 9Argelander-Institut für Astronomie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Universität Bonn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Auf dem Hügel 71,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 53121 Bonn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Germany 10Department of Physics & Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' University of Wyoming,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Laramie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' WY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 82071,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' USA 11Universität Heidelberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Zentrum für Astronomie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Institut für theoretische Astrophysik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Albert-Ueberle-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2, 69120 Heidelberg, Germany 12Departamento de Física de la Tierra y Astrofísica, Universidad Complutense de Madrid, E-28040 Madrid, Spain 13Instituto de Física de partículas y del Cosmos, IPARCOS, E-28040 Madrid, Spain Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' in original form ZZZ ABSTRACT Ionized nebulae provide critical insights into the conditions of the interstellar medium (ISM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Their bright emission lines enable the measurement of physical properties, such as the gas-phase metallicity, across galaxy disks and in distant galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The PHANGS–MUSE survey has produced optical spectroscopic coverage of the central star-forming discs of 19 nearby main- sequence galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Here, we use the H𝛼 morphology from this data to identify 30,790 distinct nebulae, finding thousands of nebulae per galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For each nebula, we extract emission line fluxes and, using diagnostic line ratios, identify the dominant excitation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A total of 23,244 nebulae (75%) are classified as H ii regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The dust attenuation of every nebulae is characterised via the Balmer decrement and we use existing environmental masks to identify their large scale galactic environment (centre, bar, arm, interarm and disc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Using strong-line prescriptions, we measure the gas-phase oxygen abundances (metallicity) and ionization parameter for all H ii regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' With this new catalogue, we measure the radial metallicity gradients and explore second order metallicity variations within each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' By quantifying the global scatter in metallicity per galaxy, we find a weak negative correlation with global star formation rate and stronger negative correlation with global gas velocity dispersion (in both ionized and molecular gas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' With this paper we release the full catalogue of strong line fluxes and derived properties, providing a rich database for a broad variety of ISM studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Key words: galaxies:ISM – H ii Regions – galaxies: abundances 1 INTRODUCTION Emission lines from ionized nebulae play a fundamental role in our understanding of galaxy evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Apart from their use in deter- mining spectroscopic redshifts, emission lines have been used to determine star-formation rates, the presence of active galactic nuclei (AGN), galaxy dynamics, gas-phase metal abundances and more (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' However, in most extragalactic studies, the emission lines have been typically measured from a single spectrum of the entire galaxy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' SDSS, Abazajian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' GAMA, Driver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' VVDS, Le Fevre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Resolved maps of nearby ★ email:brent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='groves@uwa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='au emission line galaxies in strong spectral lines has been available with narrow filters, slit spectra on bright H ii regions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Croxall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Moustakas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2010) and even spectrally resolved with Fabry-Perot surveys (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Veilleux 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Epinat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Moiseev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Sil’chenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2019), but it is only with the advent of integral field spectrographs (IFS) that maps of multiple emission lines across thousands of galaxies have become common, with most achieving kpc scale sampling (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' CALIFA, Sánchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' VENGA, Blanc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' MaNGA, Bundy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' SAMI, Croom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' However, the individual ionized nebulae that the emission lines originate from (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' H ii regions, supernova remnants, planetary neb- ulae) are typically <100 pc in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This means that in these large © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='03811v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='GA] 10 Jan 2023 2 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' surveys, the nebulae cannot be spatially distinguished from each other or the faint surrounding diffuse ionized gas (DIG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Reynolds 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' While there have been efforts to account for the DIG (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Espinosa-Ponce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2020), only in nearby galaxies (or rare lensed systems) are we able to achieve the 10–100 pc resolution required to separate out the individual ionized nebula (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' H ii re- gions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Kennicutt 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Once imaged in multiple emission lines, it is then possible to use line ratios to distinguish nebular emission in H ii regions from other nebular sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' supernovae (SNe), planetary nebulae (PNe) Ciardullo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Long-slit studies have compiled growing samples of emission line spectroscopy for H ii regions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Pilyugin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2020), yet generally these studies pre-select for the brightest regions, which introduces biases in the sampling across galaxy disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' With the advent of new instruments and approaches, wide-area and high angular resolution spectral maps are now available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This is made possible through long-slit spectral stepping (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' the TY- PHOON survey Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2017), imaging Fourier transform spec- trographs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' the SIGNALS survey with SITELLE on the CFHT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Rousseau-Nepton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2019) or IFS (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' MAD or TIMER on the MUSE/VLT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Erroz-Ferrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Gadotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Such maps not only make it possible for individual nebulae to be identi- fied, but the integrated emission line fluxes within each nebula mean that they can be classified and their key physical properties measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Furthermore, sensitive data can also examine the properties of the intervening diffuse ionized gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Full spectral maps can also be used to understand the properties of the underlying stellar populations, including the stars that are the potential sources of ionizing photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Surveys using spectral maps have already identified leaking radiation from H ii regions as the dominant ionizing source of the DIG (Della Bruna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Belfiore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022), quantified the impact of the DIG on the measurement of the gas-phase metallicity (Poetrodjojo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2019), and determined that metallicity variations exist within galaxies on top of the well known radial gradients (Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Metha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Sánchez-Menguiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' It is within this context that we describe here the PHANGS–MUSE Nebulae Catalogue of over 30,000 nebulae across 19 galaxies, the largest catalogue of high-resolution (< 100 pc) extragalactic nebulae with homogeneous optical spectroscopic coverage currently avail- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' First introduced by Santoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2022) when fitting for the H ii region luminosity functions, the nebula catalogue has already been used in a number of papers for a range of science topics, in- cluding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' quantifying the pre-supernova feedback within 6000 of the H ii regions (Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2021), modelling of line ratios for diffuse ionized gas surrounding these H ii regions (Belfiore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022), and interpolating metallicities measured at each H ii region to construct full coverage metallicity maps (Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' With this paper we release a full catalogue of emission line properties and derived physical properties associated with the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This catalogue is based on the mosaicked MUSE IFS observations from the Physics at High-Angular resolution in Nearby GalaxieS program (PHANGS–MUSE survey Emsellem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022), which are drawn from the larger PHANGS survey (Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2021, PHANGS)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We summarise this survey and the specific subsam- ple targeted with MUSE in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We present our methods of constructing our catalogue of nebular emitting objects in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We describe how derived properties are obtained and included in the catalogue as value-added products in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We present results 1 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='phangs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='org focused on the metallicity measurements in those objects classified as H ii regions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We discuss the interpretation of the metal- licity variation we observe in Section 6, as well as other technical aspects of our catalogue, and conclude in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2 THE PHANGS–MUSE SURVEY The PHANGS survey was designed specifically to resolve galaxies into the individual elements of the star-formation process: molec- ular clouds, H ii regions, and stellar clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Driven by this aim, the full PHANGS sample was originally determined by select- ing southern-sky accessible (−75◦ ≤ 𝛿 ≤ +25◦, for ALMA & MUSE), low inclination (𝑖 < 75◦), massive star-forming galax- ies (log(𝑀∗/𝑀⊙) > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='75 and log(𝑠𝑆𝐹𝑅/yr−1) > −11) within ∼ 23 Mpc, such that 1” < 100 pc (described in full detail Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2021, as the PHANGS–ALMA sample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In addition to ALMA data, there is a wealth of data from other telescopes such as the Hub- ble Space Telescope (the PHANGS-HST survey Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022), and the MUSE IFS on ESO’s VLT (Bacon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2010), known as the PHANGS–MUSE survey (Emsellem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The PHANGS–MUSE survey is an ESO large program (∼ 170 h, PI Schinnerer) aimed at spectroscopically mapping the discs of 19 nearby star-forming galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This subsample of PHANGS (orig- inally selected to align with the PHANGS–ALMA pilot surveys) covers a broad range in stellar mass, but is biased somewhat to main- sequence massive galaxies (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Further details on the sample, observations, reduction and MUSE data products are described in Emsellem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Here we summarise the sample and data, and refer the reader to Emsellem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2022) for the full details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Public data products, including data cubes and line maps, are available at the ESO archive 2 and CADC3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The physical properties of these galaxies are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As all our galaxies are within ∼20 Mpc, the typical seeing (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='′′91) of our MUSE observations means that all structures down to 100 pc can be isolated within the disk environment, with a median physi- cal resolution of ∼70 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The galaxy distances we use are the latest compilation from Anand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2021), including new tip of the red giant branch distances from the PHANGS–HST observations and new planetary nebula luminosity function distances measured from the PHANGS–MUSE data itself (Scheuermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The in- clination and position angle of the galaxies were determined by Lang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2020) from the PHANGS ALMA CO rotation curves analysis, or near-IR imaging when the CO data was not available or the fit to the CO velocity field was deemed unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The position angle and inclinations were used to determine the deprojected radial dis- tances used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The listed stellar masses and star formation rates are global measures from UV and IR photometry (Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Representative disc scale lengths are provided, both the 25th magnitude B-band isophotal radius (R25) from RC3 (de Vaucouleurs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 1991 via HyperLEDA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Makarov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2014) and the effective radius containing half of the stellar mass of the galaxy (reff).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' These quantities are compiled and computed in Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Due to their proximity, the stellar structures of the galaxies, such as spiral arms and bars, are clearly resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Within this work we use the structural morphology masks created by Querejeta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2021) using Spitzer 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 𝜇m imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Querejeta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2021) used 2 https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='org/scienceportal/home?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='data_ collection=PHANGS 3 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='canfar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='net/storage/vault/list/phangs/ RELEASES/PHANGS-MUSE MNRAS 000, 1–27 (2022) PHANGS Nebular Catalogue 3 photometric fitting that decomposes the galaxies into bulges and discs (Salo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2015), then further divided the structures visually and through fitting into centres, bars (with the bar mask defined by a fitted ellipse), rings and lenses, and spiral arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Spiral arms were fitted as logarithmic spiral curves with widths fitted to both the stellar and molecular gas surface density, and the inter-arm region was considered to be any region in the disc outside of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We follow Querejeta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2021) in distinguishing between inner bar and outer disc regions when considering the arm and inter-arms and use their notations (their Figure 2 and Table 1, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' These morphological masks allow us to determine the influence of local environment on the nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Given our bias towards massive main sequence galaxies, spiral or disk features are seen in all, and only four out of the 19 galaxies are not barred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The PHANGS–MUSE large program was observed over several semesters and includes data from other programs and includes MUSE observations in both ground layer AO (adaptive optics) and non AO mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Combined with variations in seeing, this means that the point spread function (PSF) varied between galaxies and among point- ings within the same galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' To account for this variation between pointings in the same galaxy, we created mosaicked datacubes with a consistent PSF, where all pointings in a single galaxy were convolved to a single Gaussian PSF, whose size was determined by the pointing with the worst (largest) PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We used this optimised convolution data (copt) to identify the nebulae for the catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The consistent PSF across the mosaic avoids issues with variable nebulae sizes across a single galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The mosaicked MUSE datacubes were then passed through a data analysis pipeline (DAP) to provide maps of value-added products such as emission lines, mean stellar properties, gas and stellar kine- matics and more (as detailed in Emsellem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As the emis- sion lines form the key data for this paper, we briefly describe the analysis here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The DAP uses the penalised pixel fitting method (pPXF Cappellari 2017) to derive both the stellar continuum and emission lines properties within the spectral range 4850–7000 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' F Before any fitting, the MUSE data is corrected for foreground Galactic extinc- tion, using the Cardelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (1989) extinction law and the 𝐸(𝐵 −𝑉) attributed to the Milky Way foreground from Schlafly & Finkbeiner (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' To fit the stellar continuum and derive the stellar properties, the datacubes are first spatially Voronoi-binned (using the vorbin pack- age Cappellari & Copin 2003) to achieve a minimum S/N of 35 in the 5300 − 5500 Å wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The continuum between 4850−7000 Å in each bin is then fit with a combination of E-MILES simple stellar population model templates (Vazdekis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2016) generated with a Chabrier (2003) initial mass function and BaSTI isochrones (Pietrinferni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The Na I D absorption dou- blet (already removed in AO observations) are masked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The higher spectral resolution templates are convolved to the resolution of the MUSE data before fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The spectra is first fit to determine the stellar kinematics using a smaller set of model templates sampled at eight ages (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='15 − 14 Gyr, logarithmically sampled in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='22 dex), and four metallicities ([Z/H] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='35;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='06;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' To the fit for the stellar population parameters we fix the kinematics and use a larger set of templates sampled at ages = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='08, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='40, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='60, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='75, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5] Gyr and [Z/H] = [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='49, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='96, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='35, +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='06, +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='26, +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' When fitting for the stel- lar population properties we also constrain the average attenuation of the stellar continuum, parametrized by the Calzetti (2001) curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' To fit the emission lines we rerun pPXF on the mosaicked cubes at an individual spaxel level, with the emission lines treated as ad- ditional Gaussian components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The underlying stellar continuum is fit using the smaller set of E-MILES templates and the derived kine- matics of the Voronoi bin that contained the individual spaxel, with the inclusion of an 8th-order multiplicative polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We fit all strong emission lines and tie the kinematics (velocity and velocity dispersion) in three groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Hydrogen lines (H𝛼, H𝛽), low ioniza- tion lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', [N ii]𝜆6583, [S ii]𝜆𝜆6716, 6731), and high ionization lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', [O iii]𝜆5007, [S iii]𝜆6312).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' These maps of stellar kine- matics and mean properties, emission line fluxes, and gas kinematics form the key part of the data analysis pipeline and the PHANGS– MUSE release, as described in Emsellem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 3 METHODS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 Nebular catalogue construction The PHANGS–MUSE galaxies are replete with emission lines, with more than 95% of our 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='′′2 spaxels within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5𝑅25 containing H𝛼 emission at a > 3𝜎 level (see figure 20 in Emsellem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' With such filled maps, distinguishing individual nebulae from each other and the diffuse ionized gas is difficult, even with a median physical resolution of 70 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Therefore, to identify the nebulae we require an unbiased and robust region identifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' While several such methods exist and have been applied previously (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Clumpfind;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (1994);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2016) or pyHIIExtractor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Lugo- Aranda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2022)), we chose to use HIIphot, a code specifically built to identify and characterise H ii regions with their irregular morphology (Thilker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We use a slightly altered version HIIphot to work on the H𝛼 maps created from IFS data, first used in Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Originally designed to be applied to narrow band imaging data centred on the H𝛼 line, HIIphot used the associated broad band data used for continuum subtraction from the narrow band data to determine the significance of the H𝛼 detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' However, IFS can spectrally resolve any underlying stellar continuum and subtract this as done within the data analysis pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Therefore HIIphot was altered to work on H𝛼 maps alone with the associated fitting error map to identify the nebulae and determine their boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' However the main algorithm in nebulae identification is still as described in Thilker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The key to nebulae identification is to first distinguish individual nebulae, then grow these up to a given termination criterion defining the edges of the nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' While a classical photoionized nebula has a clear boundary defined by the edge of the Strömgren sphere, real nebulae may be centrally concentrated or appear as rings, or have several peaks and a diffuse boundary due to density variations within the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The angular resolution of our observations means that we only resolve the largest of H ii region complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In most H ii regions our resolution smooths any features and boundaries and, a potentially larger problem we discuss in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4, merge proximate nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Therefore the choice of controlling parameters is driven by both the dataset and the physics of nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As described in Santoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2022), to identify the nebulae we first require to detect the peaks in H𝛼 emission, or ‘seed regions’, above the diffuse background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We set this background for each galaxy to be the median of all H𝛼 pixels within the MUSE FoV with ΣH𝛼 < 1 × 10−17 erg s−1 cm−2 arcsec−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This ranges from ΣH𝛼 = 1 to 3 × 10−18 erg s−1 cm−2 arcsec−2 across our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The detection threshold within HIIphot was set to 3𝜎 above this background, where 𝜎 is the standard deviation of the background pixels and typically around the same level as the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Given the diverse morphologies of H ii regions (and other ionized MNRAS 000, 1–27 (2022) 4 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' General properties of the PHANGS-MUSE galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Name Distance𝑎 𝑣𝑏 sys 𝑃𝐴𝑐 𝑖𝑐 log10 𝑀 𝑑 ∗ 𝑅𝑏 25 𝑟eff 𝐸 (𝐵 − 𝑉 )𝑒 MW resolution Mpc km s−1 deg deg M⊙ arcmin arcmin mag pc IC5332 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 699 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='67 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='014 45 NGC0628 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='8 651 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='34 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='061 42 NGC1087 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 1502 359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='93 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='030 71 NGC1300 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1545 278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='026 62 NGC1365∗ 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 1613 201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='99 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 𝑓 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='018 84 NGC1385 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 1477 181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='017 96 NGC1433∗ 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 1057 199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='87 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='008 83 NGC1512 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='8 871 261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='71 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='009 96 NGC1566∗ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 1483 214.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='086 33 NGC3351 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 775 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='36 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='024 43 NGC3627∗ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 715 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='83 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='029 69 NGC4254 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 2388 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='42 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='033 61 NGC4303∗ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1560 312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='52 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='019 96 NGC4321 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 1572 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='023 59 NGC4535 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='8 1954 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='53 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='017 80 NGC5068 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 667 342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='090 23 NGC7496∗ 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 1639 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='008 104 𝑎From the compilation of Anand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 𝑏From LEDA (Makarov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 𝑐From Lang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2020), based on CO(2–1) kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 𝑑Derived by Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2021), using GALEX UV and WISE IR photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 𝑒From Schlafly & Finkbeiner (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 𝑓 Due to AGN bias, derived from the scale length (l∗) as reff = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='41 l∗ following Equation 5 in Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' ∗Classified as an AGN by Véron-Cetty & Véron (2010) nebulae), HIIphot performs iterative Gaussian smoothing on the H𝛼 maps, merging connecting features to create the nebulae ‘footprints’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' To avoid the detection of regions with unphysical sizes, we limit the spatial smoothing to three iterations, each time increasing the smoothing kernel (starting from the original resolution image) by 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' These footprints are then further trimmed to a ‘seed’ with a consistent isophotal boundary defined by 50% of the median within the footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Once detected, we further cleaned the seed sample to avoid artefacts due to noise by imposing a S/N cut of 50 above the H𝛼 error maps for the integrated flux values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Defining the boundary edges of nebulae is challenging, with many criteria existing in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' H𝛼 surface brightness, line ratios, H𝛼 equivalent widths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' By using HIIphot we chose to use the spatial gradient of the H𝛼 surface brightness to define the boundaries of our nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As discussed in Thilker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2000), the choice of terminal gradient is ambiguous, with flatter values leading to larger H ii regions that can include the diffuse ionized gas directly associated with the H ii region (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Belfiore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022, for the level of association) but also lead to a more contiguous map of nebulae (Figure 5 in Thilker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The spatial resolution of the data also impacts the exact boundaries, smoothing edges and potentially merging adjoining nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We chose to use a single termination gradient of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 EM pc−1 (where the emission measure, EM, is in cm−6 pc) for all galaxies (corresponding to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='43 × 10−16 erg s−1 arcsec−2 pc−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This value is similar to that used in other nearby galaxy studies (Oey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2017, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' ), and visually provided the best balance in terms of capturing the total H𝛼 flux for each nebula, while limiting the size growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We chose to use a single termination gradient rather than one for each galaxy for consistency in nebulae identification, even given the factor of ∼4 difference in physical resolution across the PHANGS–MUSE sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' These steps lead to 31,497 identified nebulae with defined bound- aries across our sample of 19 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For each nebula we report the central position in both RA and Dec, weighted by H𝛼 intensity, and their position relative to the galaxy centres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We quantify the area encompassed by the nebulae in pixels, however as most of our re- gions are unresolved or only marginally resolved (see also Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2), we do not provide size measurements (though see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 where we do present 10%, 50%, and 90% circularized radial sizes for the overall distributions in each galaxy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As described in Em- sellem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2022, particularly §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3), foreground star masks were generated for all PHANGS–MUSE galaxies based on the Gaia DR2 catalogue (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We exclude 98 sources whose footprint falls within the star masks and are likely impacted by artefacts from incorrect stellar continuum subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We also flag 609 nebulae with centres within 1 PSF FWHM of the edges of our PHANGS–MUSE galaxy footprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' While the emission lines in these regions are likely correctly measured, their proximity to the edges mean that their boundaries are potentially incorrectly defined and that their integrated line fluxes may not represent the total emis- sion of the nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For larger nebulae where the distinct ionized zones can be distinguished (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' the S++ and S+ zones are resolved), the emission line ratios measured for these regions are potentially incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' While these flagged nebulae are included in the full cat- alogue, we exclude them from our further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For the results presented in this paper, we focus on the remaining 30,790 nebulae (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Spatial masks corresponding to the locations of each identified nebula are released as data products accompanying this paper, and can also be found via the PHANGS webpage4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' An image atlas show- ing the footprints of all nebulae in each galaxy is included in Ap- pendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 Emission line measurements With the footprints of all nebulae defined, we integrated the original MUSE spectra within each nebula and re-fit using the same data 4 https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='com/view/phangs/home/data MNRAS 000, 1–27 (2022) PHANGS Nebular Catalogue 5 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A visualisation of the two dimensional spatial extent and distribution of all nebulae in the galaxy NGC 1433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The background greyscale image shows the H𝛼 emission in log scale, and the colour of the nebulae indicates their intrinsic (dust corrected) H𝛼 luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A full atlas of each galaxy is available in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' analysis pipeline (DAP) described in Emsellem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2022) used to create the original H𝛼 maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We do this to increase the signal- to-noise of our emission lines and to detect faint spectral features, such as the temperature sensitive auroral lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The only changes to the pipeline are to extend the wavelength range fitted to include the [S iii]𝜆9069 Å emission line and, when integrating the spectra, we use the unconvolved, native resolution mosaicked data cubes to minimise the impact of PSF smearing at nebula boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The latter only has a small impact due to the consistency in seeing between observations, but for some galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' NGC1365, as seen in Table A1 in Emsellem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022) the PSF variation can be a factor of two within the different pointings of the mosaic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As with the global PHANGS–MUSE DAP, we fit all emis- sion lines simultaneously, but also include lines that are fainter (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' [N ii]𝜆5754) and at longer wavelengths (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' [S iii]𝜆9069).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The full list of lines released in this catalogue is given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Sim- ilarly, when fitting lines we assume single Gaussian profiles and tie the kinematics (velocity and velocity dispersion) in three groups: hydrogen lines, low ionization lines, and high ionization lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Line velocities are reported relative to the systemic velocity of the galaxy, provided in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In our fit we account for a Milky Way foreground extinction, assuming the 𝐸(𝐵 − 𝑉) values provided by Schlafly & Finkbeiner (2011) (also listed in Table 1) and an O’Donnell (1994) extinction law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In Figure 2, we show a typical nebular spectrum from IC5332 (ID:38, 𝐿H𝛼 = 1036.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='8 erg s−1), along with the best fitted spectrum from the data analysis pipeline overlaid, and the relative residual from the fit at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' What is clear from this spectrum are the emission lines, and how well we reproduce these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Also clear are the strong sky line residuals (especially beyond 6800Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The underlying continuum is dominated by stellar light, although at the scaling in this figure, only certain stellar absorption features are visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As an insert we show a zoom-in of the H𝛼 region of the spec- trum with the strong [N ii] and [S ii] lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Also shown is the stellar continuum fit in green, clearly demonstrating the H𝛼 absorption feature (H𝛼𝑎𝑏𝑠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In the brightest nebulae, weaker lines such as the [O ii]𝜆7319, 7330Å doublet are clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In some nebulae, faint residuals around bright lines are visible, suggesting more complex kinematics than can be modelled by a single Gaussian component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' However typically these residuals are still at the level of the spectral uncertainty propagated from the MUSE spectral cubes (cyan lines in residual plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In the example shown in Figure 2, 89% of all pixels shown have residuals within 3𝜎 of the errors, with sky residuals dominating the outlying pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We also determine the Balmer emission line equivalent width (EW), as both an indication of the relative brightness of the nebula to the underlying stellar population, and a proxy for the local spe- cific star formation rate surface density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We calculate EW(H𝛼) and EW(H𝛽) following the procedure described in Westfall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2019), as applied within the MaNGA survey for calculation of emission-line moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For EW(H𝛼), we integrate the flux over a central band from 6557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 – 6571.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='35Å in the rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We calculate the continuum flux using determining the median over a blue (6483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0–6513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0Å) and red (6623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0–6653.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0Å) channel and then determine the mean between these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For EW(H𝛽), we use the same approach, with the H𝛽 line determined over the 4847.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 – 4876.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6Å band and the continuum flux using 4827.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 – 4847.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9Å and 4876.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 – 4891.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6Å for the blue and red channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We calculate the equivalent widths in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The first approach is to use the direct nebulae spectrum, with the line flux simply the integral over the central band minus the determined con- tinuum, what we call here the ‘raw’ EW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' However, this ignores the impact of the underlying stellar absorption feature visible in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The second method is to integrate over the spectrum once the best-fitting continuum fit from pPXF has been subtracted or the ‘fit’ MNRAS 000, 1–27 (2022) 47°12\'00" 39 30" 13\'00" 38 Dec 30" 37 14\'00" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='C 30" 36 3h42m12s 06s 00s 41m54s 48s R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 356 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Wavelengths and ionisation potential of the relevant ion for each emission line included in the public catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' All lines are corrected for the Milky Way foreground dust extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Wavelengths are taken from the National Institute of Standards and Technology (NIST;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' https://physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='gov/ PhysRefData/ASD/lines_form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='html), and are Ritz wavelengths in air (consist with wavelengths in the public data release) except for the H Balmer lines, in which case we use the ‘observed ’ wavelength in air as reported in NIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The DAP string name is used to identify the correct extension in the PHANGS-MUSE MAPS files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Ionisation potentials are taken from Draine (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Line name Wavelength String ID Ionisation potential Fixed ratio (air) [Å] [eV] Hydrogen Balmer lines H𝛽 4861.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='35 HB4861 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='60 no H𝛼 6562.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='79 HA6562 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='60 no Low ionisation lines [O i]𝜆6300 6300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='30 OI6300 — no [N ii]𝜆6548 6548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='05 NII6548 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='34 [N ii]𝜆6584 [N ii]𝜆6584 6583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='45 NII6583 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='53 no [S ii]𝜆6717 6716.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='44 SII6716 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='36 no [S ii]𝜆6731 6730.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='82 SII6730 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='36 no High ionization lines [O iii]𝜆4959 4958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='91 OIII4958 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='35 [O iii]𝜆5007 [O iii]𝜆5007 5006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='84 OIII5006 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='12 no [S iii]𝜆9068 9068.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 SIII9068 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='34 no EW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This accounts for the underlying absorption feature, however requires sufficient S/N in the data to get a good fit to the continuum and is poorly determined in low spectral resolution data, such as narrow-band imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In both cases the EW is then the determined line flux over the mean continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The median value across our full catalogue is EW(H𝛼)raw ∼ 20Å and EW(H𝛽)raw ∼ 1Å, with ∼ 43% of nebulae having EW(H𝛽)raw < 0 due to the underlying stellar absorption feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' While the median difference between EW(H𝛼)raw−EW(H𝛼)fit is typically only −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2Å, the difference is relatively stronger for the weaker H𝛽 line at −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' While this presents the EW in a standard format, suitable for com- parison with previous work, it is clear that our nebulae are sitting within the central stellar disk of each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Due to this, our stel- lar continuum band naturally suffers from a significant contribution of light from old stellar population, which is not associated with the young nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The impact of this background contribution is explored in Scheuermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (submitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Given that nebular objects can be marginally resolved in our data, with H ii regions displaying a variety of H𝛼 morphologies, deter- mination of the completeness of our catalogue by quantifying the recovery rate of artificial source injection is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In Santoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2022), the completeness for our catalogue was es- timated in an empirical way by considering the H𝛼 line emission outside of the region masks and measuring the H𝛼 surface bright- ness at the 90th percentile level of the surface brightness distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This surface brightness was then converted to a luminosity assum- ing an unresolved point source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' By this metric, typical completeness limits are 1036 – 1037 erg s−1, which are roughly equivalent to the ionizing flux of a single O7V star (Vacca 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We refer the reader to Santoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2022) for further details and a complete table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For objects classified as H ii regions (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2), in Table 3 we quantify the 10th, 50th and 90th percentile extinction corrected H𝛼 luminosities and physical sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 4 VALUE-ADDED PRODUCTS Given the large suite of emission lines measured within our nebulae, there are multiple physical properties that can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We in- clude these value-added properties in the nebulae catalogue, though note that different calibrations can be used for many of these prop- Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Distribution of H ii region attenuation-corrected H𝛼 luminosities (L𝐻 𝛼) and sizes (𝑟circ) for each galaxy, listing the 10%, 50% and 90% values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Galaxy log(L𝐻 𝛼 [erg s−1]) 𝑟circ [pc] 10% 50% 90% 10% 50% 90% IC5332 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 27 33 46 NGC0628 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='8 31 35 58 NGC1087 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 51 55 98 NGC1300 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 58 62 88 NGC1365 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 76 82 126 NGC1385 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='8 45 51 103 NGC1433 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 58 62 85 NGC1512 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='8 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 79 83 115 NGC1566 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 48 53 100 NGC1672 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 64 70 121 NGC2835 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 48 52 84 NGC3351 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='8 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 36 39 58 NGC3627 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 41 49 96 NGC4254 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 41 48 91 NGC4303 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 46 56 107 NGC4321 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 59 64 96 NGC4535 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 32 40 71 NGC5068 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='8 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 19 23 45 NGC7496 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 56 61 96 erties resulting in systematically different results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For all properties below we only present the results where the relevant lines have a S/N greater than 5, and when the nebular classification is appropriate (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' metallicities are only calculated for H ii regions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A complete list of all columns contained in our nebular catalogue is provided in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 Dust Attenuation All line fluxes are provided in our catalogue as observed values, yet the derivations of physical quantities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', metallicity, ionization pa- rameter) are typically based on intrinsic line fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Therefore, before deriving any quantities, the measured fluxes need to be corrected for reddening due to dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We assume here an O’Donnell (1994) extinc- tion curve with an 𝑅𝑉 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1, that represents a small modification of the Cardelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (1989) extinction curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We then derive the reddening, 𝐸(𝐵 − 𝑉), based on this curve and assume an intrinsic MNRAS 000, 1–27 (2022) PHANGS Nebular Catalogue 7 [NII] [SII] Hα Hαabs [NII] Hα+[NII] [SII] CaT Hβabs [ΟΙII] Ηβ Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A nebula spectrum from IC5332 (region ID 38, 𝐿H𝛼 = 1036.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='8 erg s−1), showing a typical integrated spectrum (gold), including the underlying stellar (and weak nebula) continuum with clear strong emission and absorption lines (with the strong lines labelled) and visible residuals from the sky background subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Overlaid is the resulting analysis pipeline fit from pPXF (blue dashed line), from which we extract the emission line fluxes and kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The lower panel shows the relative residual of the spectral fit in indigo and relative spectral error in cyan ((Nebula-Fit)/Fit and error/fit, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In the insert we show a zoom in of the H𝛼 and [N ii] region, where the spectral lines can be seen more clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We also show the underlying continuum fit (dotted green line) revealing the underlying H𝛼abs feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Balmer ratio of H𝛼/H𝛽= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In practice the choice of extinction curve has little impact on the corrections, as extinction curves do not deviate significantly across the MUSE wavelength range and the H𝛼 and H𝛽 emission lines are bracketing most of the emission lines of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' However the derived 𝐴𝑉 , and hence line luminosity, is directly dependent upon our assumed value of 𝑅𝑉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' By using the O’Donnell (1994) extinction curve we are assuming that the nebula itself only experiences attenuation from a uniform foreground dust layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In reality, complex dust geometries within the nebulae (as seen in nearby H ii regions like the Tarantula;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' De Marchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2016) as well as blending of multiple regions along our line of sight might bias our inferred extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' However, we believe at our <100 pc scales with distinguished nebulae and the thin star-forming disk the foreground screen assumption is more justified than a mixed-media model for the majority of our nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In Figure 3, we show the distribution of V-band attenuations, 𝐴𝑉 , experienced by the nebulae with significant detections of H𝛽 (S/N > 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 31,377 objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9% of the sample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We find a median 𝐴𝑉 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='72 mag (16%–84% range is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='34–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 mag) and a tail of highly attenuated nebulae (5% of objects have 𝐴𝑉 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 mag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' When we weight the 𝐴𝑉 ’s linearly by the intrinsic H𝛼 luminosity (attenuation corrected), we find a median 𝐴𝑉 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 mag (16%–84% range is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='8–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 mag), a significant increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' While it does appear in the sample (and as suggested by Figure 3) that the brightest H ii regions are more attenuated, this difference in median 𝐴𝑉 is also caused by the highly obscured faint H ii regions being undetected in H𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Also visible in this figure are a small subset (<3%) with unphysical attenuation (𝐴(𝑉) < 0, meaning H𝛼/H𝛽 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='86) even with a S/N > 5 in both H𝛼 and H𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 80% of these are consistent within the 3𝜎 line flux uncertainties with a value of H𝛼/H𝛽 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The remainder are typically found in nebulae with low H𝛼 equivalent widths, as can be seen in the central plot in Figure 3, suggesting that for most of these nebulae the underlying Balmer absorption features are incorrectly subtracted leading to an overestimated H𝛽 flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' However, it may also be that the intrinsic Balmer ratio for some of these nebulae is less than our fiducial value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='86 due to physical MNRAS 000, 1–27 (2022) 8 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Columns in the catalogue Column Unit Description gal_name galaxy name region_ID region ID cen_ra deg RA (J2000) center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' weighted by H𝛼 intensity cen_dec deg Dec (J2000),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' weighted by H𝛼 intensity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='flag_edge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='flag set to 1 if within one PSF of the field edge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='flag_star ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='flag set to 1 if overlapping with a star ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='deproj_r_R25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='R25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='Deprojected distance from galaxy center in units of R25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='deproj_r_reff ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='reff ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='Deprojected distance from galaxy center in units of reff ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='deproj_phi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='deg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='Deprojected position angle within galaxy disk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='region_area ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='pixels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='H ii region area ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='emline*_FLUX† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1e-20 erg/s/cm2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='emission line fluxes (see Table 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='emline*_FLUX_CORR† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1e-20 erg/s/cm2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='attenuation-corrected emission line fluxes (see Table 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='assuming an O’Donnell (1994) extinction curve and R𝑉 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 emline*_VEL† km/s line velocity relative to v𝑠𝑦𝑠 (Table 1) emline*_SIGMA† km/s line velocity dispersion, corrected for instrumental broadening AV† mag A𝑉 , V-band attenuation derived from the Balmer decrement assuming an O’Donnell (1994) extinction curve and R𝑉 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 EW_HA6562_raw† Å Equivalent width of H𝛼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' measured directly EW_HB4861_raw† Å Equivalent width of H𝛽,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' measured directly EW_HA6562_fit† Å Equivalent width of H𝛼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' measured after stellar continuum subtracted EW_HB4861_fit† Å Equivalent width of H𝛽,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' measured after stellar continuum subtracted HA6562_LUM_CORR erg/s attenuation corrected H𝛼 luminosity BPT_NII BPT flag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' see Table 5 BPT_SII BPT flag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' see Table 5 BPT_OI BPT flag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' see Table 5 met_scal† Metallicities determined using the Scal prescription (Pilyugin & Grebel 2016) Delta_met_scal Offset in metallicity relative to the radial gradient (Table 9) logU† Ionization parameter derived from [S iii]/[S ii] using the prescription in Diaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 1991 Environment Environment flag, as in Table 8 ∗emission lines are listed in Table 2 †Note that corresponding errors are included as *_ERR reasons associated with the nebula itself (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Planetary Nebulae are both faint and typically several thousand Kelvin hotter than H ii regions and therefore have an intrinsically lower ratio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For these unphysical attenuations we set 𝐴𝑉 = 0 mag when considering the reddening correction of the line ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' All emission lines included in our catalogue are also provided as corrected values (’*_CORR’;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' see Table 4) by applying our determined 𝐴𝑉 and chosen extinction curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 Emission line diagnostic classifications and H ii region catalogue construction In defining the nebulae catalogue, we have used the HIIphot code (Thilker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' However, within the PHANGS–MUSE data we also clearly see H𝛼 emission associated with supernova rem- nants, planetary nebulae, and regions ionized by active galactic nuclei (AGN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As a first pass at separating these regions we use emission line ratio diagrams (commonly called BPT diagrams after their in- troduction in Baldwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 1981) and use the diagnostic curves described in Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2001) and Kauffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2003) to classify the nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We note that while the Kauffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2003) curve is derived empirically from global spectra, it still provides a useful constraint on whether ionization by processes other than UV photons from OB-stars are playing a role in the nebulae (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' shocks, AGN, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Belfiore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022) We use the three strong line diagnostic diagrams (Figure 4) to classify our nebulae;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' [O iii]/H𝛽 versus [N ii]𝜆6584/H𝛼, [O iii]/H𝛽 versus [S ii]𝜆6717, 6731/H𝛼, and [O iii]/H𝛽 vs [O i]𝜆6300/H𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We note that different galaxies follow tracks that are slightly offset and correlate with the total stellar mass of the galaxy (and presumably its metallicity), with all galaxies shown individually in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For each diagnostic, we flag the nebulae with S/N < 5 in any of the lines used in the diagnostic, then mark the remaining as H ii regions, composites, clear AGN impact or LINER-like (indicative of shocks or strong contributions from more diffuse ionized gas) spectra (Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We construct an H ii region sample from those objects classified as H ii regions by all three diagnostics (20,577 objects), as well as objects where [O i] is not detected with S/N > 5 but they are otherwise consistent with the H ii region BPT diagnostics (2,667 objects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This results in a total of 23,244 (74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0%) of objects that are consistent with photoionization by massive stars, and we consider this to be our full H ii region catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This sample would increase by about 1500 objects if we reduced our S/N requirement to 3, and it would only increase by about ∼100 objects if we included objects that fall below the BPT demarcations when accounting the line flux uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Given the factor of two variation in distance and moderate variation in sensitivity between galaxies, we do not achieve uniform detection thresholds across the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In addition, the various H ii region morphologies considered by HIIphot do not lead to homogeneous luminosity thresholds in our source identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' To provide a gen- eral quantification of our typical H ii region properties per galaxy, we quote the 10th, 50th and 90th percentiles in both attenuation- corrected H𝛼 luminosity and H ii region size in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Here, the size is taken as the circularized radius that results in an equal area to the area of the H ii region mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We note that the vast majority of our regions are unresolved, as reflected by the close agreement in 10% and 50% sizes, along with the clear correlation with galaxy distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Because of this, we purposefully do not include size measurements MNRAS 000, 1–27 (2022) PHANGS Nebular Catalogue 9 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The distribution of nebulae attenuations, 𝐴𝑉 , derived from the Balmer decrement and the O’Donnell (1994) attenuation curve with an 𝑅𝑉 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1, compared to the H𝛼 emission line equivalent width determined from the integrated spectra (EW(H𝛼)raw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The central plot shows the 2D histogram of 𝐴𝑉 with the measured EW(H𝛼)raw, while the outer histograms show the 1D distributions of both properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The Balmer decrement (H𝛼/H𝛽) is also shown directly on the top axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' BPT flags included in the catalog Column name Value Meaning BPT_NII 0 star formation −1 low S/N < 5 1 composite 3 AGN BPT_SII 0 star formation −1 low S/N < 5 2 LI(N)ER 3 AGN BPT_OI 0 star formation −1 low S/N < 5 3 AGN in our catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The 10th percentile luminosities provide a general idea of the completeness limits for each galaxy, and we refer to San- toro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2022) for more detailed discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Future work will aim to map out of the H ii region selection function more completely and provide homogenised 150 pc scale catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 Gas-phase metallicities To derive the gas-phase metallicity there are a wide range of prescrip- tions in the literature that can be applied to the nebulae classified as H ii regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Systematic differences between these prescriptions are well known in the literature, and routinely produce absolute measure- ments that differ by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 dex, and even up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 dex, in 12+𝑙𝑜𝑔(O/H) for the same H ii regions (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Peimbert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2019, for reviews on this problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' While qualitatively the difference between individual H ii regions is typically maintained (metal-poor remain poor), the scale in these differences can also be markedly different, as shown by Kewley & Ellison (2008) in SDSS galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We demonstrate these differences in Figure 5, where we apply eight different prescriptions from the literature (Table 6) to our 23,244 H ii regions and compare the resulting metallicity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We note that, as the wavelength coverage of our galaxies by MUSE ex- cludes emission lines below 4800 Å, some of the standard metallicity prescriptions using lines such as the [O ii]𝜆3727 doublet cannot be applied here (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Kobulnicky & Kewley 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Pilyugin & Thuan 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In a pair-wise comparison, we compute a linear conversion between prescriptions, and tabulate the fits (shown in red in Figure 5) in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Note that the number of H ii regions in each panel differs slightly depending upon the detection (𝑆/𝑁 > 5) of the lines involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Most values show a positive correlation, though the strength of the correlation and the scatter between measurements vary wildly, with offsets of up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 dex and scatter of up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 dex apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This should serve as a warning that when comparing metallicity measurements in the literature, it is important to ensure consistent prescriptions are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The conversions we provide are MNRAS 000, 1–27 (2022) Hα/Hβ 3 4 5 6 7 2000 N 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 log(N) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 0 2000 Av [mag] N10 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The distribution of our nebulae on the three BPT (Baldwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 1981) diagnostic diagrams considered when flagging for ionization mechanism (Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' From left to right, we show [O iii]𝜆5007/H𝛽 versus [N ii]𝜆6583/H𝛼, [S ii]𝜆𝜆6716,6730/H𝛼, and [O i]𝜆6300/H𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Overlaid are the Kauffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2003) diagnostic curve (dashed line) and Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2006) diagnostic curves (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Non-stellar ionizing sources are typically found above these lines, and objects between the two demarcations are considered composites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Greyscale demonstrates the distribution of nebulae in these diagrams, and 79% of nebulae are consistent with photoionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Coloured lines (scaled to the galaxy stellar mass) indicate galaxy trends (binned along the y-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' [N ii]/H𝛼 and [S ii]/H𝛼 show clear secondary correlations with stellar mass (corresponding to trends in metallicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Metallicity prescriptions considered in Figure 5 Abreviation Lines used Reference Scal-PG16 H𝛽, [O iii], [N ii], [S ii] Pilyugin & Grebel (2016) O3N2-M13 H𝛼, [O iii], H𝛼, [N ii] Marino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2013) O3N2-PP04 H𝛽, [O iii], H𝛼, [N ii] Pettini & Pagel (2004) N2-M13 H𝛼, [N ii] Marino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2013) N2S2-D16 H𝛼, [N ii], [S ii] Dopita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2016) O3S2-C20 H𝛽, [O iii], H𝛼, [S ii] Curti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2020) RS32-C20 H𝛽, [O iii], H𝛼, [S ii] Curti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2020) R3-C20 H𝛽, [O iii] Curti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2020) only applicable over the metallicity range covered by our sample, defined as the 5–95 percentiles of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In some cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' N2-M13 versus R3-C20) no clear correlation between the pre- scriptions is observed over the narrow metallicity range covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As described in Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2019), we favour the S calibration (Scal) prescription defined in Pilyugin & Grebel (2016), hereafter Scal-PG16, and include these calculated metallicities in our value- added catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The Scal-PG16 prescription was empirically cali- brated against a sample of 313 H ii regions where direct auroral line detections provided measurements of the electron temperature, and hence more robust determination of 12+log(O/H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As it relies on a larger number of emission lines than other prescriptions (Table 6), it is less biased by ionization parameter variations, which can cause line ratio variations and results in degeneracies in the metallicity determination when only one or two line ratios are considered for the prescription (Kewley & Dopita 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' However, note that for the range of metallicities encountered in our sample the calibration is based only on a small fraction of H ii regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The Scal-PG16 prescription relies on three standard diagnostic line ratios: N2 = ([N ii]𝜆6548 + 𝜆6584)/H𝛽, S2 = ([S ii]𝜆6717 + 𝜆6731)/H𝛽, R3 = ([O iii]𝜆4959 + 𝜆5007)/H𝛽, (1) where attenuation corrected line fluxes are used (and therefore im- plicitly includes the ratio of Balmer lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' It is defined separately over the upper and lower branches in log N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The upper branch (log N2 ≥ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6) is calculated as 12 + log(O/H) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='424 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='030 log(R3/S2) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='751 log N2 + (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='349 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='182 log(R3/S2) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='508 log N2) × log 𝑆2 (2) and the lower branch (log 𝑁2 < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6) is calculated as 12 + log(O/H) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='072 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='789 log(R3/S2) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='726 log N2 + (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='069 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='170 log(R3/S2) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='022 log N2) × log S2 (3) The Scal-PG16 prescription is highly correlated with the Dopita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2016) N2S2-D16 prescription, which is calibrated against photoionization models but has similarly been designed to minimise degeneracies with ionization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We calculate the uncertain- ties associated with the Scal-PG16 metallicity by Monte Carlo error propagation of the emission line flux errors, with 1000 samples used to determine the 1𝜎 distribution corresponding to each measured metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Our H ii regions cover a range of metallicities from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 < 12+log(O/H) < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 with typical statistical uncertainties of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='01 dex (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In an independently constructed catalogue, also identified using the HIIphot package, metallicity measurements from 8,914 H ii regions in eight of these galaxies have been published previously (Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Line flux measurements were made using a different data reduction pipeline and different analysis approach, such that the physical extent of the regions may differ and the fitting of the under- lying stellar continuum has changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Comparing our new catalogue with the previous one, we find that ∼7000 H ii regions cross-match within 1′′, and for these objects the median difference in metallicity is negligible (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0003 dex) and the standard deviation is small (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='02 dex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' MNRAS 000, 1–27 (2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 H/[IO] 0 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 10 : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='8 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content="2 0- '0- 9'0- 8'0- 0'- 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 log[NIl/Hα log [Sll]/Hα log [Ol]/HaPHANGS Nebular Catalogue 11 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A comparison of 12+log(O/H) metallicities across eight different prescriptions (listed in Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The corresponding 1-to-1 line (dashed) is shown in each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A linear fit is given for each (red line), with fit parameters listed in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Systematic offsets of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 dex and scatter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 dex is apparent between prescriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Note that the O3N2-PP04 and O3N2-M31 rely on exactly the same line ratios and hence show a perfect correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Our favoured prescription is the Pilyugin & Grebel (2016) Scal, which uses 3 different line ratios (and implicitly the Balmer decrement) to remove degeneracies due to ionization parameter variations, and shows a strong correlation with N2S2-D16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 Ionization Parameter In addition to the actual metal abundance, another parameter of the ISM that regulates line emissivity is the ionization parameter, the ratio of the density of ionizing photons to the number den- sity of hydrogen atoms in the gas (𝑈).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Empirically, this is typically parametrized by line ratios of different ions arising from the same element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Based on theoretical photoionization models (Kewley & Dopita 2002), [O iii]/[O ii] is sensitive to changes in 𝑈 but has a strong secondary dependence on metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' [S iii]/[S ii] shows very little dependence on metallicity but is less commonly used to trace 𝑈 due to the difficulty in observing the red [S iii]𝜆𝜆 9068,9532 lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' MUSE now gives us coverage of the [S iii]𝜆9068 emission line, en- abling us to explore ionization parameter variations more directly using the [S iii]/[S ii] line ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' However, while the [S iii]/[S ii] line ratio appears to be a robust tracer of 𝑈 in theoretical models, comparisons have uncovered large offsets between model predictions for the ratio and empirical results (Mingozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We apply the prescription of Diaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (1991) to determine 𝑈 from this diagnostic ratio as log(𝑈) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='684 log([S iii]𝜆𝜆9068, 9532/[S ii]𝜆𝜆6716, 6730) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (4) MNRAS 000, 1–27 (2022) 6 2-D1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 M1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 O3N2-PP04 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 12+log(0/H) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 N2-M13 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 0 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 log(N) 2 O3S2-C2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 RS32-C20 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 R 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 N2S2-D16 O3N2-PP04 N2-M13 O3S2-C20 Scal-PG16 O3N2-M13 RS32-C20 12+log(O/H)12 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Linear fit parameters for converting metallicities, see red lines in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The range (5–95 percentiles) over which this conversion holds is shown in square brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For each conversion we provide in parenthesis the following quantities: intercept, slope, scatter about the conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' N2S2-D16 [8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9] (-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9, 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='02) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='07) (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='04) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='03) Scal-PG16 N2S2-D16 O3N2-M13 O3N2-PP04 N2-M13 O3S2-C20 RS32-C20 [8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6] [8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9] [8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6] [8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9] [8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6] [8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='8] [8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9] Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Distribution of errors in 12+log(O/H) as a function of offset from the radial gradient in each galaxy (Δ(O/H)) based on the Scal-PG16 prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' These errors are based on Monte Carlo resampling of the associated line flux uncertainties, and do not take into account systematic uncertainties related to the calibration of the metallicity prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Typical 1𝜎 uncertainties are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='01 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Here we measure only the shorter wavelength [S iii]𝜆9068 emis- sion line but assume a ratio of [S iii]𝜆9532 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 × [S iii]𝜆9068 fixed by atomic physics (Osterbrock & Ferland 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Tayal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We require a S/N > 5 in all lines, and are able to compute 𝑈 for 20,781 objects (66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2% of the sample), nearly all of which (20,083 objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 97% of objects with measured 𝑈) are classified as H ii regions (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In Figure 7 we see that our H ii regions cover a range of ion- ization parameters from −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 < log𝑈 < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5, with no systematic differences by galaxy stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Typical uncertainties are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='04 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Apparent within each galaxy is a positive correlation between 𝑈 and 12+log(O/H), first reported in luminous star-forming galaxies (Do- pita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2014), and recently identified as a robust trend across H ii region samples (Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Grasha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This is con- trary to theoretical predictions by Massey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2005), and is also not well reflected in photoionization models (Ji & Yan 2022), indicating a need for additional model development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' No clear correlation of 𝑈 with stellar mass of the galaxy due to the scatter within each galaxy and the differing slopes of 𝑈 versus (O/H) found between galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 Nebulae Environments The nebulae we identify do not exist in isolation, but rather are part of the larger scale structure of our galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Therefore we also include in our catalogue parameters that trace the different galactic environments in which they occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The galaxies in our sample show visible structures (centre, bar, spiral arm, interarm, disc) that may reflect differences in dynamical conditions and star formation histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' To define the nebular envi- ronments, we use the stellar morphological masks that have recently been identified systematically in Querejeta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2021) based on Spitzer 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 𝜇m images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We locate each of our nebulae with respect to the simplified environments defined in that paper, as summarised in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In Figure 8 we show both the absolute number of H ii regions within each environment (left) as well as the surface density of all objects in each environment (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In absolute numbers, most of our H ii regions (∼40%) are located in interarm regions, however this environment also makes up the largest area in our fields and so correspondingly there is a relatively low number density of nebulae in this environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In contrast, we identify only a small number (∼ 300, ∼2%) of H ii regions in the galaxy centres, but this corresponds to a high number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We find the highest H ii region number density within spiral arm environments, and the lowest H ii region number density within bar environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This reflects the fact that star formation is typically concentrated into spiral arms, and bar environments (excluding bar ends) potentially suppress star formation via bar-driven dynamics (James et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Looking at the full object catalogue, including objects that do not meet our H ii region classification criteria, the number density of objects in the bar approximately doubles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The centre environments also contain a high number density of objects that are not classified as H ii regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This is expected if AGN and dynamical shocks are significantly contributing to the gas ionization in these environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 5 RESULTS While the total stellar mass and integrated star formation rate are global properties that correlate with the integrated properties of the ISM (Tremonti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Sánchez 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Pessa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2021), sig- nificant secondary correlations are identified within galaxies once their nebular emission is spatially resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The most well known MNRAS 000, 1–27 (2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='07 3 r in 12+log(O/H) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='05 3g 2 (N)60l 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='03 error L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='00 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='20 △ (O/H)PHANGS Nebular Catalogue 13 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Metallicity (12+log(O/H)) as a function of ionization parameter (U) for H ii regions within each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Galaxies are sorted by stellar mass from low (top left) to high (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Linear fits are performed in each galaxy, demonstrating the prevalence of positive correlations across the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Apart from the well-established mass-metallicity relation, no obvious trends are seen as a function of total galaxy stellar mass (red colour scale, bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Our Environmental flags, based on simplified assignments from Querejeta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=" Label Querejeta2021 Environment Environment 1 Centre −→ Centre 2 Bar (excluding bar ends) � Bar 3 Bar ends 5 Spiral arms inside interbar (𝑅gal < 𝑅bar) � Arm 6 Spiral arms (𝑅gal > 𝑅bar) 4 Interbar (𝑅gal < 𝑅bar) � Interarm 7 Interarm (𝑅gal > 𝑅bar) 8 Outer disc (𝑅gal > spiral arm ends) 9 Disc (𝑅gal > 𝑅bar) in galaxies without spiral masks −→ Disc MNRAS 000, 1–27 (2022) 'NGC5068 IC5332 NGC1087 NGC1385 8." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 NGC2835 NGC0628 NGC4254 NGC4303 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 12+log(0/H) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 NGC4535 NGC1300 NGC1672 NGC432 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 NGC1 NGC1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 log(M*) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 10 11 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0-2.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 log U14 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 0 5 10 15 20 25 30 35 40 fraction [%] Centre Bar Arm Interarm Disc 0 2000 4000 6000 8000 10000 N Centre Bar Arm Interarm Disc 0 2 4 6 8 10 12 14 16 18 N / kpc2 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Distribution of H ii region detections across different galactic environments, both in absolute numbers (left) and in surface density (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Unfilled bars (right) indicate the environmental distribution of all nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Interarm regions contain the largest number of H ii regions (∼40% of the sample) but cover a large area of our fields and as such reflect a relatively low surface density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In contrast, we identify only a small number (∼ 300) of H ii regions in the galaxy centres, but this corresponds to a high surface density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Nebulae not classified as H ii regions are over-represented in centre and bar environments, where shock excitation is more likely to contribute to the gas ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' of these is the metallicity gradient (for recent reviews see Maiolino & Mannucci 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' However, gas-phase metal- licity has also been shown to spatially vary with both stellar mass surface density (Barrera-Ballesteros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2016) and gas-mass sur- face density (Barrera-Ballesteros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Once this radial gra- dient is removed, higher-order variations of the metallicity are seen, along spiral arms (Sánchez-Menguiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2017) and across discs (Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We revisit here that work of Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2019), an earlier analysis of spatial metallicity varia- tions in a subset (8 out of 19) of the PHANGS–MUSE galaxies using data from an earlier version of our reduction and analysis pipeline (see also Emsellem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For this analysis, we consider our H ii region sample to be those objects that are fully contained in the field of view (‘flag_edge’ = 0), are consistent with photoionization (‘BPT_NII’ = ‘BPT_SII’ = 0 and ‘BPT_OI’ <= 0), and where we have high confidence in our metallicity measurement (‘met_scal_err’ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='04 dex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' see Figure 6 and Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' By including a cut on metallicities with large uncertainties, we exclude only ∼1000 regions that have an average uncertainty of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='06 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We also exclude six regions with metallicity values 12+log(O/H) < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 as we believe these are spurious and they significantly bias our statistics (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For the following sections, our catalogue consists of 22,318 H ii regions, with between 477–2556 H ii regions per galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 Radial Gradients As described above, it has been clearly established that galaxies in the local universe systematically have a lower metallicity with increasing radius (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Moustakas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Pilyugin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Sánchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' These radial trends neglect prominent morphological features (spiral arms, stellar bars), though do appear to show variations in the inner and outer parts of galaxies (Sánchez-Menguiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The metallicity gradients vary with stellar mass (Sánchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2014) and with radius at a given mass (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Boardman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2021), and are thought to chart the typical inside-out growth of most disc galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In Figure 9 we show radial trends for a few of the key ISM diagnos- tics available in our H ii region catalogue, for a representative sample of four galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' These are trends in the H𝛼 luminosity (L(H𝛼)) of in- dividual H ii regions, attenuation derived from the Balmer decrement (A𝑉 ), metallicity (12+log(O/H)), and ionization parameter (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Ra- dial trends for all 19 galaxies are shown in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' These four galaxies in Figure 9 represent a low stellar mass (IC 5332) and high stellar mass (NGC 1365) galaxy, and systems with a regular spiral pattern and no bar (NGC 0628) or strong bar and widely separated arms (NGC 7496).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For each galaxy we show the radial trends scaled to reff, to normalise the sample, and note that in most cases our cov- erage is limited to the inner parts of each galaxy (<2 reff).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We fit each gradient with a linear relation that neglects the uncertainties, as variations in these properties are expected to reflect local variations in the physical conditions and not uncertainties in the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Representative values at 1 reff are also given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The individual radial gradients we determine for our PHANGS–MUSE sample fall largely within the range found from the much larger IFS surveys such as CALIFA (Espinosa-Ponce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022) and MaNGA (Barrera-Ballesteros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Beyond these radial trends, how- ever, we see a large scatter of these properties within each galaxy due to the higher spatial resolution of our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In general, galaxies show relatively flat or slightly negative slopes in L(H𝛼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In many cases central starbursts are also apparent, cov- ering scales of a few 100 pc and exhibiting high H𝛼 luminosities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' in NGC 1365), though we note that these measurements could be biased by our ability to deblend neighbouring H ii regions given our angular resolution (∼70 pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' There is also a floor imposed by our region identification methods and sensitivity limits, visible as a relatively sharp lower bound to our region luminosities, and tracking the variations in galaxy distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Most (80%) of the H ii regions we detect have L(H𝛼) < 1038 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' About ∼800 (∼3%) of our H ii regions might be categorized as ‘giant H ii regions’ (L(H𝛼) > 1039 erg s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' H ii region luminosity functions for each galaxy are presented in Santoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We find flat or slightly negative slopes in A𝑉 , with typical values ranging from 0–2 mag of extinction, and a median value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='75 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We emphasise that these measurements do not represent an unbiased view of dust in the disc, as most of our nebulae are associated with star-forming regions that are expected to be dustier (Calzetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We also potentially miss high attenua- tion, heavily embedded regions where H𝛽 or even H𝛼 may not be visible, though the incidence of such obscured nebulae in the lo- cal Universe is small (Prescott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2007), and likely even rarer in our low-inclination galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Any such obscured population will be constrained with our upcoming PHANGS–JWST observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' MNRAS 000, 1–27 (2022) PHANGS Nebular Catalogue 15 Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Linear fit parameters for radial gradients in 12+log(O/H) using the Scal prescription, see Figure 10 Galaxy intercept slope [dex/r𝑒 𝑓 𝑓 ] value at r𝑒 𝑓 𝑓 𝜎(O/H) IC5332 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='475 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='173 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='002 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='302 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='066 NGC0628 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='533 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='054 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='001 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='478 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='048 NGC1087 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='479 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='070 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='010 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='409 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='032 NGC1300 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='617 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='079 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='001 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='537 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='042 NGC1365 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='666 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='188 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='005 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='477 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='040 NGC1385 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='459 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='038 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='014 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='421 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='033 NGC1433 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='569 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='013 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='001 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='556 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='051 NGC1512 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='581 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='016 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='001 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='565 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='042 NGC1566 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='613 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='037 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='004 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='576 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='037 NGC1672 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='566 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='013 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='000 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='553 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='033 NGC2835 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='555 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='157 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='002 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='398 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='040 NGC3351 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='579 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='007 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='001 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='587 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='044 NGC3627 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='538 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='006 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='004 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='544 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='033 NGC4254 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='590 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='028 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='003 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='562 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='030 NGC4303 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='613 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='032 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='006 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='580 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='034 NGC4321 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='592 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='028 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='004 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='564 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='036 NGC4535 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='580 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='040 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='003 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='541 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='039 NGC5068 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='412 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='094 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='001 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='318 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='054 NGC7496 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='588 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='081 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='003 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='507 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='045 As in Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2019), we do not see significant radial gradi- ents in the ionization parameter for any of our galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This reflects that the localized ionization state of the gas is sensitive mainly to changes in the ionizing source and local gas density at the cloud interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The flat radial gradient also increases our confidence in ra- dial trends we uncover in metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Ionization parameter variations can influence diagnostic line ratios and introduce biases depending on the metallicity prescription used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Our preferred metallicitiy pre- scription, the Pilyugin & Grebel (2016) S-calibration, is designed to minimize this bias but it is reassuring that we also do not observe any radial trends in ionization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Our most pronounced radial trends are apparent in the H ii region metallicities (12+log(O/H)), and we show the radial metallicity gra- dients for all 19 galaxies in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A simple linear fit (solid line) shows very good agreement with the median value in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 reff wide bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For the radial fit, we exclude the central 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 reff as suggested by Sánchez-Menguiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We have also not included the un- certainties when performing our linear fit, as we allow for an intrinsic scatter due to physical variations in the gas conditions in excess of our estimated uncertainties (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='01 dex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For each bin we also track the 1𝜎 scatter (outer lines), and find very good agreement between these binned radial trends and the linear fits, suggesting that to first order a linear fit describes the data well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This finding has been more thoroughly quantified in Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Our galaxies clearly reflect the well-established mass-metallicity relation (Tremonti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2004), with less massive galaxies exhibiting system- atically lower metallicities (bottom right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For 10 of the 19 galaxies, measurements of the corotation radius are available (Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This is the location where the gas rotational dynamics and spiral pattern speed are matched, and can only be robustly measured through analysis of the stellar kinemat- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Theoretical work has suggested that at this dynamical location, metallicity variations are predicted to be amplified due to the lower relative velocity between the gas and spiral pattern overdensity (Spi- toni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We see no obvious change in the metallicity scatter at the locations of corotation, or radially with respect to corotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Radial gradients for all galaxies are reported in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Gradients derived using alternative metallicity prescriptions are also provided in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 Correlations with Global Properties Based on the radial trends, we explore correlations of representative derived properties with global galaxy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In particular, we explore trends with total stellar mass, star formation rate (SFR) and gas fraction (calculated as the sum of the Hi and H2 gas mass rela- tive to the total gas and stellar mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' These are all global properties that are typically associated with the regulation of galaxy evolution (Genzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Our galaxies span just over an order of magni- tude dynamic range in these key properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' While our sample size is small compared to integral field spectral galaxy surveys like CALIFA (Sánchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2012), MaNGA (Bundy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2015), or SAMI (Croom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2021), our ability to robustly isolate individual H ii regions pro- vides a novel opportunity to cleanly consider trends relating small scale properties to global differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As both extinction (A𝑉 ) and ionization parameter (U) show no clear radial trends, we consider the median value measured across the galaxy disk (Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A𝑉 shows modest trends for higher values at higher stellar mass and SFR, consistent with an increased amount of gas (and hence dust) associated with these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We see no trend with gas fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Ionization parameter shows no trends with global properties, indicating that it is regulated by local physical conditions in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Using our radial metallicity gradient fits, we calculate a repre- sentative metallicity at 1 reff for each galaxy, and consider global trends in metallicity and metallicity slope (Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As expected, we recover the mass-metallicity relation (Tremonti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2004), with more massive galaxies systematically exhibiting higher metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Secondary dependencies have been reported with SFR (Mannucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Lara-López et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' de los Reyes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2015), with galaxies at lower stellar mass exhibiting lower metallicities at fixed SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This effect is broadly seen in our small galaxy sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' From gas-equilibrium models, this trend has been proposed to derive pri- marily from a decreasing gas fraction corresponding to high SFRs (Peeples et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2008, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Bothwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2013), and to some de- gree this is also reflected in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' However, we are generally in agreement with the larger CALIFA sample of Alvarez-Hurtado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2022) in that, once the stellar mass metallcity correlation is removed, the other global properties show no obvious trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Trends for steeper metallicity gradients in more massive galaxies are reported in large galaxy surveys (Belfiore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Poetrodjojo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2018), but in fact we observe the opposite trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This could be due to the radial coverage of our sample being limited, or the predominance of bar-dominated systems (these have been observed to exhibit flatter metallicity gradients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Zurita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' With those caveats, we also see trends for flatter slopes at high SFR and low gas fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 Global metallicity variations As the linear gradient in metallicity represents the dominant first order trend, we follow the approach developed in Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2019) to fit and subtract this radial gradient and examine the second order variations in metallicity, Δ(O/H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We further quantify the 1𝜎 scatter in Δ(O/H) over the entire galaxy as 𝜎(O/H) for each galaxy (Table 9), to understand whether the second order variations in metallicity are driven by global galaxy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We find 𝜎(O/H) varies across the galaxy sample, with values rang- ing from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='03 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='06 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' These values do not change significantly (<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='005 dex) if we impose a stricter cut on our metallicity uncertain- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In the top panels of Figure 13 we show how 𝜎(O/H) correlates with different global galaxy properties, including total stellar mass MNRAS 000, 1–27 (2022) 16 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Radial gradients for four representative galaxies in key ISM diagnostics (from top to bottom): H𝛼 luminosity (L(H𝛼)), extinction derived from the Balmer decrement (A𝑉 ), metallicity (12+log(O/H)), and ionization parameter (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For each galaxy we show the radial trends normalized to reff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' These four galaxies represent low stellar mass (IC 5332) and high stellar mass (NGC 1365) galaxies, and systems with a regular spiral pattern and no bar (NGC 0628) or strong bar and widely separated arms (NGC 7496).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Most trends are flat or mildly negative, except for the metallicity gradients which show the strongest negative trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (M∗), total star formation rate (SFR), and gas fraction (calculated as the sum of the Hi and H2 gas mass relative to the gas plus stellar mass), all properties which might be expected to regulate mixing in the disc (Krumholz & Ting 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We find a weak correlation with M∗ and SFR, and no correlation with gas fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2020), the mixing scale for metals, as quantified via the two point correlation function, was found to display similar weak correlations with SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' However, those authors found the most pronounced correlation with the gas velocity dispersion, indicating that the homogeneity of the metal distribution in the gas (and cor- responding mixing scale length) was regulated by gas turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' To test this, we consider 𝜎(O/H) as a global measure of this metal distribution homogeneity and compare it with three tracers of the multi-phase gas velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In the bottom panels of Figure 13, we show two constraints from the ionized gas: the median ion- ized gas velocity dispersion, measured across the entire MUSE map (𝜎H𝛼,disc) and the median ionized gas velocity dispersion measured only within the H ii regions (𝜎H𝛼,HII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For both of these we con- sider only pixels or regions where the H𝛼 emission achieves a S/N > 20, to minimize uncertainties introduced by the low spectral resolu- tion of MUSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We also correct for the instrumental dispersion (∼49 km s−1 at H𝛼, as reported in Bacon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The disc as a whole shows typically higher dispersions (∼30–35 km s−1) compared to the H ii regions (∼20–30 km s−1), reflecting elevated gas dispersion in the diffuse ionized gas (Moiseev & Lozinskaya 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Moiseev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Della Bruna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Both show positive correlations with 𝜎(O/H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The tightest correlation is seen when considering the molecular gas velocity dispersion (𝜎CO), measured from the median value within the ‘strict’ second moment maps (Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' These values are significantly smaller (∼2–4 km s−1), reflecting the thin mid-plane distribution of this colder and dense ISM component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' These correlations between the global scatter in metallicities (rela- tive to the radial gradients) and the turbulent state of the ISM (in both the ionized and molecular material) demonstrate convincingly MNRAS 000, 1–27 (2022) IC5332 NGC0628 NGC7496 NGC1365 log L(Ha) [erg/s] 42 41 40 39 38 31 36 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 Av [mag] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 12+log(0/H) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 8:1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 U 601 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 R / reffPHANGS Nebular Catalogue 17 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Metallicity 12+log(O/H) radial gradients based on the H ii regions within each of the 19 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Galaxies are sorted by the stellar mass, from low (top left) to high (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For each galaxy we show the radial trends normalized to reff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A simple linear fit (central line) shows very good agreement with the median value in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 reff wide bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For each bin we also track the 1𝜎 scatter (outer lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' These radial trends are compared to the locations of co-rotation (blue vertical lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2021), as measured from the stellar kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The bottom right panel shows the linear trends for all galaxies overplotted, with the colour-scale indicating the total stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' that the ISM dynamics play a critical role in regulating the mixing of metals across galaxy discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 Local metallicity variations In Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2019) we identified a strong correlation between Δ(O/H) and ionization parameter (as traced by [S iii]/[S ii]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Here, we revisit secondary correlations between Δ(O/H) and other local ISM physical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure 14 compares Δ(O/H) with the H𝛼 velocity dispersion measured across the integrated H ii region spec- tra and with the A𝑉 measured via the Balmer decrement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' There is a weak negative correlation with velocity dispersion echos the result found globally in Figure 13, and is reflected systematically within in- dividual galaxies (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Given the low instrumental velocity resolution (∼49 km s−1 at H𝛼) and moderately large integrated scales (∼70 pc), we expect that our determined ionized gas velocity disper- sion traces predominantly the larger scale ISM turbulence rather than local cloud turbulence (likely contributing only ∼10 km s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Relaño et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Medina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2014), though we cannot exclude the possibility that some of these systems experience strong stellar winds (which can contribute to expansion velocities by as much as ∼60 km s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Egorov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2014, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We also identify a positive correlation between Δ(O/H) and A𝑉 , which is again present within individual galaxies (dashed lines) though with more variations be- tween galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' However, both of these correlations could also arise from the correlation of Δ(O/H) with H𝛼 luminosity (see Figure 5 in Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In Figure 15 we qualitatively examine the locations of regions with particularly high and low metallicity (relative to the radial gradient) within the galaxy discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Here, we make a simple cut and highlight MNRAS 000, 1–27 (2022) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 NGC5068 IC5332 NGC1087 NGC1385 NGC7496 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 NGC2835 NGC0628 NGC4254 NGC4303 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 2 +log(O/H) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 NGC4535 NGC1300 NGC1512 NGC1672 NGC4321 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 NGC1566 NGC3627 NGT1433 NGC1365 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 log(M) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 9 10, 11 3 0 3 4 0 1 2 3 4 0 4 2 3 4 R / reff18 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 109 1010 1011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 median AV [mag] r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='43 100 101 r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='35 r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='01 109 1010 1011 M ∗ [M ⊙ ] −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 median log(U) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='26 100 101 SFR [M ⊙ / year] r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='35 gas fraction r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='24 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Comparison of median extinction (A𝑉 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' top) and median ionization parameter (𝑈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' bottom) as a function of global galaxy properties (stellar mass, star formation rate, gas fraction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In the top left of each plot we show the Pearson correlation coefficient of those quantities, 𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A𝑉 shows modest correlations with stellar mass and SFR, reflecting an increased gas content (and hence dust content) in these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 𝑈 shows no correlations, reflecting that it is set primarily by local ISM conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Comparison of parameters derived from the radial metallicity gradients with global galaxy properties (stellar mass, star formation rate, gas fraction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' These include the metallicity measured at 1 reff(top), and the metallicity slope (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Points are colour coded by the total stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' MNRAS 000, 1–27 (2022) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='65 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='60 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='00 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='55 1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='75 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='50 (H/0)60I+2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='45 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='50 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='40 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='35 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='25 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='30 [m 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='25 109 1010 1011 100 101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='35 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='00 (M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='05 log10( 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='00 slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='50 etallicity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='15 met 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='20 109 1010 1011 100 101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='35 M*[M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='] SFR [M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' / year] gas fractionPHANGS Nebular Catalogue 19 109 1010 1011 M ∗ [M ⊙ ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='07 σ (O/H) r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='22 100 101 SFR [M ⊙ / year] r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='35 gas fraction r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='23 20 25 30 35 40 σHα, disc [km/s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='07 σ (O/H) r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='63 20 25 30 35 40 σHα, HII [km/s] r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='65 0 1 2 3 4 5 6 7 8 σCO [km/s] r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='67 Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Correlations between the global scatter in metallicity (𝜎(O/H)), after removing the first order radial gradient, as a function of galaxy integrated properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In the top right of each plot we show the Pearson correlation coefficient of those quantities, 𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The total stellar mass (M∗), total star formation rate (SFR), and gas fraction show weak or absent trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' More pronounced correlations are seen with the ionized gas velocity dispersion, as measured across the full disc (𝜎H𝛼,disc) or just the H ii regions (𝜎H𝛼,HII), as well as with the molecular gas velocity dispersion (𝜎CO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' H ii regions with Δ(O/H) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='05 in red, and H ii regions with Δ(O/H) < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='05 in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' While in some cases the enriched regions appear concentrated along spiral arms or at bar-ends, they can also be found distributed across the galaxy discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Similarly, the regions with de- creased abundances show some clustering (reflecting the homogene- ity on kpc scales quantified in Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2020), but no obvious patterns with galaxy environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' There is no clear difference in Δ(O/H) between the different environmental masks of Querejeta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2021), as was also shown by Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This reflects a complicated relation between enrichment patterns and individual galaxy dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A more detailed analysis of a larger sample of H ii regions with a larger dynamic range in these quantities will be need to disentangle what drives these second-order metallicity variations in galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 6 DISCUSSION 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 Metallicity variations As reported in Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2022), our metallicity gradients are dominated by linear radial trends (Figure 10) following earlier works on larger samples (Sánchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Espinosa-Ponce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Barrera-Ballesteros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Following Sánchez-Menguiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2018) we do not measure the gradient within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5reff, however unlike that work we see no consistent indication for a flattening within our galaxy sample at larger radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Our smaller sample as compared to the 109 galaxies in Sánchez-Menguiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2018) mean its harder to draw conclusions why this is, but the higher physical resolution of the PHANGS–MUSE sample (∼ 70 pc) as compared to theirs (∼ 460 pc) could be one possible reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Second-order variations are small, typically below ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='05 dex, and reflect a remarkable level of homogeneity in the metallicity distribution across galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' They do not show any correlation with co-rotation radius, which has been predicted by simulations to influence the efficiency of mixing within the disc (Spitoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We establish that there is a weak global correlation between the magnitude of metallicity variations (𝜎(O/H)) and star formation rate (Figure 13, top), but tighter correlations are observed with measures of the global median gas velocity dispersion (Figure 13, bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This holds when considering both ionized and molecular gas phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This is consistent with more turbulent ISM conditions leading to mixing on larger scales, resulting in overall more homogeneity in the metal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We do not aim to investigate the source of turbulence in this paper, and careful work will be needed to disentangle the effects of star formation, secular dynamical processes and external gas accretion processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We identify H ii regions with metallicities significantly different from the linear radial gradients, and observe that on local scales this correlation holds, with enriched H ii regions correlating with lower local ionized gas velocity dispersions (Figure 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' These also correlate with dustier local environments, as traced by the Balmer decrement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Assuming gas and dust are well mixed, this would suggest that more enriched regions are associated with higher density gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Together with the trend established with H𝛼 velocity dispersion, this leads to a picture where a denser, calmer ISM facilitates localized pockets of enrichment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' At the other end of the spectrum, relatively more metal poor gas is associated with lower gas densities and in- creased turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The local metallicity variations we report are MNRAS 000, 1–27 (2022) 20 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Metallicity variations measured after removing the radial gradient (Δ(O/H)) show a negative correlation with the H𝛼 velocity dispersion (𝜎𝐻 𝛼) measured across the H ii region (left), and a positive correlation with the A𝑉 measured from the Balmer decrement (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The greyscale indicates the distribution density, and linear fits performed for individually galaxies are shown as dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Relative to a perfectly linear metallicity gradient (blue line), enriched regions are found at low velocity dispersions and dusty environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' consistent with the picture developed in Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2020), which quantified the mixing scales within eight of these 19 galaxies and determined that metallicity variations are likely reflecting dilution rather than pollution of the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Another key result from Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2019) in their initial study of eight of the PHANGS–MUSE targets is the identification of sys- tematic azimuthal variations in the metallicity distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This has been confirmed for the full sample of 19 galaxies in Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2022), who analyzed interpolated metallicity maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' What remains less clear is how these variations may or may not correlate with galaxy environments (centre, bar, spiral arm, interarm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Looking at integrated environments across the sample, Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2022) were unable to recover any systematic trends aside from systematic enrichment of galaxy centres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This is somewhat in conflict with pre- vious results on individual galaxies (Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Vogt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2018), and claims based on growing samples (Sánchez- Menguiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2019) for correlations in metallicity variations with spiral arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In the initial sample of eight galaxies of Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2019), half were found to have variations correlating with spiral structure but often in only a single spiral arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We do not revisit this interesting topic as we believe it requires careful dynamical considerations, tailored to each galaxy, but it demonstrates the complicated abundance patterns in relation to the galaxy environments (Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' While in some galaxies the en- riched regions (in red) appear to strongly trace the spiral pattern (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' NGC 1365, NGC 1566, NGC 1672), they can also generally be found throughout the entire disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Qualitatively, the regions with reduced metallicity (in blue) often appear somewhat clustered and located at bar-ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' These maps highlight the challenges in estab- lishing the role of galaxy environment and the role of gas flows in regulating the enrichment patterns in galaxy discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 Missing nebulae and objects failing the BPT cuts Our catalogue consists of objects that are selected to be bright in H𝛼, but as is apparent in the BPT diagrams (Figure 4) these are not all nebulae where the ionization is dominated by photoionization from young massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Based on our consideration of diagnostic line ratios (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2), this results in a sample of 23,244 nebulae that we classify as H ii regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' However, it leaves 7,546 objects in our catalogue for which we provide no definitive classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' These could be H ii regions blended with strong DIG or AGN emission, SNRs or PNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Note that 609 objects are already excluded entirely from this analysis as they fall at the field edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Of the unclassified objects, we find that 4,688 are labelled as ‘composite’ based on the [O iii]/H𝛽 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' [N ii]/H𝛼 BPT diagnostic (BPT_NII = 1), and are quite likely H ii regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The commonly used BPT demarcation empirically established by Kauffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2003) was developed for classification of central kpc-scale and in- tegrated galaxy spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Recent work has begun to explore whether this parameter space is sufficiently represented once outer disc envi- ronments are considered, and wider parameter space including kine- matic diagnostics are included (Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' With our work, we consider even smaller physical scales (<100 pc), and indeed recent modelling has shown that individual H ii regions throughout their evolution may populate the ‘composite’ regions of the BPT diagram (falling between the Kauffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2003 and Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2001 demarcations) for short periods during their earliest phase of evolu- tion (Pellegrini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' One of the main long-term science goals for producing this catalogue is to provide the necessary database of high-quality emission-line fluxes necessary to continue such detailed comparisons with cutting-edge models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Within PHANGS, ongoing work applies a bayesian framework to match line ratios in emission line objects with different model grids, with the goal of establishing new classification methods (Congiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' MNRAS 000, 1–27 (2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='20 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='05 2 / / II (N)60) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='05 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='20 0 0 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 [km/s] Ay [mag] OHαPHANGS Nebular Catalogue 21 Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Maps of the H𝛼 emission overlaid with locations of H ii regions with particularly high (red) or low (blue) metallicities compared to the radial gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Here we have required low statistical uncertainties (<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='01 dex) and be offset from the radial trend by more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='05 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' While patterns appear in some galaxies relative to the spiral pattern (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' NGC 1672, NGC 1365), and spatial clustering is apparent in many systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' NGC 2835, NGC 3627), variations in the metallicity patterns are not systematically observed in all galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Active Galactic Nuclei (AGN) provide another potential source of gas excitation, and are present in 7 (37%) of our galaxies (as labelled in Table 1), with four of these AGN hosting molecular gas outflows (three galaxies without AGN also host molecular gas outflows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Stuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In some cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' NGC 1365;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Venturi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2018) they represent a remarkable dominant source of ionization, with [O iii] bright ionization cones visible across the central kpc of the galaxy and extending over nearly the full MUSE field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In cases of lower luminosity AGN (NGC 1433, NGC 4303, NGC 7496), it can be difficult to spatially isolate any AGN contributions, as they appear as extended ionized structures associated with (presumably) outflowing material, and seen in projection with H ii regions will bias the emission line diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Visualization of the emission line maps for all galaxies is available in Emsellem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' MNRAS 000, 1–27 (2022) ngc1087 ngc0628 ngc1300 ngct kpc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' kpc 1 kpc ngc1385 ngc1566 ngc1512 ngc1433 167 kpo 1kpc 1kpc ngc3627 ngc2835 ngc4303 1 kpc kpc ngc4535 ngc7496 ngc5068 ngc432 "kpo22 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Supernova remnants (SNRs) are the network of shocks caused by supernova explosions as they expand into the surrounding ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' They are often identified via their strong [S ii] and H𝛼 emission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As predicted by shock models (Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2008), their diagnostic line ratios populate regions of the BPT diagrams that partially overlap with photoionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' With typical sizes of less than 100 pc, most of these objects are expected to be unresolved in our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Ongoing projects within the PHANGS collaboration are building catalogues of SNRs, focusing on detection of these objects via their distinctive line ratios and broadened line kinematics (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' in prep, Congiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Preliminary results suggest between 1000–5000 SNRs are present in our data, and may make up a significant fraction of the unclassified objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Planetary nebulae (PNe) are shells of gas expelled by intermediate mass stars (1-8 M⊙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Parker 2022) and ionized by the central source in the end phase of their life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' They are particular bright in [O iii] line emission (Parker 2022), but can also emit strongly in H𝛼, and are un- resolved at the distances of our target galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Given that the central stars exhibit a harder ionizing spectrum than typical H ii regions, they could be expected to fall in regions of the BPT diagrams tradition- ally populated by AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Recently, Scheuermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2022) used the PHANGS–MUSE observations to identify PNe, with selection based on [O iii] emission and source classification confirmed based on diagnostic line ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' They find a total of 899 PNe across these 19 galaxies, 193 of which are within 1′′ of sources in our nebular cata- logue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 158 of these (82%) are unclassified, failing our BPT cuts, and making up a very small fraction of the unclassified objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Overall, these make up a very small fraction of our nebular catalogue, which is unsurprising as many of the PNe are faint or undetected in H𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 Odd/misclassfied nebulae Because of our automated approach to object identification, line fitting, and object classification, our catalogue contains objects that appear as unusual outliers in critical diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We examine the spectra of a subset of these objects to understand if there are any systematic problems and provide some explanation for these outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' One category of unusual object are H ii regions identified as having particular large H𝛼 velocity dispersions, and we look at the 12 objects 𝜎𝐻 𝛼 > 200 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' One object (NGC1385, #12) clearly corresponds to a foreground star missed by our flagging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We find that the central AGN in both NGC 1365 and NGC 1566 end up classified as H ii regions because we perform only single Gaussian fits to our emission lines, and the underlying broad line component results in a biased fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Two other objects (#589 in NGC 1300 and #35 NGC 3627) appear to be a result of poorly constrained continuum fits and the misclassification of noise as spectral lines within the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The remaining seven objects clearly contain a broader secondary line component that is coincident with the H𝛼 line emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Whether this is a signature of massive stars driving strong winds or due to a background galaxy seen in projection is difficult to distinguish with these spectra alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' PHANGS–HST imaging reveals unresolved bright sources coincident with two of the nebulae, but no obvious background galaxy or bright stellar source at the other positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Six objects that we identify as H ii regions result in inferred metal- licity measurements that are unusually low, 12+log(O/H) < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0, and in fact all appear to be spurious sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' On closer inspection, two are faint foreground stars that had not been identified, one corresponds to an object where the stellar continuum is poorly constrained and noise peaks are fit as emission lines, and the remaining three show secondary broadened Gaussian profiles that bias the H𝛼 fit and skew the resulting line ratio diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Many of these exhibit S/N∼10 in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A comparison of MUSE and HST H𝛼 imaging in a star-forming complex of NGC 1672 (Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' While it is clear that MUSE does separate distinct complexes, the high ∼8 pc HST resolution reveals complicated H𝛼 morphologies including clustered compact regions which cannot be distinguished from neighbouring brighter regions at the 70 pc MUSE resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' fainter lines ([S ii], [O iii]), suggesting more stringent S/N cuts would be effective in excluding spurious sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Finally, our observation of NGC 1672 happened to occur only shortly after a supernova event in the galaxy, AT2017gax, and as a result is poorly fit in our catalogue and has ID #429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 Impact of Resolution Given that star formation is often observed to be clustered (that is, star formation occurs in high gas density regions like spiral arms), one limiting factor in constructing our nebular catalogue is the phys- ical resolution of our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In Table 1 we list the physical resolution within each galaxy, which reflects both the seeing condi- tions during the observation and the distance to the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Values range from 23–104 pc, sufficient to identify star forming regions that are isolated but likely insufficient to separate clustered star-forming complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In comparison with HST H𝛼 narrowband imaging (Fig- ure 16), it is apparent that objects identified as a single source in MUSE can be comprised of several neighbouring complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We are not able to fully account for this effect without additional HST H𝛼 imaging, which is currently only available for a handful of our targets (NGC 628, NGC 1672, NGC 3351) but will be available for the full sample with an upcoming PHANGS HST narrow band survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The effect of blending of objects within this catalogue is discussed in Santoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2022) in relation to its impact on the H ii region luminosity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Those authors estimate the mean separation between H ii regions per galaxy and find that it shows only a modest correlation with changes in the H ii region luminosity function slope and cannot be responsible for differences in luminosity function slope that are observed between galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Another location where blending becomes a clear issue is in the centres of galaxies hosting starbursting rings, where the extreme spa- tial concentration of bright H ii regions make decomposition chal- lenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Region identification is further complicated by the increased levels of diffuse ionized gas, associated with the high stellar den- sity contributing an additional ionization component (see Section MNRAS 000, 1–27 (2022) MUSE Nebulae MUSE HST 500pc Ha Ha 500pcPHANGS Nebular Catalogue 23 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Belfiore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Three galaxies (Figure 17) host particu- larly bright nuclear star-forming rings (NGC 1672, NGC 3351, NGC 4321), and we caution against over-interpretation of these integrated nebular fluxes without more careful spatial decomposition or de- blending of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' While the HST comparisons and H ii region luminosity functions suggest blending of nebulae is not a huge issue (excluding starburst- ing rings), even if it is occurring, the resolutions we achieve with MUSE are generally insufficient for accurate determination of H ii region sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This has lead us to exclude any size determination from our catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This issue was been highlighted in comparisons of MUSE H ii regions with HST narrow band H𝛼 imaging (Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022), where it is apparent that at ∼70 pc scales MUSE is unable to provide robust quantification of sizes or morphologies (Hannon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In our catalogue, we therefore provide the number of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2′′× 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2′′pixels associated with the spatial mask of each H ii re- gion, as a way of distinguishing the larger objects and derive surface brightnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 Diffuse Ionized Gas A prominent feature in our deep emission line maps is the pervasive diffuse ionized gas (DIG) component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As was shown in Zurita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2000) and more recently revisited for the PHANGS–MUSE galaxies (Belfiore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022), the bulk of this emission is spatially correlated with H ii region locations, and can be explained well by a model where ionizing photons leak from their H ii regions and propagate out to ∼kpc scales in the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' However, with an additional consideration of the emission line ratios in the gas, it is clear that an additional contribution from hot low-mass evolved stars (HOLMES) is required (Belfiore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Systematic differences in line ratios observed in H ii regions com- pared to the DIG were first established in Milky Way observations (Haffner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2009), and have the potential to significantly bias line flux measurements particularly for [S ii] and [N ii] which are emitted strongly in the DIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' However, the irregular spatial distribution and low-surface brightness of the DIG makes it difficult to model and sub- tract this component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In our construction of the nebular catalogues, we have not applied corrections for the DIG along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As most of our objects are bright, exhibiting median H𝛼 luminosities of 2×1037 ergs s−1 and H𝛼 surface brightnesses of 2×1039 ergs s−1 kpc−2, we expect that the DIG should not strongly impact our line ratios or derived quantities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' dust attenuation, metallicity, ion- ization parameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' However, we still caution that physical quantities interpreted from integrated line fluxes may still be impacted, espe- cially for H ii regions with low surface brightnesses or in crowded environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 7 CONCLUSIONS We present a new catalogue of 30,790 nebulae, selected morpholog- ically in H𝛼 images, that are fully contained within the PHANGS– MUSE coverage of 19 nearby star-forming spiral galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For all nebulae, we construct an integrated spectrum and measure integrated line fluxes and line kinematics for strong lines across the optical (4800–9300Å) spectrum (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We calculate derived properties, including dust attenuation via the Balmer decrement, characterise their galactic environments (centre, bar, spiral arm, interarm, disk), and classify objects based on their emission line (BPT) diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' For the 23,244 nebulae that we classify as H ii regions (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2), we calculate the gas-phase metallicity and ionization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A full list of properties characterised in our catalogue is included in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We examine the radial gradients of the nebula physical properties within galaxies, finding that the metallicity gradients demonstrate the most pronounced trends (Table 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We also look in more detail at the residual variation in metallicity (Δ(O/H)), after fitting and sub- tracting the linear radial trends in log(O/H), and quantify the global scatter in this residual metallicity within each galaxy (𝜎(O/H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As was previously shown using a sub-sample of our galaxies (Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2019), we recover small scatters with 𝜎(𝑂/𝐻) =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='03 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='07 dex, and show that this scatter correlates weakly with the global star formation rate and correlates strongly with both ionized and molecu- lar gas velocity dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Local metallicity variations show further support for these trends with the turbulent condition of the ISM, and we find that enriched regions preferentially show lower H𝛼 velocity dispersion and higher dust attenuation, reflecting a correspondence with calmer pockets of the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' These correlations between the metallicity variations (relative to the radial gradients) and the turbu- lent state of the ISM (in both the ionized and molecular material) demonstrate convincingly that the ISM dynamics play a critical role in regulating the mixing of metals across galaxy discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Correlations with galaxy environment are qualitatively apparent in some galax- ies, but systematic trends are less clear and more detailed dynamical modelling is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' With this catalogue, we do not yet include a full analysis of the temperature sensitive auroral emission lines ([N ii]5755, [S iii]6313, [O ii]7320,7330), which are contained within our wavelength range and detected in about a thousand H ii regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Careful fitting of these faint lines (∼1% of the intensity of H𝛼) is required to determine robust line fluxes, and will be the subject of future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In combination with the collisionally excited strong-lines, we aim to derive electron temperatures and simultaneously (using the [S ii] density diagnostic) determine electron densities for a sub-sample of our catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This catalogue of the young, ionized nebulae represents a key pa- rameter when developing a model of the baryon cycle within galaxies on resolved (<100 pc) scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Within the PHANGS collaboration, we aim to quantify key stages in this process by characterising with PHANGS–ALMA the molecular gas (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2020) and individ- ual giant molecular clouds (GMCs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Rosolowsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' in prep), and with PHANGS–HST the individual star clusters and stellar associations (Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Larson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Comparison of the H ii region and GMC distributions reveal rela- tively little spatial overlap (Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2018), indicative of short cloud disruption timescales (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Chevance et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Ongoing work links individual H ii regions with their parent GMC, to determine if stellar feedback has a measurable impact on molec- ular cloud properties (Zakardjian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' in prep), and searches for super-bubble morphologies in the molecular gas, to quantify feed- back energetics (Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Cross-matching our H ii re- gion catalogue with young stellar associations provides quantitative constraints on the mass and age of the stars powering these ionized nebulae, uniquely enabling us to constrain the evolutionary sequence and link the ionizing photon budget to the ionization state of the gas (Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2022, Scheuermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' submitted, Egorov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In the near future, these results will be further complemented by PHANGS–JWST maps of the earliest embedded phases of star formation and constraints on the dust chemistry, further completing our view of the complete star-formation cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' MNRAS 000, 1–27 (2022) 24 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Galaxy centres from NGC 1672, NGC 3351 and NGC 4321, which host nuclear star-forming rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' H𝛼 emission (greyscale) shows clustered and clumpy star-forming regions, which are only moderately well separated into individual H ii regions (red contours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Region sizes in this particular environment are also significantly overestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This work was carried out as part of the PHANGS collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Based on observations collected at the European Southern Ob- servatory under ESO programmes 1100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='B-0651, 095.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='C-0473, and 094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='C-0623 (PHANGS–MUSE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' PI Schinnerer), as well as 094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='B- 0321 (MAGNUM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' PI Marconi), 099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='B-0242, 0100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='B-0116, 098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='B- 0551 (MAD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' PI Carollo) and 097.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='B-0640 (TIMER;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' PI Gadotti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' KK, FS, and OE gratefully acknowledge funding from the Ger- man Research Foundation (DFG) in the form of an Emmy Noether Research Group (grant number KR4598/2-1, PI Kreckel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' KK, EJW and SCOG acknowledge support from the Deutsche Forschungs- gemeinschaft (DFG, German Research Foundation) – Project-ID 138713538 – SFB 881 (“The Milky Way System”, subprojects B1, B2, B8 and P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' SCOG also acknowledges funding from the European Research Council via the ERC Synergy Grant “ECOGAL – Under- standing our Galactic ecosystem: From the disk of the Milky Way to the formation sites of stars and planets” (project ID 855130) and from the Heidelberg Cluster of Excellence (EXC 2181 - 390900948) “STRUCTURES: A unifying approach to emergent phenomena in the physical world, mathematics, and complex data”, funded by the German Excellence Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' ATB and FB would like to acknowl- edge funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation pro- gramme (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='726384/Empire).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' FS, ES, and TGW acknowledge funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 694343).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' acknowledge sup- port from ANID Basal projects ACE210002 and FB210003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' PSB acknowledge support from the project project PID2019-107427-GB- 31 funded by the MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='13039/50110001103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California In- stitute of Technology, under contract with NASA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' It also made use of a number of python packages, namely the main astropy package (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2013, 2018), numpy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2020) and matplotlib (Hunter 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' DATA AVAILABILITY The MUSE data underlying this work are presented in Emsellem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2022) and are available at the ESO archive 5 and CADC archive 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The catalogue of all identified nebulae, along with fits files containing masks of the individual nebulae locations, are published along with this paper and available in the online supplementary material of the journal as well as at the CADC archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' REFERENCES Abazajian K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2009, ApJS, 182, 543 Allen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Groves B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Dopita M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Sutherland R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Kewley L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2008, ApJS, 178, 20 Alvarez-Hurtado P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Barrera-Ballesteros J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Sánchez S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Colombo D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', López-Sánchez A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Aquino-Ortíz E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2022, ApJ, 929, 47 Anand G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2021, MNRAS, 501, 3621 Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2013, A&A, 558, A33 Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2018, AJ, 156, 123 Bacon R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2010, in Ground-based and Airborne Instrumentation for Astronomy III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 773508, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='856027 Bacon R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2017, A&A, 608, A1 Baldwin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Phillips M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Terlevich R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 1981, PASP, 93, 5 Barnes A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2021, MNRAS, 508, 5362 Barnes A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='05679 Barrera-Ballesteros J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2016, MNRAS, 463, 2513 Barrera-Ballesteros J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2018, ApJ, 852, 74 Barrera-Ballesteros J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='07058 Belfiore F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2017, MNRAS, 469, 151 Belfiore F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2022, A&A, 659, A26 Berg D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Pogge R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Skillman E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Croxall K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Moustakas J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Rogers N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Sun J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2020, ApJ, 893, 96 Blanc G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2013, AJ, 145, 138 Boardman N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Zasowski G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Newman J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Sanchez S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Schaefer A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Lian J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Bizyaev D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Drory N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2021, MNRAS, 501, 948 Bothwell M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Maiolino R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Kennicutt R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Cresci G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Mannucci F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Marconi A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Cicone C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2013, MNRAS, 433, 1425 Bundy K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2015, ApJ, 798, 7 Calzetti D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2001, PASP, 113, 1449 Calzetti D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Kinney A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Storchi-Bergmann T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 1994, ApJ, 429, 582 5 https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='org/scienceportal/home?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='data_collection=PHANGS 6 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='canfar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='net/storage/vault/list/phangs/RELEASES/PHANGS- MUSE MNRAS 000, 1–27 (2022) NGC1672 NGC3351 NGC4321 15°49\'35" 00 11°42\'25" 59°14\'40" 30\' 20" 25" (ICRS) 15" (ICRS) 50 Dec ( 20" Dec C 10" C 55" 15" , 8 05 15\'00" 10″ 00" 1 kpc 05" h45m44 43s 42s 4is 10h43m58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5s 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0s 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='55 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='05 2h22m55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='05 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='09 RA (ICRS) RA (ICRS) RA (ICRS)PHANGS Nebular Catalogue 25 Cappellari M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2017, MNRAS, 466, 798 Cappellari M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Copin Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2003, MNRAS, 342, 345 Cardelli J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Clayton G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Mathis J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 1989, ApJ, 345, 245 Chabrier G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2003, PASP, 115, 763 Chevance M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2022, MNRAS, 509, 272 Ciardullo R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Feldmeier J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Jacoby G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Kuzio de Naray R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Laychak M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Durrell P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2002, ApJ, 577, 31 Croom S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2021, MNRAS, 505, 991 Croxall K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', van Zee L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Lee H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Skillman E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Lee J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Côté S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Kennicutt Robert C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Miller B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2009, ApJ, 705, 723 Curti M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Mannucci F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Cresci G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Maiolino R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2020, MNRAS, 491, 944 De Marchi G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2016, MNRAS, 455, 4373 Della Bruna L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2020, A&A, 635, A134 Diaz A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Terlevich E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Vilchez J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Pagel B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Edmunds M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 1991, MNRAS, 253, 245 Dopita M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Rich J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Vogt F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Kewley L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Ho I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Basurah H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Ali A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Amer M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2014, Ap&SS, 350, 741 Dopita M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Kewley L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Sutherland R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Nicholls D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2016, Ap&SS, 361, 61 Draine B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2011, Physics of the Interstellar and Intergalactic Medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Princeton University Press Driver S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2009, Astronomy and Geophysics, 50, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='12 Egorov O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Lozinskaya T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Moiseev A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Smirnov-Pinchukov G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2014, MNRAS, 444, 376 Egorov O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Lozinskaya T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Moiseev A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Shchekinov Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2017, MNRAS, 464, 1833 Emsellem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2022, A&A, 659, A191 Epinat B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Amram P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Marcelin M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2008, MNRAS, 390, 466 Erroz-Ferrer S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2019, MNRAS, 484, 5009 Espinosa-Ponce C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Sánchez S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Morisset C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Barrera-Ballesteros J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Galbany L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', García-Benito R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Lacerda E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Mast D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2020, MNRAS, 494, 1622 Espinosa-Ponce C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Sánchez S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Morisset C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Barrera-Ballesteros J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Galbany L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', García-Benito R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Lacerda E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Mast D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2022, MNRAS, 512, 3436 Gadotti D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2019, MNRAS, 482, 506 Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2018, A&A, 616, A1 Genzel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2015, ApJ, 800, 20 Grasha K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2022, ApJ, 929, 118 Haffner L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2009, Reviews of Modern Physics, 81, 969 Hannon S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2022, MNRAS, 512, 1294 Harris C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2020, Nature, 585, 357 Ho I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2017, ApJ, 846, 39 Ho I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2018, A&A, 618, A64 Hunter J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2007, Computing in Science & Engineering, 9, 90 James P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Bretherton C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Knapen J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2009, A&A, 501, 207 Ji X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Yan R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2022, A&A, 659, A112 Kauffmann G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2003, MNRAS, 346, 1055 Kennicutt Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 1989, ApJ, 344, 685 Kewley L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Dopita M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2002, ApJS, 142, 35 Kewley L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Ellison S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2008, ApJ, 681, 1183 Kewley L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Heisler C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Dopita M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Lumsden S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2001, ApJS, 132, 37 Kewley L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Groves B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Kauffmann G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Heckman T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2006, MNRAS, 372, 961 Kewley L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Nicholls D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Sutherland R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2019, ARA&A, 57, 511 Kim J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2021, MNRAS, 504, 487 Kobulnicky H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Kewley L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2004, ApJ, 617, 240 Kreckel K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2013, ApJ, 771, 62 Kreckel K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Blanc G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Schinnerer E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Groves B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Adamo A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Hughes A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Meidt S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2016, ApJ, 827, 103 Kreckel K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2018, ApJ, 863, L21 Kreckel K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2019, ApJ, 887, 80 Kreckel K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2020, MNRAS, 499, 193 Krumholz M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Ting Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2018, MNRAS, 475, 2236 Lang P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2020, ApJ, 897, 122 Lara-López M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2010, A&A, 521, L53 Law D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2021, ApJ, 915, 35 Le Fevre O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2003, in Guhathakurta P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 4834, Discov- eries and Research Prospects from 6- to 10-Meter-Class Telescopes II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' pp 173–182, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='457547 Lee J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2022, ApJS, 258, 10 Leroy A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2019, ApJS, 244, 24 Leroy A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2021, ApJS, 257, 43 Long K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2010, ApJS, 187, 495 Long K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Blair W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Winkler P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Della Bruna L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Adamo A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', McLeod A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Amram P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2022, ApJ, 929, 144 Lugo-Aranda A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Sánchez S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Espinosa-Ponce C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', López-Cobá C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Gal- bany L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Barrera-Ballesteros J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Sánchez-Menguiano L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Anderson J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2022, RAS Techniques and Instruments, 1, 3 Maiolino R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Mannucci F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2019, A&ARv, 27, 3 Makarov D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Prugniel P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Terekhova N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Courtois H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Vauglin I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2014, A&A, 570, A13 Mannucci F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Cresci G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Maiolino R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Marconi A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Gnerucci A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2010, MN- RAS, 408, 2115 Marino R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2013, A&A, 559, A114 Massey P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Puls J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Pauldrach A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Bresolin F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Kudritzki R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Simon T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2005, ApJ, 627, 477 Medina S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Arthur S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Henney W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Mellema G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Gazol A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2014, MNRAS, 445, 1797 Metha B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Trenti M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Chu T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2021, MNRAS, 508, 489 Mingozzi M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2020, A&A, 636, A42 Moiseev A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Lozinskaya T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2012, MNRAS, 423, 1831 Moiseev A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Tikhonov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Klypin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2015, MNRAS, 449, 3568 Moustakas J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Kennicutt Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Tremonti C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Dale D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Smith J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Calzetti D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2010, ApJS, 190, 233 O’Donnell J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 1994, ApJ, 422, 158 Oey M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2007, ApJ, 661, 801 Osterbrock D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Ferland G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2006, Astrophysics of gaseous nebulae and active galactic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' University Science Books Parker Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2022, Frontiers in Astronomy and Space Sciences, 9, 895287 Peeples M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Pogge R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Stanek K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2008, ApJ, 685, 904 Peeples M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Pogge R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Stanek K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2009, ApJ, 695, 259 Peimbert M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Peimbert A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Delgado-Inglada G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2017, PASP, 129, 082001 Pellegrini E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Rahner D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Reissl S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Glover S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Klessen R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Rousseau-Nepton L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Herrera-Camus R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2020, MNRAS, 496, 339 Pessa I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2021, A&A, 650, A134 Pettini M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Pagel B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2004, MNRAS, 348, L59 Pietrinferni A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Cassisi S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Salaris M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Castelli F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2004, ApJ, 612, 168 Pilyugin L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Grebel E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2016, MNRAS, 457, 3678 Pilyugin L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Thuan T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2005, ApJ, 631, 231 Pilyugin L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Grebel E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Kniazev A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2014, AJ, 147, 131 Poetrodjojo H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2018, MNRAS, 479, 5235 Poetrodjojo H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', D’Agostino J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Groves B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Kewley L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Ho I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Rich J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Madore B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Seibert M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2019, MNRAS, 487, 79 Prescott M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2007, ApJ, 668, 182 Querejeta M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2021, A&A, 656, A133 Relaño M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Beckman J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Zurita A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Rozas M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Giammanco C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2005, A&A, 431, 235 Reynolds R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 1990, ApJ, 349, L17 Rosolowsky E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2021, MNRAS, 502, 1218 Rousseau-Nepton L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2019, MNRAS, 489, 5530 Salo H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2015, ApJS, 219, 4 Sánchez S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2020, ARA&A, 58, 99 Sánchez-Menguiano L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2016, ApJ, 830, L40 Sánchez-Menguiano L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2018, A&A, 609, A119 Sánchez-Menguiano L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Sánchez Almeida J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Muñoz-Tuñón C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Sánchez S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Filho M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Hwang H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Drory N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2019, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='03930 Sánchez-Menguiano L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Sánchez S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Pérez I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Ruiz-Lara T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Galbany L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Anderson J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Kuncarayakti H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2020, MNRAS, 492, 4149 Sánchez S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2012, A&A, 538, A8 Sánchez S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2014, A&A, 563, A49 Santoro F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2022, A&A, 658, A188 Scheuermann F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2022, MNRAS, 511, 6087 Schlafly E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Finkbeiner D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2011, ApJ, 737, 103 MNRAS 000, 1–27 (2022) 26 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Sil’chenko O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Moiseev A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Egorov O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2019, ApJS, 244, 6 Smith R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Points S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Chu Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Winkler P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Aguilera C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Leiton R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', MCELS Team 2005, in American Astronomical Society Meeting Abstracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='07 Spitoni E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Cescutti G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Minchev I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Matteucci F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Silva Aguirre V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Martig M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Bono G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Chiappini C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2019, A&A, 628, A38 Stuber S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2021, A&A, 653, A172 Sun J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2020, ApJ, 892, 148 Tayal S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Zatsarinny O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Sossah A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2019, ApJS, 242, 9 Thilker D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Braun R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Walterbos R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2000, AJ, 120, 3070 Tremonti C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2004, ApJ, 613, 898 Turner J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2021, MNRAS, 502, 1366 Vacca W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 1994, ApJ, 421, 140 Vazdekis A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Koleva M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Ricciardelli E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Röck B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Falcón-Barroso J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2016, MNRAS, 463, 3409 Veilleux S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2002, in Rosada M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Binette L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Arias L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', eds, Astronomical Society of the Pacific Conference Series Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 282, Galaxies: the Third Dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 293 Venturi G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2018, A&A, 619, A74 Véron-Cetty M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Véron P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2010, A&A, 518, A10 Vogt F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Pérez E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Dopita M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Verdes-Montenegro L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Borthakur S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2017, A&A, 601, A61 Westfall K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2019, AJ, 158, 231 Williams J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', de Geus E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Blitz L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 1994, ApJ, 428, 693 Williams T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2021, AJ, 161, 185 Williams T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2022, MNRAS, 509, 1303 Zhang K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2017, MNRAS, 466, 3217 Zurita A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Rozas M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Beckman J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2000, A&A, 363, 9 Zurita A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Florido E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Bresolin F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Pérez I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Pérez-Montero E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2021, MNRAS, 500, 2380 de Vaucouleurs G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', de Vaucouleurs A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Corwin Herold G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Buta R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Paturel G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', Fouque P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 1991, Third Reference Catalogue of Bright Galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Springer Science+Business Media de los Reyes M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=', 2015, AJ, 149, 79 MNRAS 000, 1–27 (2022) PHANGS Nebular Catalogue 27 Table C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Representative values of key physical parameters at 1 reff Galaxy A𝑉 log EW(H𝛼) log L(H𝛼) log U [mag] [Å] [erg s−1] NGC0628 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 NGC1087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 NGC1300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='97 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 NGC1365 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='69 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 NGC1385 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 NGC1433 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='69 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 NGC1512 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='71 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 NGC1566 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 NGC1672 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 NGC2835 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 NGC3351 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 NGC3627 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 NGC4254 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 NGC4303 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 NGC4321 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='3 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 NGC4535 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 NGC5068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='9 NGC7496 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='74 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 IC5332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 APPENDIX A: IMAGE ATLAS To provide a visual impression of the size and distribution of our identified nebulae, we provide images (Figures A1 - A16) of the H𝛼 emission in all 19 galaxies with the nebular catalogue masks overplotted with the colours indicating the intrinsic (dust corrected) H𝛼 luminosity of each nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' APPENDIX B: BPT DIAGRAMS FOR ALL GALAXIES BPT (Baldwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 1981) diagnostics are commonly used to distin- guish the ionization source both in individual ionized nebulae and across integrated galaxy spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' We demonstrate the range of line ratios observed within our complete nebular catalogue in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 and Figure 4, and include here the breakdown for each individ- ual galaxy (Figures B1-B3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In all diagrams, regions consistent with photoionization across all three diagnostics are marked in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The complete nebular catalogue is shown in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Overlaid are the Kauff- mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2003) diagnostic curve (dashed line) and Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2006) diagnostic curves (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' APPENDIX C: RADIAL GRADIENTS FOR ALL GALAXIES To give a more comprehensive look at the radial variations in nebu- lar properties, we show the dust extinction derived from the Balmer decrement (A𝑉 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure C1), the extinction corrected H𝛼 luminosity (Figure C2), the ionization parameter (U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure C3), and the equiv- alent width of H𝛼 (EW H𝛼;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure C4) as a function of effective radius (reff) for all galaxies in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' In Table C1 we tabulate a representative value for each based on linear radial fits at 1 reff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' APPENDIX D: RADIAL METALLICITY GRADIENTS FOR ALTERNATE PRESCRIPTIONS A zoomed in view of the radial metallicity gradients for each galaxy is shown for metallicities calculated using the Scal-PG16 (Figure D1), O3N2-M13 (Figure D2), and N2S2-D16 (Figure D3) prescriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Table D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Linear fit parameters for radial gradients in 12+log(O/H) using the O3N2 (Marino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2013) prescription, see Figure D3 Galaxy intercept slope [dex/r𝑒 𝑓 𝑓 ] value at r𝑒 𝑓 𝑓 𝜎(O/H) IC5332 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='528 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='043 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='000 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='485 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='063 NGC0628 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='605 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='041 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='002 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='564 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='046 NGC1087 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='525 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='032 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='007 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='494 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='041 NGC1300 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='649 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='071 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='000 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='579 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='049 NGC1365 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='671 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='186 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='003 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='486 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='054 NGC1385 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='495 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='001 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='009 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='494 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='039 NGC1433 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='605 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='005 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='001 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='051 NGC1512 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='584 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='006 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='002 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='590 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='042 NGC1566 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='607 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='023 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='003 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='583 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='040 NGC1672 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='568 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='009 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='000 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='559 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='045 NGC2835 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='581 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='093 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='001 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='488 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='063 NGC3351 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='563 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='038 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='001 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='601 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='055 NGC3627 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='581 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='015 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='003 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='596 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='036 NGC4254 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='623 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='030 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='002 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='594 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='039 NGC4303 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='623 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='029 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='004 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='594 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='039 NGC4321 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='580 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='007 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='002 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='587 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='040 NGC4535 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='611 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='011 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='003 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='599 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='050 NGC5068 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='547 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='066 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='001 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='481 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='062 NGC7496 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='624 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='081 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='002 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='543 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='050 Table D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Linear fit parameters for radial gradients in 12+log(O/H) using the N2S2 (Dopita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 2016) prescription, see Figure D3 Galaxy intercept slope [dex/r𝑒 𝑓 𝑓 ] value at r𝑒 𝑓 𝑓 𝜎(O/H) IC5332 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='575 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='263 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='003 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='312 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='120 NGC0628 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='740 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='119 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='002 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='622 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='103 NGC1087 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='574 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='110 ± 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='101 Corresponding linear radial fits are provided for the Scal-PG16 (Table 9), O3N2-M13 (Table D1), and N2S2-D16 (Table D2) prescriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' MNRAS 000, 1–27 (2022) 28 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The spatial distribution of nebulae in the galaxy IC5332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The background greyscale image shows the H𝛼 emission in log scale, and the colour of the nebulae indicates their intrinsic (dust corrected) H𝛼 luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' MNRAS 000, 1–27 (2022) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 36°05\'00" 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 + 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 30" 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 Dec .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='.00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='90 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 30" 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 07\'00" 1 kpc 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 30" 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 23h34m32s 30s 28s 26s 24s R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='PHANGS Nebular Catalogue 29 Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As in figure A1 but for NGC 628.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=" MNRAS 000, 1–27 (2022) 15°49' 39 48' 38 47' Dec 37 46' 36 1 kpc 45' 35 1h36m50s 45s 40s 35s R." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='30 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As in figure A1 but for NGC 1087 MNRAS 000, 1–27 (2022) 40 0°28\'30" 29\'00" 39 30" log1o(L(Hα) [erg s-1] 38 Dec 30\'00" 37 30" 36 31\'00"- I kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 35 2h46m28s 26s 24s 22s R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='PHANGS Nebular Catalogue 31 Figure A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As in figure A1 but for NGC 1300 MNRAS 000, 1–27 (2022) 39 19°23\'30" 24\'00" 30" Dec 25\'00" 37 30"- 1 kpc 26\'00" 36 3h19m50s 45s 40s 35s R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 3532 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As in figure A1 but for NGC 1365 MNRAS 000, 1–27 (2022) 41 36°07\'00" 40 30" log1o(L(Hα) [erg s-1] 08\'00" 39 Dec 30" 38 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='.00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='60 37 30″- 1 kpc 36 3h33m48s 42s 36s 30s R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 35PHANGS Nebular Catalogue 33 Figure A6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As in figure A1 but for NGC 1385 MNRAS 000, 1–27 (2022) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 24°29\'00".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 30" 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 Dec 30\'00" 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 30" 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 31\'00"- 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1 kpc 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 30" 3h37m32s 30s 28s 26s R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='34 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure A7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As in figure A1 but for NGC 1512 MNRAS 000, 1–27 (2022) 43°19\'30" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 39 20\'00" 30" 38 Dec 21\'00" 37 30" 36 22\'00" 1 kpc 35 4h04m00s 03m56s 52s 48s R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='PHANGS Nebular Catalogue 35 Figure A8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=" As in figure A1 but for NGC 1566 MNRAS 000, 1–27 (2022) 54°54' 40 39 55' 38 Dec 56' 37 57' 36 1 kpc 35 4h20m10s 05s 00s 19m55s R." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='36 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure A9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As in figure A1 but for NGC 1672 MNRAS 000, 1–27 (2022) 40 59°14\'00"- 39 30" 38 Dec 15\'00" 37 30" 1 kpc 36 4h45m54s 48s 42s 36s 30s R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 35PHANGS Nebular Catalogue 37 Figure A10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As in figure A1 but for NGC 2835 MNRAS 000, 1–27 (2022) 22°20\'00" 39 30" 38 log1o(L(Hα)) [erg s-1] 21\'00" Dec 30" 37 22\'00" 36 1 kpc 30" 35 gh17m57s 54s 51s 48s R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='38 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure A11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As in figure A1 but for NGC 3351 MNRAS 000, 1–27 (2022) 40 11°43\'30" 39 1 00" 38 42\'30" Dec 00" ": 37 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 41\'30" + 36 1 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 00" 35 10h44m03s 0os 43m57s 54s R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='PHANGS Nebular Catalogue 39 Figure A12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=" As in figure A1 but for NGC 3627 MNRAS 000, 1–27 (2022) 40 13°00' 39 log1o(L(Hα) [erg s-1] 38 12°59' Dec 37 58' 36 1 kpc 57' 35 11h20m18s 16s 14s 12s R." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='40 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure A13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=" As in figure A1 but for NGC 4254 MNRAS 000, 1–27 (2022) 14°27' 39." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content="5 26' 38." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 Dec 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content="5 25' 37." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content="5 24' kp 36." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 12h18m57s 54s 51s 48s 45s R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='PHANGS Nebular Catalogue 41 Figure A14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As in figure A1 but for NGC 4535 MNRAS 000, 1–27 (2022) 8°13\'00" 40 12\'30" 39 00" log1o(L(Hα) [erg s-1] +++ 38 Dec 11\'30" 37 00" 36 10\'30" 1 kpc 35 12h34m24s 22s 20s 18s R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='42 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure A15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As in figure A1 but for NGC 5068 MNRAS 000, 1–27 (2022) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content="5 21°01' 38." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content="5 02' Dec 37." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content="5 03' 36." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 I kpc 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content="5 04' 13h19m00s 18m56s 52s 48s R." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='PHANGS Nebular Catalogue 43 Figure A16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' As in figure A1 but for NGC 7496 MNRAS 000, 1–27 (2022) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 43°24\'30" 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 : 25\'00" 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' log1o(L(Hα)) [erg s-1] 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 30" Dec 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 26\'00" 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 30" 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' kpc 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 27\'00" 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 23h09m52s 50s 48s 46s 44s R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='44 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' [O iii]/H𝛽 as a function of [N ii]/H𝛼 for each galaxy individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Galaxies are ordered by stellar mass from low (top left) to high (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The full nebular catalogue is shown in grey, H ii regions identified as consistent with photoionization across all three BPT diagnostics (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 and Table 5) are marked in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Overlaid are the Kauffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2003) diagnostic curve (dashed line) and Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2006) diagnostic curves (solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' MNRAS 000, 1–27 (2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 log [SI]/Ha46 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' [O iii]/H𝛽 as a function of [O i]/H𝛼 for each galaxy individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Galaxies are ordered by stellar mass from low (top left) to high (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' The full nebular catalogue is shown in grey, H ii regions identified as consistent with photoionization across all three BPT diagnostics (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='2 and Table 5) are marked in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Overlaid is the Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' (2006) diagnostic curve (solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' MNRAS 000, 1–27 (2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 NGC5068 IC5332 NGC1087 NGC1385 NGC7496 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 qH/[IIO] f 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 NGC2835 NGC4254 NGC4303 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 601 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 NG NGC1300 NGC1512 NGC1672 NGC4321 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 NGC1566 NGC36 NGC1365 log [O1]/HaPHANGS Nebular Catalogue 47 Figure C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' A𝑉 radial gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' MNRAS 000, 1–27 (2022) NGC5068 IC5332 NGC1087 NGC1385 NGC7496 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 NGC2835 NGC0628 NGC3351 NGE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 [mag] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 Av NGC1300 NGC1512 NGC1672 NGC4321 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 NGC1566 NGC1433 NGC1365 log(M) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0 4 4 0 R / reff48 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' L(H𝛼) radial gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' MNRAS 000, 1–27 (2022) 41 NGC5068 IC5332 NGC1087 NGC1385 NGC7496 40 39 38 37 36 35 41 NGC2835 NGC0628 NGC3351 NGC4254 NGC4303 40 39 [erg/s] 38 37 36 L(Hα) 35 41 NGC4535 NGC1300 NGC1512 NGC1672 NGC4321 60l 40 39 38 37 36 35 41 NGC1566 NGC3627 NGC1433 NGC1365 log(M) 40 9 10 11 39 38 37 36 35 2 3 4 4 0 4 R / reffPHANGS Nebular Catalogue 49 Figure C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Ionization parameter (U) radial gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' MNRAS 000, 1–27 (2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 NGC5068 NGC08 NGC1385 NGC7496 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 U g 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 NGC1566 log(M) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 10 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 4 0 R / reff50 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' EW(Ha) radial gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' MNRAS 000, 1–27 (2022) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 NGC5068 IC5332 NGC1087 NGC1385 NGC7496 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 NGC2835 NGC0628 NGC3351 NGC4254 NGC4303 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 EW(Hα)raw 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 NGC4535 NGC1300 NGC1512 NGC1 NGC4321 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 g 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 NGC1566 NGC3627 NGC1433 NGC1365 log(M) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 LU 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 4 0 R / reffPHANGS Nebular Catalogue 51 Figure D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Metallicity radial gradients, using the Scal-PG16 prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' All galaxies show the same dynamic range in 12+log(O/H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' MNRAS 000, 1–27 (2022) IC5332 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='7 NGC0628 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='6 NGC1087 8.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content='0 R / reff52 Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Figure D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' Metallicity radial gradients, using the O3N2-M13 prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfUQd7/content/2301.03811v1.pdf'} +page_content=' All galaxies show the same dynamic range in 12+log(O/H).' metadata={'source': 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+Amritapuri, India +malavikasnair@am.students.amrita.edu +Pramod P Nair +Department of Mathematics +Amrita Vishwa Vidyapeetham +Amritapuri, India +pramodpn@am.amrita.edu +Anil Lal S +Department of Mechanical Engineering +Amrita Vishwa Vidyapeetham +Amritapuri, India +anillals@am.amrita.edu +Abstract—Deep learning techniques with neural networks have +been used effectively in computational fluid dynamics (CFD) to +obtain solutions to nonlinear differential equations. This paper +presents a physics-informed neural network (PINN) approach to +solve the Blasius function. This method eliminates the process of +changing the non-linear differential equation to an initial value +problem. Also, it tackles the convergence issue arising in the +conventional series solution. It is seen that this method produces +results that are at par with the numerical and conventional +methods. The solution is extended to the negative axis to show +that PINNs capture the singularity of the function at η = −5.69. +Index Terms—Blasius equation, physics-informed neural net- +works, automatic differentiation, computational fluid dynamics, +boundary layer flow, singularity. +I. INTRODUCTION +Machine learning helps learn patterns from the data to make +predictions. The three basic requirements for machine learning +are data, theory, and hardware to overcome computational +difficulties using better GPUs and new approaches. Computa- +tional science is an essential tool that we can use to incorporate +physical invariances into learning. For example, the laws that +govern the conservation of momentum, mass, and energy. To +quote Dr.Tinsley Oden - “Computational Science can analyze +past events and look into the future. It can explore the effects +of thousands of scenarios instead of actual experiments and +be used to study events beyond the reach of expanding the +boundaries of experimental science”. +Neural networks(NNs) are the most used Machine Learning +technique. It offers a set of robust tools for solving various +supervised and unsupervised problems in pattern recognition, +data processing, and non-linear control, which can be regarded +as complementary to the conventional approaches [1]. It can +be considered a functional approximator as it maps a set of +input variables to a set of output variables. A set of parameters +called weights manages this mapping. These weights are +updated to train the network that fetches the required output. +A polynomial can be viewed as a function that transforms +a single input variable into a single output variable. The +coefficients in the polynomial are comparable to the weights +in a neural network. Determining these coefficients helps in +evaluating the solution. +A. Physics Informed Neural Networks +Physics-informed neural networks (PINNs) play a huge role +where fewer data and the system’s conventional physics are +known to get the solutions to a differential equation. NN can +approximate the error to ground truth function, generalize to +unseen data and train the model. The contribution of PINNs is +to obtain a neural network that knows about the physics hidden +behind the equation and thus efficiently solves differential +equations. The earlier numerical methods for solving them are +finite difference, finite elements, spectral elements, and finite +volume. When working with PINNs, there is data efficiency +from a machine learning perspective since it is regularized +heavily with physics. The required derivatives can be calcu- +lated using automatic differentiation at the end of the network. +It then finds a loss function corresponding to the differential +equation and boundary conditions. +B. Boundary Layer Theory +The boundary layer of a flowing fluid is a thin layer near +a solid surface, and the flow near the solid surface is known +as the boundary layer flows. Ludwig Prandtl is credited for +developing the boundary layer theory. In 1904, he published a +paper titled ”On the Motion of a Fluid with Very Small Vis- +cosity” [2]. He laid out the mathematical foundation for flows +and condensed the two-dimensional Navier-Stokes equations +(NSE) into the boundary layer equations. This publication +made understanding fluid motion physics possible, which is +regarded as the beginning of contemporary fluid mechanics. +The introduction of the similarity variables and the trans- +formation of the PDEs into nonlinear ODEs in one coordinate +allowed the successful resolution of the problem. Still, specific +methods were needed to handle the unbounded boundary +arXiv:2301.00106v1 [cs.LG] 31 Dec 2022 + +conditions. The Blasius benchmark problem has been resolved +[3] using the trial function method put out by Lagaris [4] or +a hybrid approach [5]. The solution function is seen to have +a singularity on the negative real axis at approximately -5.69. +The Blasius equation is a single equation that solely models +the viscous boundary layer and is one of CFD’s fundamental +models. +In this paper, we propose solving the Blasius equation +using PINNs and comparing the solution we obtain with +the best-known solutions available in the literature. We shall +also extend our solution to the negative real axis to locate +the singularity existing for the function. In the next section +of the paper, we shall present a brief literature review on +the Blasius equation and PINNs. In Section III, we present +the methodology that has been adopted to solve the Blasius +Equation. Section IV discusses the results and analysis of the +proposed algorithm. The conclusions are discussed in the last +section. +II. LITERATURE REVIEW +A. Physics Informed Neural Networks +Neural networks can be used to attain solutions to ODEs +and PDEs by reducing them to an optimization problem +instead of numerically solving the equations. The application +of NNs in fluid mechanics began in the 1990s. Raissi and his +team [6], [7] developed a technique called Physics informed +neural network (PINN) where the loss function defined in the +corresponding NN is extracted from the physics behind the +PDE and related equations. In earlier years, Computational +fluid dynamics (CFD) had been a great relief in numerically +solving the compressible and incompressible Na¨ıve Stokes +equations. Minimizing the loss function is challenging in +PINNs, as it can be highly complex. Nevertheless, PINNs +have been proven to be more accurate than conventional CFD +techniques with lesser computations [8]. +Although numerical discretization of the Navier-Stokes +equations (NSE) has made significant progress in simulating +flow problems over the past 50 years, the present algorithms +cannot solve governed by high-parametrized NSE. Addition- +ally, it is expensive to solve inverse flow problems due to their +complexity, expensive formulations, and need for new algo- +rithms. PINNs are expanded to fractional PINNs (fPINNs) that +explore their convergence methodically to solve space-time +fractional advection-diffusion equations (fractional ADEs) [9]. +A hybrid method for building the residual in the loss function +that utilizes both automated differentiation for the integer- +order operators and numerical discretization for the fractional +operators is a novel component of the fPINNs. +Wide varieties of PINNs have been found in the literature +in recent times. can-PINNs [10] link derivative terms with +nearby support points, which generally apply to Taylor series +expansion-based numerical systems. Apart from demonstrating +good dispersion and dissipation characteristics, they are highly +trainable and require four to sixteen times fewer collocation +points than original PINNs. Auxiliary PINNs (A-PINN) is +a technique for solving forward and inverse problems of +non-linear integrodifferential equations [11]. ViscoelasticNet +is another PINN framework for stress discovery and model +selection [12]. PINNs can also be used to solve Reynolds- +averaged Navier-Stokes equations [13], full waveform seismic +inversions in 2D acoustic media, and wave propagation as +it seamlessly handles boundary conditions and physical con- +straints [14]. In addition to addressing ill-posed problems be- +yond the scope of conventional computing techniques, PINNs +can also close the discrepancy between computational and +experimental heat transfer [15]. +B. Blasius Equation +Blasius equation is a third order non-linear ordinary differ- +ential equation of the form f ′′′+ 1 +2ff ′′ = 0 with the boundary +conditions f(0) = 0, f ′(0) = 0, f ′(∞) = 1. It governs the +boundary layer flow over a semi-infinite flat plate. Suppose +that the u−velocity, the velocity parallel to the surface, is +much greater than the v−velocity, perpendicular to the surface, +and the changes in the perpendicular direction to the surface +are much greater than changes parallel to the surface. The +boundary layer equations consists of conservation of mass +(II.1), conservation of x-momentum (II.2), and conservation +of y-momentum. In a flat plate boundary layer, the pressure +gradient term appearing in the x-momentum equation becomes +zero (II.2). This leads to the hydrodynamic solution for the +flat plate boundary layer in a laminar flow called the Blasius +solution. +∂u +∂x + ∂v +∂y = 0 +(II.1) +u∂u +∂x + v ∂u +∂y = ν ∂2u +∂y2 +(II.2) +Blasius’ analysis focus on the laminar boundary layer +forming on a flat plate. The main aspect of the work is the +transformation of the PDE for a flat plate boundary layer +with zero pressure gradient into a single ordinary differential +equation(ODE) by considering the velocity components that +satisfy equation II.1. +u ≡ ∂ψ +∂y +v ≡ −∂ψ +∂x +(II.3) +The stream function ψ = U∞ +� +νx +U∞ f(η), is directly pro- +portional to the function f(η) called the Blasius function. +Here, U∞ is the free stream velocity, and η is a transformed +coordinate called the similarity parameter. Here, the velocity +components are proportional to the first derivative of f(η), +and the second and third derivatives of f are proportional to +the first and second derivatives of velocity. Substituting these +relations into the momentum equation (II.2), the final form of +the Blasius boundary layer equation f ′′′ + 1 +2ff ′′ = 0, for a +flat plate can be obtained. The first and second derivatives of +f(η) are given by f ′ = +u +U∞ (Non-dimensional velocity profile) +and f ′′ = +1 +U∞ +� +νx +U∞ +∂u +∂y (quantity related to shear stress). The +boundary conditions are set considering the laminar flow on + +a flat plate, the no-slip condition, and free-stream velocity +outside the boundary layer. Hence we have +f(0) = 0 +f ′(0) = 0 +f ′(∞) = 1 +(II.4) +Since the value of f ′′ at η = 0 is unknown, it cannot be +considered an initial value problem. +A power series solution to the boundary layer equation +of flow across a flat plate was proposed by Blasius [16]. +Schmidt and Beckmann [17] and Ostrach [18] conducted the +most important work on the topic, conducting theoretical and +experimental research on the free convection flow of air around +a vertical flat plate under the influence of gravity. Boyd solved +the equation using an analytical series solution technique [19]. +Nowadays, with the availability of computers, we can obtain +a numerical solution to this equation and calculate it with a +very high degree of accuracy. To solve the Blasius equation +numerically, a possible method is to use the shooting algorithm +to find what value would satisfy the boundary condition at +η = 0. The first step is to guess a value at the wall. Then +solving the ODE along the non-dimensional coordinate until +the first derivative of f stops changing. This checks if the +first derivative satisfies the given boundary condition; instead, +adjust the guessed value to decrease or increase if the first +derivative is higher or lower than one. The algorithm is +repeated until the boundary condition is finally satisfied to +reach the final solution. Howarth [20] found the solution using +a numerical method that accurately predicted the value of +f ′′(0) to be 0.332. +This differential equation is a direct representation of the +velocity profile inside the boundary layer. Once the solution +is obtained, the velocity components can be calculated using +these relations. Hence the boundary layer thickness can be +calculated located at the point where the velocity is 99% of +the free stream velocity. It was found that this occurs approx- +imately at η equal to 5. The displacement and momentum +thickness can also be estimated using the solution. The wall +shear stress based on the velocity gradient at the wall can be +estimated, and the friction and drag coefficients can be calcu- +lated. Note that the expression is only accounting for one side +of the plate. Comparing the exact solution with the approxi- +mation solution from the integral analysis is interesting. The +relations are close to each other. Indeed the integral analysis +is within 10% of the exact solution; unlike Blasius’ solution, +these values were obtained without complicated math. The +Blasius solution provides a self-similar solution meaning that +the solution is the same if the independent and dependent +variables of the governing equations are appropriately scaled. +This can be seen in comparing experimental results with the +Blasius solution. Different Reynolds numbers (Re) provide +the same profile when the variables on the two axes are +appropriately scaled. +The solution to the Blasius equation can be found in [21], +[22], where the accurate benchmark results of the Blasius +boundary layer problem using a leaping Taylors series that +converges for all real values. There have been different meth- +ods to solve the Blasius equation, such as the Topher transfor- +mation, which is executed using inverse transformation, [23]– +[25]. Runge-Kutta, incorporated with the shooting method, +finds the solution numerically, while Adomian Decomposition +Method [26] finds the solution analytically. +III. PROPOSED METHODOLOGY +The ODE representing the Blasius equation derived from the +PDE is considered in the present work. We follow a similar +methodology proposed for the general form of generalized +non-linear ODE by Raissi [6]. The workflow diagram of the +present method is shown in Fig.1. +Fig. 1. Workflow diagram to solve for f(η) +The input values η are discrete and are equally distributed +between the boundary points η0 and ηm. The beauty of PINNs +lies in loss function as it incorporates the boundary conditions +and the differential equation and thus includes the physics +information. The loss function is the total loss from given +ODE Lo, initial conditions Li, and boundary Lb conditions. +L = Lo + Lb + Li +(III.1) +Lo = +� +η +[ ˆf ′′′ + 1 +2 +ˆf ˆf ′′]2 +(III.2) +Li = [ ˆf(η0) − f(η0)]2 + [ ˆf ′(η0) − f(η0)]2 += [ ˆf(η0) − 0]2 + [ ˆf ′(η0) − 0]2 +(III.3) +Lb = [ ˆf(ηm) − f(ηm)]2 = [ ˆf(ηm) − 1]2 +(III.4) +From the definition of the loss function, it is clear that the total +loss will be zero if the function f(η) is exact. Here we update +the weights of the neural network in each iteration such that the +loss function is minimized. Supplying the independent input +values representing discrete spatial coordinates η ranging from +zero to ηm into the neural network is sufficient to solve the +ODE via PINNs. The neural network maps the input η to f(η), +which is the estimated solution to the stream function. Unlike +standard neural network techniques that approximate the value +of f(η) in a heuristic manner from sample output values, +PINNs obtain a solution function that minimizes the loss +function, which is a combination of the differential equation +and boundary values. +Several combinations of the number of hidden layers and +the number of nodes in each layer were tested on a trial- +and-error basis. We see that the results obtained are at par + +Forward pass +Minimization and update weights +Done +dr +ODE +LOSS +On2 +Boundary +Loss< +conditions +a3 +On3 +N +Hidden layers +Backward passwith that of [22] when initializing the neural network using +two fully connected hidden layer structures and setting each +hidden layer’s width to 100 neurons. The learning rate set in +this case is 0.96. Some of the relatively good results obtained +for various combinations of hidden layers and nodes, along +with the changes in the learning rate of the algorithms, are +presented in table I. The solution fi(η) for each of the cases +in table I is graphically presented in Fig.2. +TABLE I +VARIATIONS IN f′′(0) AND LOSS FUNCTION +Layers +Neurons +f”(0) +Loss +Solution +2 +100 +0.33165 +1.67x10−6 +f1(η) +3 +100 +0.32154 +1.10x10−4 +f2(η) +4 +90 +0.16021 +1.95x10−3 +f3(η) +5 +90 +0.0.5071 +5.58x10−3 +f4(η) +Fig. 2. Solutions obtained for various widths and depths of NN +In earlier methods proposed by Lagaris [4], all the boundary +conditions had to be found before moving to find the solution. +In this method, it is only required to set an appropriate finite +value for the η at ∞, which we assumed to be ηm = 8 in our +case. Although higher values could be set, we see from the +literature and our results that f ′(η) takes the value very close +to one when η = 5. Hence the justification for taking the value +of η at ∞ as ηm = 8. Another benefit of using PINNs is that +we can considerably reduce the number of collocation points +and still attain the same level of accuracy. In our method, we +have considered a total of 100 equidistant collocation points +of η between 0 and 8. The network trains with this input data +and finds the derivatives using the automatic differentiation +tool to calculate the loss Lo. +The Adam optimizing algorithm is incorporated into the +PINNs methodology to set the model’s adaptive learning rates. +It adds momentum as the estimate of the first-order moment of +the gradient and includes bias corrections to the estimates of +first and second-order moments. The second-order method to +train the network was the Limited memory Broyden Fletcher +Goldfarb Shanno (L-BFGS) algorithm. The cost of memory +has been decreased by avoiding the hessian approximation +of BFGS and replacing it with an identity matrix. The loss +function is optimized using ADAM and L-BFGS until it +converges. +The method was extended to the negative axis setting η0 +to -5.69 and ηm to 7. The value of the f ′′(0) found from +the previous program is incorporated into the existing loss +function. Our purpose in doing this was to check if our +method could capture the singularity of the solution function +mentioned in the literature. +IV. RESULTS AND DISCUSSION +The proposed PINNs method can reduce computational +time in solving the Blasius equation over conventional CFD +techniques as it reduces the burden of finding the value of +f ′′(0) and then solving the initial value problem. The neural +Fig. 3. Estimated results f(η), f′(η), f′′(η) obtained from PINNs and those +obtained by numerical methods +network constructed with two hidden layers with 100 neurons +each is considered for our results and discussions. The input +of η ranges from 0 to 8 (ηm). The learning rate for adam +optimizer is assigned the value of 0.96. From Fig.3, it is clear +that the results obtained from PINNS perfectly match with +numerical techniques’ results [21]. +In the present method, the value of f ′′(η) at 0 is not +considered for finding the solution. Further, once f(η) is +obtained, we can also find the functions corresponding to f ′(η) +and f ′′(η). The value of the second derivative at the wall +is observed from the PINNs method to be 0.33165, which +aligns approximately with those found by Howarth [20]. The +loss generated from this method is 1.67x10−6 that is of order +10−6 indicating the results to be accurate with sufficiently less +error value. As proposed by Gaurav Pandey and Ambedkar +Dukkipati [27], increasing the width (number of neurons) +and depth (number of hidden layers) of the network can be +beneficial. But in our case, we see that the function ˆf(η) +tends to become more vulnerable to a high amount of bias + +4.0 +f(1) +f(2) +3.5 +f(3) +f(4) +3.0 +2.5 +f +2.0 +1.5 +1.0 +0.5 +0.0 +11 +F2 +0 +4 +5 +9 +7 +8 +n4.0 +PINN f(n) +Num f(n) +3.5 +PINN f '(n) +Num f'(n) +3.0 +PINN f "(n) +Num f"(n) +2.5 +± 2.0 +1.5 +1.0 +0.5 +0.0 +0 +2 +3 +4 +5 +6 +7 +8 +nby choosing a deeper network. Thus, the initial conditions +must be chosen much more accurately for the loss function +to converge to the global minimum. Further, fixing the width +and depth of the network to 100 and 2, respectively, depicts +accurate results; hence, the need for a more complex network +was overridden. +The velocity components are directly proportional to f ′(η). +Thus, the proposed method’s solution helps in mapping non- +dimensional coordinate η with f ′(η). This is the velocity +profile inside the boundary layer. The values of velocity +components can also be calculated from f ′(η) and f ′′(η) From +the graph, it is visually appealing that when η approaches five, +the f ′(η) goes to 1. Hence the boundary layer thickness is +calculated where the velocity is 99% of free stream velocity. +Fig. 4. Graph of f(η) on the negative η−axis +In 1999, Boyd [19] suggested that convergence of his power +series method to solve the Blasius equation is limited by a +singularity on the negative η−axis at η = −5.6900380545. +Anil and Milan [22] considered a leaping Taylor’s series +solution for the Blasius equation to overcome this singularity. +In the second part of our investigation, we extended the +collocation points to the negative portion to check for the +solution of the Blasius equation on the negative η−axis. The +solution obtained from PINNs also shows that the function +f(η) increases rapidly near η = −5.7 signifying the presence +of a singularity near the point. A graph of f(η) including the +negative η−axis is shown in Fig.4. +V. CONCLUSION +PINNs use knowledge of the governing equation in deep +learning and find a solution to the differential equation by +minimizing a loss function, including the physics information. +The proposed methodology using PINNs is free from mesh +generation, which is an integral part of conventional CFD +techniques. Here, we have obtained the solutions to the Blasius +equations, which agree with all the numerical methods in +the literature. Further, the solution captures the singularity +mentioned while solving the differential equation analytically. +REFERENCES +[1] C. M. Bishop, “Neural networks and their applications,” Review of +scientific instruments, vol. 65, no. 6, pp. 1803–1832, 1994. +[2] L. Prandtl, “ ¨Uber flussigkeitsbewegung bei sehr kleiner reibung,” Ver- +handl. III, Internat. Math.-Kong., Heidelberg, Teubner, Leipzig, 1904, +pp. 484–491, 1904. +[3] H. Mutuk, “A neural network study of blasius equation,” Neural Pro- +cessing Letters, vol. 51, no. 3, pp. 2179–2194, 2020. +[4] I. E. Lagaris, A. Likas, and D. I. Fotiadis, “Artificial neural networks for +solving ordinary and partial differential equations,” IEEE transactions +on neural networks, vol. 9, no. 5, pp. 987–1000, 1998. +[5] I. Ahmad and M. Bilal, “Numerical solution of blasius equation through +neural networks algorithm,” American Journal of Computational Math- +ematics, vol. 4, no. 3, p. 223, 2014. +[6] M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics-informed +neural networks: A deep learning framework for solving forward and +inverse problems involving nonlinear partial differential equations,” +Journal of Computational physics, vol. 378, pp. 686–707, 2019. +[7] ——, “Physics informed deep learning (part i): Data-driven so- +lutions of nonlinear partial differential equations,” arXiv preprint +arXiv:1711.10561, 2017. +[8] H. Bararnia and M. Esmaeilpour, “On the application of physics in- +formed neural networks (pinn) to solve boundary layer thermal-fluid +problems,” International Communications in Heat and Mass Transfer, +vol. 132, p. 105890, 2022. +[9] G. Pang, L. Lu, and G. E. Karniadakis, “fpinns: Fractional physics- +informed neural networks,” SIAM Journal on Scientific Computing, +vol. 41, no. 4, pp. A2603–A2626, 2019. +[10] P.-H. Chiu, J. C. Wong, C. Ooi, M. H. Dao, and Y.-S. Ong, “Can-pinn: +A fast physics-informed neural network based on coupled-automatic– +numerical differentiation method,” Computer Methods in Applied Me- +chanics and Engineering, vol. 395, p. 114909, 2022. +[11] L. Yuan, Y.-Q. Ni, X.-Y. Deng, and S. Hao, “A-pinn: Auxiliary physics +informed neural networks for forward and inverse problems of nonlinear +integro-differential equations,” Journal of Computational Physics, vol. +462, p. 111260, 2022. +[12] S. Thakur, M. Raissi, and A. M. Ardekani, “Viscoelasticnet: A physics +informed neural network framework for stress discovery and model +selection,” arXiv preprint arXiv:2209.06972, 2022. +[13] H. Eivazi, M. Tahani, P. Schlatter, and R. Vinuesa, “Physics-informed +neural networks for solving reynolds-averaged navier–stokes equations,” +Physics of Fluids, vol. 34, no. 7, p. 075117, 2022. +[14] M. Rasht-Behesht, C. Huber, K. Shukla, and G. E. Karniadakis, +“Physics-informed neural networks (pinns) for wave propagation and full +waveform inversions,” Journal of Geophysical Research: Solid Earth, +vol. 127, no. 5, p. e2021JB023120, 2022. +[15] S. Cai, Z. Wang, S. Wang, P. Perdikaris, and G. E. Karniadakis, “Physics- +informed neural networks for heat transfer problems,” Journal of Heat +Transfer, vol. 143, no. 6, 2021. +[16] H. Blasius, Grenzschichten in Fl¨ussigkeiten mit kleiner Reibung. Druck +von BG Teubner, 1907. +[17] E. Pohlhausen, E. See Schmidt, and W. Beckmann, “Das temperatur-und +geschwindigkeitsfeld vor einer w¨arme abgebenden senkrechten platte bei +nat¨urlicher konvektion,” Tech. Mech. u. Thermodynamik, vol. 1, p. 341, +1930. +[18] S. Ostrach, “An analysis of laminar free-convection flow and heat +transfer about a flat plate parallel to the direction of the generating +body force,” National Aeronautics and Space Administration Cleveland +Oh Lewis Research Center, Tech. Rep., 1952. +[19] J. P. Boyd, “The blasius function in the complex plane,” Experimental +Mathematics, vol. 8, no. 4, pp. 381–394, 1999. +[20] L. Howarth, “On the solution of the laminar boundary layer equations.” +Proceedings of the Royal Society of London. Series A-Mathematical and +Physical Sciences, vol. 164, no. 919, pp. 547–579, 1938. + +35 +30 +25 +20 +15 +10 +5 +0 +-6 +-4 +-2 +0 +2 +4 +6 +n[21] S. Anil Lal and M. Neeraj Paul, “An accurate taylors series solution +with high radius of convergence for the blasius function and parameters +of asymptotic variation,” Journal of Applied Fluid Mechanics, vol. 7, +no. 4, pp. 557–564, 2014. +[22] L. S. Anil and M. Martin, “Accurate benchmark results of blasius +boundary layer problem using a leaping taylors series that converges +for all real values,” arXiv preprint arXiv:2204.02215, 2022. +[23] F. M. White and J. Majdalani, Viscous fluid flow. +McGraw-Hill New +York, 2006, vol. 3. +[24] R. Cortell, “Numerical solutions of the classical blasius flat-plate prob- +lem,” Applied Mathematics and Computation, vol. 170, no. 1, pp. 706– +710, 2005. +[25] T. Fang, W. Liang, and F. L. Chia-fon, “A new solution branch +for the blasius equation—a shrinking sheet problem,” Computers & +Mathematics with Applications, vol. 56, no. 12, pp. 3088–3095, 2008. +[26] K. Abbaoui and Y. Cherruault, “New ideas for proving convergence of +decomposition methods,” Computers & Mathematics with Applications, +vol. 29, no. 7, pp. 103–108, 1995. +[27] G. Pandey and A. Dukkipati, “To go deep or wide in learning?” in +Artificial Intelligence and Statistics. +PMLR, 2014, pp. 724–732. + diff --git a/JdAyT4oBgHgl3EQfTfdZ/content/tmp_files/load_file.txt b/JdAyT4oBgHgl3EQfTfdZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..627562696758a8c539d320ebb88a72bb4a40ced1 --- /dev/null +++ b/JdAyT4oBgHgl3EQfTfdZ/content/tmp_files/load_file.txt @@ -0,0 +1,426 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf,len=425 +page_content='Physics-informed Neural Networks approach to solve the Blasius function Greeshma Krishna Department of Mathematics Amrita Vishwa Vidyapeetham Amritapuri, India greeshmakrishna@am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='amrita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='edu Malavika S Nair Department of Mathematics Amrita Vishwa Vidyapeetham Amritapuri, India malavikasnair@am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='amrita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='edu Pramod P Nair Department of Mathematics Amrita Vishwa Vidyapeetham Amritapuri, India pramodpn@am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='amrita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='edu Anil Lal S Department of Mechanical Engineering Amrita Vishwa Vidyapeetham Amritapuri, India anillals@am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='amrita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='edu Abstract—Deep learning techniques with neural networks have been used effectively in computational fluid dynamics (CFD) to obtain solutions to nonlinear differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' This paper presents a physics-informed neural network (PINN) approach to solve the Blasius function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' This method eliminates the process of changing the non-linear differential equation to an initial value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Also, it tackles the convergence issue arising in the conventional series solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' It is seen that this method produces results that are at par with the numerical and conventional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The solution is extended to the negative axis to show that PINNs capture the singularity of the function at η = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Index Terms—Blasius equation, physics-informed neural net- works, automatic differentiation, computational fluid dynamics, boundary layer flow, singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' INTRODUCTION Machine learning helps learn patterns from the data to make predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The three basic requirements for machine learning are data, theory, and hardware to overcome computational difficulties using better GPUs and new approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Computa- tional science is an essential tool that we can use to incorporate physical invariances into learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' For example, the laws that govern the conservation of momentum, mass, and energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' To quote Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='Tinsley Oden - “Computational Science can analyze past events and look into the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' It can explore the effects of thousands of scenarios instead of actual experiments and be used to study events beyond the reach of expanding the boundaries of experimental science”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Neural networks(NNs) are the most used Machine Learning technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' It offers a set of robust tools for solving various supervised and unsupervised problems in pattern recognition, data processing, and non-linear control, which can be regarded as complementary to the conventional approaches [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' It can be considered a functional approximator as it maps a set of input variables to a set of output variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' A set of parameters called weights manages this mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' These weights are updated to train the network that fetches the required output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' A polynomial can be viewed as a function that transforms a single input variable into a single output variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The coefficients in the polynomial are comparable to the weights in a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Determining these coefficients helps in evaluating the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Physics Informed Neural Networks Physics-informed neural networks (PINNs) play a huge role where fewer data and the system’s conventional physics are known to get the solutions to a differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' NN can approximate the error to ground truth function, generalize to unseen data and train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The contribution of PINNs is to obtain a neural network that knows about the physics hidden behind the equation and thus efficiently solves differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The earlier numerical methods for solving them are finite difference, finite elements, spectral elements, and finite volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' When working with PINNs, there is data efficiency from a machine learning perspective since it is regularized heavily with physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The required derivatives can be calcu- lated using automatic differentiation at the end of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' It then finds a loss function corresponding to the differential equation and boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Boundary Layer Theory The boundary layer of a flowing fluid is a thin layer near a solid surface, and the flow near the solid surface is known as the boundary layer flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Ludwig Prandtl is credited for developing the boundary layer theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' In 1904, he published a paper titled ”On the Motion of a Fluid with Very Small Vis- cosity” [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' He laid out the mathematical foundation for flows and condensed the two-dimensional Navier-Stokes equations (NSE) into the boundary layer equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' This publication made understanding fluid motion physics possible, which is regarded as the beginning of contemporary fluid mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The introduction of the similarity variables and the trans- formation of the PDEs into nonlinear ODEs in one coordinate allowed the successful resolution of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Still, specific methods were needed to handle the unbounded boundary arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='00106v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='LG] 31 Dec 2022 conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The Blasius benchmark problem has been resolved [3] using the trial function method put out by Lagaris [4] or a hybrid approach [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The solution function is seen to have a singularity on the negative real axis at approximately -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The Blasius equation is a single equation that solely models the viscous boundary layer and is one of CFD’s fundamental models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' In this paper, we propose solving the Blasius equation using PINNs and comparing the solution we obtain with the best-known solutions available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' We shall also extend our solution to the negative real axis to locate the singularity existing for the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' In the next section of the paper, we shall present a brief literature review on the Blasius equation and PINNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' In Section III, we present the methodology that has been adopted to solve the Blasius Equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Section IV discusses the results and analysis of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The conclusions are discussed in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' LITERATURE REVIEW A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Physics Informed Neural Networks Neural networks can be used to attain solutions to ODEs and PDEs by reducing them to an optimization problem instead of numerically solving the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The application of NNs in fluid mechanics began in the 1990s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Raissi and his team [6], [7] developed a technique called Physics informed neural network (PINN) where the loss function defined in the corresponding NN is extracted from the physics behind the PDE and related equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' In earlier years, Computational fluid dynamics (CFD) had been a great relief in numerically solving the compressible and incompressible Na¨ıve Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Minimizing the loss function is challenging in PINNs, as it can be highly complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Nevertheless, PINNs have been proven to be more accurate than conventional CFD techniques with lesser computations [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Although numerical discretization of the Navier-Stokes equations (NSE) has made significant progress in simulating flow problems over the past 50 years, the present algorithms cannot solve governed by high-parametrized NSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Addition- ally, it is expensive to solve inverse flow problems due to their complexity, expensive formulations, and need for new algo- rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' PINNs are expanded to fractional PINNs (fPINNs) that explore their convergence methodically to solve space-time fractional advection-diffusion equations (fractional ADEs) [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' A hybrid method for building the residual in the loss function that utilizes both automated differentiation for the integer- order operators and numerical discretization for the fractional operators is a novel component of the fPINNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Wide varieties of PINNs have been found in the literature in recent times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' can-PINNs [10] link derivative terms with nearby support points, which generally apply to Taylor series expansion-based numerical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Apart from demonstrating good dispersion and dissipation characteristics, they are highly trainable and require four to sixteen times fewer collocation points than original PINNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Auxiliary PINNs (A-PINN) is a technique for solving forward and inverse problems of non-linear integrodifferential equations [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' ViscoelasticNet is another PINN framework for stress discovery and model selection [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' PINNs can also be used to solve Reynolds- averaged Navier-Stokes equations [13], full waveform seismic inversions in 2D acoustic media, and wave propagation as it seamlessly handles boundary conditions and physical con- straints [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' In addition to addressing ill-posed problems be- yond the scope of conventional computing techniques, PINNs can also close the discrepancy between computational and experimental heat transfer [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Blasius Equation Blasius equation is a third order non-linear ordinary differ- ential equation of the form f ′′′+ 1 2ff ′′ = 0 with the boundary conditions f(0) = 0, f ′(0) = 0, f ′(∞) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' It governs the boundary layer flow over a semi-infinite flat plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Suppose that the u−velocity, the velocity parallel to the surface, is much greater than the v−velocity, perpendicular to the surface, and the changes in the perpendicular direction to the surface are much greater than changes parallel to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The boundary layer equations consists of conservation of mass (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='1), conservation of x-momentum (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='2), and conservation of y-momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' In a flat plate boundary layer, the pressure gradient term appearing in the x-momentum equation becomes zero (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' This leads to the hydrodynamic solution for the flat plate boundary layer in a laminar flow called the Blasius solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' ∂u ∂x + ∂v ∂y = 0 (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='1) u∂u ∂x + v ∂u ∂y = ν ∂2u ∂y2 (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='2) Blasius’ analysis focus on the laminar boundary layer forming on a flat plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The main aspect of the work is the transformation of the PDE for a flat plate boundary layer with zero pressure gradient into a single ordinary differential equation(ODE) by considering the velocity components that satisfy equation II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' u ≡ ∂ψ ∂y v ≡ −∂ψ ∂x (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='3) The stream function ψ = U∞ � νx U∞ f(η), is directly pro- portional to the function f(η) called the Blasius function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Here, U∞ is the free stream velocity, and η is a transformed coordinate called the similarity parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Here, the velocity components are proportional to the first derivative of f(η), and the second and third derivatives of f are proportional to the first and second derivatives of velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Substituting these relations into the momentum equation (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='2), the final form of the Blasius boundary layer equation f ′′′ + 1 2ff ′′ = 0, for a flat plate can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The first and second derivatives of f(η) are given by f ′ = u U∞ (Non-dimensional velocity profile) and f ′′ = 1 U∞ � νx U∞ ∂u ∂y (quantity related to shear stress).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The boundary conditions are set considering the laminar flow on a flat plate, the no-slip condition, and free-stream velocity outside the boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Hence we have f(0) = 0 f ′(0) = 0 f ′(∞) = 1 (II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='4) Since the value of f ′′ at η = 0 is unknown, it cannot be considered an initial value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' A power series solution to the boundary layer equation of flow across a flat plate was proposed by Blasius [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Schmidt and Beckmann [17] and Ostrach [18] conducted the most important work on the topic, conducting theoretical and experimental research on the free convection flow of air around a vertical flat plate under the influence of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Boyd solved the equation using an analytical series solution technique [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Nowadays, with the availability of computers, we can obtain a numerical solution to this equation and calculate it with a very high degree of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' To solve the Blasius equation numerically, a possible method is to use the shooting algorithm to find what value would satisfy the boundary condition at η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The first step is to guess a value at the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Then solving the ODE along the non-dimensional coordinate until the first derivative of f stops changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' This checks if the first derivative satisfies the given boundary condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' instead, adjust the guessed value to decrease or increase if the first derivative is higher or lower than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The algorithm is repeated until the boundary condition is finally satisfied to reach the final solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Howarth [20] found the solution using a numerical method that accurately predicted the value of f ′′(0) to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' This differential equation is a direct representation of the velocity profile inside the boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Once the solution is obtained, the velocity components can be calculated using these relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Hence the boundary layer thickness can be calculated located at the point where the velocity is 99% of the free stream velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' It was found that this occurs approx- imately at η equal to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The displacement and momentum thickness can also be estimated using the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The wall shear stress based on the velocity gradient at the wall can be estimated, and the friction and drag coefficients can be calcu- lated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Note that the expression is only accounting for one side of the plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Comparing the exact solution with the approxi- mation solution from the integral analysis is interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The relations are close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Indeed the integral analysis is within 10% of the exact solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' unlike Blasius’ solution, these values were obtained without complicated math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The Blasius solution provides a self-similar solution meaning that the solution is the same if the independent and dependent variables of the governing equations are appropriately scaled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' This can be seen in comparing experimental results with the Blasius solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Different Reynolds numbers (Re) provide the same profile when the variables on the two axes are appropriately scaled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The solution to the Blasius equation can be found in [21], [22], where the accurate benchmark results of the Blasius boundary layer problem using a leaping Taylors series that converges for all real values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' There have been different meth- ods to solve the Blasius equation, such as the Topher transfor- mation, which is executed using inverse transformation, [23]– [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Runge-Kutta, incorporated with the shooting method, finds the solution numerically, while Adomian Decomposition Method [26] finds the solution analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' PROPOSED METHODOLOGY The ODE representing the Blasius equation derived from the PDE is considered in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' We follow a similar methodology proposed for the general form of generalized non-linear ODE by Raissi [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The workflow diagram of the present method is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Workflow diagram to solve for f(η) The input values η are discrete and are equally distributed between the boundary points η0 and ηm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The beauty of PINNs lies in loss function as it incorporates the boundary conditions and the differential equation and thus includes the physics information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The loss function is the total loss from given ODE Lo, initial conditions Li, and boundary Lb conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' L = Lo + Lb + Li (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='1) Lo = � η [ ˆf ′′′ + 1 2 ˆf ˆf ′′]2 (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='2) Li = [ ˆf(η0) − f(η0)]2 + [ ˆf ′(η0) − f(η0)]2 = [ ˆf(η0) − 0]2 + [ ˆf ′(η0) − 0]2 (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='3) Lb = [ ˆf(ηm) − f(ηm)]2 = [ ˆf(ηm) − 1]2 (III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='4) From the definition of the loss function, it is clear that the total loss will be zero if the function f(η) is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Here we update the weights of the neural network in each iteration such that the loss function is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Supplying the independent input values representing discrete spatial coordinates η ranging from zero to ηm into the neural network is sufficient to solve the ODE via PINNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The neural network maps the input η to f(η), which is the estimated solution to the stream function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Unlike standard neural network techniques that approximate the value of f(η) in a heuristic manner from sample output values, PINNs obtain a solution function that minimizes the loss function, which is a combination of the differential equation and boundary values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Several combinations of the number of hidden layers and the number of nodes in each layer were tested on a trial- and-error basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' We see that the results obtained are at par Forward pass Minimization and update weights Done dr ODE LOSS On2 Boundary Loss< conditions a3 On3 N Hidden layers Backward passwith that of [22] when initializing the neural network using two fully connected hidden layer structures and setting each hidden layer’s width to 100 neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The learning rate set in this case is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Some of the relatively good results obtained for various combinations of hidden layers and nodes, along with the changes in the learning rate of the algorithms, are presented in table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The solution fi(η) for each of the cases in table I is graphically presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' TABLE I VARIATIONS IN f′′(0) AND LOSS FUNCTION Layers Neurons f”(0) Loss Solution 2 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='33165 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='67x10−6 f1(η) 3 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='32154 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='10x10−4 f2(η) 4 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='16021 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='95x10−3 f3(η) 5 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='5071 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='58x10−3 f4(η) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Solutions obtained for various widths and depths of NN In earlier methods proposed by Lagaris [4], all the boundary conditions had to be found before moving to find the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' In this method, it is only required to set an appropriate finite value for the η at ∞, which we assumed to be ηm = 8 in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Although higher values could be set, we see from the literature and our results that f ′(η) takes the value very close to one when η = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Hence the justification for taking the value of η at ∞ as ηm = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Another benefit of using PINNs is that we can considerably reduce the number of collocation points and still attain the same level of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' In our method, we have considered a total of 100 equidistant collocation points of η between 0 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The network trains with this input data and finds the derivatives using the automatic differentiation tool to calculate the loss Lo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The Adam optimizing algorithm is incorporated into the PINNs methodology to set the model’s adaptive learning rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' It adds momentum as the estimate of the first-order moment of the gradient and includes bias corrections to the estimates of first and second-order moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The second-order method to train the network was the Limited memory Broyden Fletcher Goldfarb Shanno (L-BFGS) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The cost of memory has been decreased by avoiding the hessian approximation of BFGS and replacing it with an identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The loss function is optimized using ADAM and L-BFGS until it converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The method was extended to the negative axis setting η0 to -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='69 and ηm to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The value of the f ′′(0) found from the previous program is incorporated into the existing loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Our purpose in doing this was to check if our method could capture the singularity of the solution function mentioned in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' RESULTS AND DISCUSSION The proposed PINNs method can reduce computational time in solving the Blasius equation over conventional CFD techniques as it reduces the burden of finding the value of f ′′(0) and then solving the initial value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The neural Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Estimated results f(η), f′(η), f′′(η) obtained from PINNs and those obtained by numerical methods network constructed with two hidden layers with 100 neurons each is considered for our results and discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The input of η ranges from 0 to 8 (ηm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The learning rate for adam optimizer is assigned the value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='3, it is clear that the results obtained from PINNS perfectly match with numerical techniques’ results [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' In the present method, the value of f ′′(η) at 0 is not considered for finding the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Further, once f(η) is obtained, we can also find the functions corresponding to f ′(η) and f ′′(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The value of the second derivative at the wall is observed from the PINNs method to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='33165, which aligns approximately with those found by Howarth [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The loss generated from this method is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='67x10−6 that is of order 10−6 indicating the results to be accurate with sufficiently less error value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' As proposed by Gaurav Pandey and Ambedkar Dukkipati [27], increasing the width (number of neurons) and depth (number of hidden layers) of the network can be beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' But in our case, we see that the function ˆf(η) tends to become more vulnerable to a high amount of bias 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='0 f(1) f(2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='5 f(3) f(4) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='5 f 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='0 11 F2 0 4 5 9 7 8 n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='0 PINN f(n) Num f(n) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content="5 PINN f '(n) Num f'(n) 3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='0 PINN f "(n) Num f"(n) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='0 0 2 3 4 5 6 7 8 nby choosing a deeper network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Thus, the initial conditions must be chosen much more accurately for the loss function to converge to the global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Further, fixing the width and depth of the network to 100 and 2, respectively, depicts accurate results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' hence, the need for a more complex network was overridden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The velocity components are directly proportional to f ′(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Thus, the proposed method’s solution helps in mapping non- dimensional coordinate η with f ′(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' This is the velocity profile inside the boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The values of velocity components can also be calculated from f ′(η) and f ′′(η) From the graph, it is visually appealing that when η approaches five, the f ′(η) goes to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Hence the boundary layer thickness is calculated where the velocity is 99% of free stream velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Graph of f(η) on the negative η−axis In 1999, Boyd [19] suggested that convergence of his power series method to solve the Blasius equation is limited by a singularity on the negative η−axis at η = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='6900380545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Anil and Milan [22] considered a leaping Taylor’s series solution for the Blasius equation to overcome this singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' In the second part of our investigation, we extended the collocation points to the negative portion to check for the solution of the Blasius equation on the negative η−axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The solution obtained from PINNs also shows that the function f(η) increases rapidly near η = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='7 signifying the presence of a singularity near the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' A graph of f(η) including the negative η−axis is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' CONCLUSION PINNs use knowledge of the governing equation in deep learning and find a solution to the differential equation by minimizing a loss function, including the physics information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' The proposed methodology using PINNs is free from mesh generation, which is an integral part of conventional CFD techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Here, we have obtained the solutions to the Blasius equations, which agree with all the numerical methods in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Further, the solution captures the singularity mentioned while solving the differential equation analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' REFERENCES [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Bishop, “Neural networks and their applications,” Review of scientific instruments, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 65, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 1803–1832, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Prandtl, “ ¨Uber flussigkeitsbewegung bei sehr kleiner reibung,” Ver- handl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' III, Internat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='-Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=', Heidelberg, Teubner, Leipzig, 1904, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 484–491, 1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [3] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Mutuk, “A neural network study of blasius equation,” Neural Pro- cessing Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 2179–2194, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [4] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Lagaris, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Likas, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Fotiadis, “Artificial neural networks for solving ordinary and partial differential equations,” IEEE transactions on neural networks, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 987–1000, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [5] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Ahmad and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Bilal, “Numerical solution of blasius equation through neural networks algorithm,” American Journal of Computational Math- ematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 4, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 223, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Raissi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Perdikaris, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Karniadakis, “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,” Journal of Computational physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 378, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 686–707, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [7] ——, “Physics informed deep learning (part i): Data-driven so- lutions of nonlinear partial differential equations,” arXiv preprint arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='10561, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [8] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Bararnia and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Esmaeilpour, “On the application of physics in- formed neural networks (pinn) to solve boundary layer thermal-fluid problems,” International Communications in Heat and Mass Transfer, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 132, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 105890, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [9] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Pang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Lu, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Karniadakis, “fpinns: Fractional physics- informed neural networks,” SIAM Journal on Scientific Computing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 41, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' A2603–A2626, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [10] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Chiu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Wong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Ooi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Dao, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Ong, “Can-pinn: A fast physics-informed neural network based on coupled-automatic– numerical differentiation method,” Computer Methods in Applied Me- chanics and Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 395, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 114909, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [11] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Yuan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Ni, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Deng, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Hao, “A-pinn: Auxiliary physics informed neural networks for forward and inverse problems of nonlinear integro-differential equations,” Journal of Computational Physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 462, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 111260, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [12] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Thakur, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Raissi, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Ardekani, “Viscoelasticnet: A physics informed neural network framework for stress discovery and model selection,” arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='06972, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [13] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Eivazi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Tahani, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Schlatter, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Vinuesa, “Physics-informed neural networks for solving reynolds-averaged navier–stokes equations,” Physics of Fluids, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 075117, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Rasht-Behesht, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Huber, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Shukla, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Karniadakis, “Physics-informed neural networks (pinns) for wave propagation and full waveform inversions,” Journal of Geophysical Research: Solid Earth, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 127, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' e2021JB023120, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Cai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Wang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Perdikaris, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Karniadakis, “Physics- informed neural networks for heat transfer problems,” Journal of Heat Transfer, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 143, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 6, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [16] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Blasius, Grenzschichten in Fl¨ussigkeiten mit kleiner Reibung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Druck von BG Teubner, 1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [17] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Pohlhausen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' See Schmidt, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Beckmann, “Das temperatur-und geschwindigkeitsfeld vor einer w¨arme abgebenden senkrechten platte bei nat¨urlicher konvektion,” Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Thermodynamik, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 341, 1930.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Ostrach, “An analysis of laminar free-convection flow and heat transfer about a flat plate parallel to the direction of the generating body force,” National Aeronautics and Space Administration Cleveland Oh Lewis Research Center, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=', 1952.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Boyd, “The blasius function in the complex plane,” Experimental Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 381–394, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [20] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Howarth, “On the solution of the laminar boundary layer equations.” Proceedings of the Royal Society of London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Series A-Mathematical and Physical Sciences, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 164, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 919, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 547–579, 1938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 35 30 25 20 15 10 5 0 6 4 2 0 2 4 6 n[21] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Anil Lal and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Neeraj Paul, “An accurate taylors series solution with high radius of convergence for the blasius function and parameters of asymptotic variation,” Journal of Applied Fluid Mechanics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 557–564, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [22] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Anil and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Martin, “Accurate benchmark results of blasius boundary layer problem using a leaping taylors series that converges for all real values,” arXiv preprint arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content='02215, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [23] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' White and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Majdalani, Viscous fluid flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' McGraw-Hill New York, 2006, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [24] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Cortell, “Numerical solutions of the classical blasius flat-plate prob- lem,” Applied Mathematics and Computation, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 170, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 706– 710, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [25] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Fang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Liang, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Chia-fon, “A new solution branch for the blasius equation—a shrinking sheet problem,” Computers & Mathematics with Applications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 56, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 3088–3095, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [26] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Abbaoui and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Cherruault, “New ideas for proving convergence of decomposition methods,” Computers & Mathematics with Applications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 29, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 103–108, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' [27] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Pandey and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' Dukkipati, “To go deep or wide in learning?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' in Artificial Intelligence and Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' PMLR, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} +page_content=' 724–732.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfTfdZ/content/2301.00106v1.pdf'} diff --git a/K9E0T4oBgHgl3EQfzwIE/content/tmp_files/2301.02675v1.pdf.txt b/K9E0T4oBgHgl3EQfzwIE/content/tmp_files/2301.02675v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1de4357979e74460acfd889a00a8ff23d0013a52 --- /dev/null +++ b/K9E0T4oBgHgl3EQfzwIE/content/tmp_files/2301.02675v1.pdf.txt @@ -0,0 +1,2300 @@ +MNRAS 000, 1–20 (2023) +Preprint 10 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Testing AGN outflow and accretion models with C iv and He ii emission +line demographics in 𝑧 ≈ 2 quasars +Matthew J. Temple +,1★ James H. Matthews +,2,3 Paul C. Hewett +,3 Amy L. Rankine +,4 +Gordon T. Richards +,5 Manda Banerji +,6 Gary J. Ferland +,7 Christian Knigge +6 and Matthew Stepney +6 +1Instituto de Estudios Astrofísicos, Universidad Diego Portales, Av. Ejército Libertador 441, Santiago 8370191, Chile +2Department of Physics, Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK +3Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK +4Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK +5Department of Physics, Drexel University, 32 S. 32nd Street, Philadelphia, PA 19104, USA +6School of Physics & Astronomy, University of Southampton, Southampton SO17 1BJ, UK +7Department of Physics and Astronomy, The University of Kentucky, Lexington, KY 40506, USA +Submitted to MNRAS 2022 December 22 +ABSTRACT +Using ≈190,000 spectra from the seventeenth data release of the Sloan Digital Sky Survey, we investigate the ultraviolet +emission line properties in 𝑧 ≈ 2 quasars. Specifically, we quantify how the shape of C iv 𝜆1549 and the equivalent width (EW) +of He ii 𝜆1640 depend on the black hole mass and Eddington ratio inferred from Mg ii 𝜆2800. Above 𝐿/𝐿Edd ≳ 0.2, there is a +strong mass dependence in both C iv blueshift and He ii EW. Large C iv blueshifts are observed only in regions with both high +mass and high accretion rate. Including X-ray measurements for a subsample of 5,300 objects, we interpret our observations in +the context of AGN accretion and outflow mechanisms. The observed trends in He ii and 2 keV strength are broadly consistent +with theoretical qsosed models of AGN spectral energy distributions (SEDs) for low spin black holes, where the ionizing +SED depends on the accretion disc temperature and the strength of the soft excess. High spin models are not consistent with +observations, suggesting SDSS quasars at 𝑧 ≈ 2 may in general have low spins. We find a dramatic switch in behaviour at +𝐿/𝐿Edd ≲ 0.2: the ultraviolet emission properties show much weaker trends, and no longer agree with qsosed predictions, +hinting at changes in the structure of the broad line region. Overall the observed emission line trends are generally consistent +with predictions for radiation line driving where quasar outflows are governed by the SED, which itself results from the accretion +flow and hence depends on both the SMBH mass and accretion rate. +Key words: quasars: emission lines +1 INTRODUCTION +1.1 Observational context: spectroscopic properties of quasars +The spectroscopic properties of type-1 quasars have long been appre- +ciated for their potential to provide insight into the physical processes +responsible for luminous active galactic nuclei (AGN; Baldwin & +Netzer 1978; Davidson & Netzer 1979; Kwan & Krolik 1981; Krolik +& Kallman 1988; Elvis 2000). These processes include the excitation +of various line- and continuum-emitting regions, and mechanisms +for launching outflows which might ‘feed back’ energy to their host +galaxies. Such processes are ultimately powered by accretion onto +supermassive black holes (SMBHs; Lynden-Bell 1969), and thus de- +pend primarily on the mass of the SMBH and the accretion rate, with +potential second-order drivers including the spin of the SMBH and +the metal content of the accreting material. +The search for insight has gained much from identifying and ex- +★ E-mail: Matthew.Temple@mail.udp.cl +ploring the properties which are observed to vary the most. Such +diversity in the observed quantities must ultimately be driven by +some of the physics which we would like to use to better constrain +both the growth of SMBHs and their effect on the galactic ecosystems +in which they reside. The most famous result of these investigations +is arguably the identification of the so-called ‘eigenvector 1’ (EV1), +which accounts for the largest amount of correlated variance in the +optical spectra of low-redshift (𝑧 < 1) type-1 AGN spectra. Most +authors now agree that the EV1 is driven by the mass-normalised +accretion rate (the Eddington ratio), possibly convolved with some +orientation effect (Boroson & Green 1992; Wills et al. 1999; Sulentic +et al. 2000; Shen & Ho 2014; Sun & Shen 2015; Sulentic & Marziani +2015; Wolf et al. 2020). +Similarly, the ultraviolet emission features in quasar spectra also +show a rich phenomenology (Croom et al. 2002; Jensen et al. 2016; +Brodzeller & Dawson 2022). Early work by Baldwin (1977) showed +that the equivalent widths (EWs) of various ultraviolet lines, most +notably C iv 𝜆1549, were anti-correlated with the ultraviolet contin- +uum luminosity. Shang et al. (2003) showed that this ‘Baldwin effect’ +© 2023 The Authors +arXiv:2301.02675v1 [astro-ph.GA] 6 Jan 2023 + +2 +M. J. Temple et al. +was independent of EV1 in 22 quasars with 𝑧 < 0.4, implying dif- +ferent physical drivers for these correlations. Early observations also +demonstrated that the centroid of the C iv emission line is commonly +shifted to the blue (Gaskell 1982; Wilkes 1984; Richards et al. 2002). +Within a sample of 87 Palomar-Green quasars, Baskin & Laor (2004, +2005) found that large C iv blueshifts were only seen in objects with +high Eddington ratios, although not all quasars with high Eddington +ratios had large C iv blueshifts. The EV1 formalism was extended by +Bachev et al. (2004) and Sulentic et al. (2007) to include the velocity +shift of C iv, again finding that large C iv blueshifts are seen only in +so-called ‘Population A’ quasars with high Eddington ratios. +With the start of the Sloan Digital Sky Survey (SDSS; York et al. +2000), large samples of rest-frame ultraviolet quasar spectra became +available. Equally as important were methods to accurately charac- +terize the systemic redshift of each quasar (Hewett & Wild 2010), +which is necessary to infer the velocity shift of any emission features. +A notable work by Richards et al. (2011) summarized the state of +the field a decade ago at the time of the seventh data release (DR7; +Schneider et al. 2010) from SDSS. Using ≈35 000 quasar spectra, +Richards et al. (2011) confirmed the Baldwin effect and showed +that the EW of C iv line also anti-correlates with the magnitude of +the C iv blueshift: quasars with higher luminosities show, on aver- +age, weaker C iv emission which is more strongly blueshifted. C iv +blueshifts could be a signature of emission from ionized gas being +driven away from the accretion disc along the line-of-sight to the ob- +server (Leighly & Moore 2004), in which case the results of Richards +et al. (2011) can be interpreted as brighter objects showing stronger +emission from outflowing gas and weaker emission from the virial- +ized broad line region (BLR). We discuss this interpretation further +in Section 5.2.2, but do not assume anything about the origin of C iv +line shifts when presenting our observational results in Section 4.1. +Richards et al. (2011) also demonstrated that the C iv properties are +strongly correlated with the EW of the nearby He ii 𝜆1640 emission +line. More recent work by Rankine et al. (2020) has shown that +the correlations between the EW of He ii and both the EW and +blueshift of C iv are also present in quasars with broad absorption +features. We now know that the C iv and He ii properties are strongly +correlated with the properties of other ultraviolet emission features +such as Ly 𝛼, N v, Si iv and O iv] (Temple et al. 2021b), Fe iii, Al iii, +Si iii] and C iii] (Temple et al. 2020), as well as the optical [O iii] +emission (Vietri et al. 2018; Coatman et al. 2019; Vietri et al. 2020), +the strength of near infrared emission from dust at the sublimation +temperature (Temple et al. 2021a), the strength of the far infrared +emission (Maddox et al. 2017), the radio detection fraction (Rankine +et al. 2021) and the strength of the 2 keV X-ray continuum (Kruczek +et al. 2011; Zappacosta et al. 2020; Timlin et al. 2020, 2021; Lusso +et al. 2021; Marlar et al. 2022; Rivera et al. 2022). Tentative links +have also been found between the C iv blueshift and the amount of +continuum reddening ascribed to nuclear dust (Calistro Rivera et al. +2021; Fawcett et al. 2022). +The existence of such correlations - between parameters which +trace emission at different wavelengths and from different physical +regions - suggests that they are driven (either directly or indirectly) +by changes in some of the fundamental physical parameters which +govern the properties of a SMBH and its surrounding regions, such as +the SMBH mass, spin, and accretion rate. The space spanned by C iv +blueshift and C iv EW therefore appears to be just as important as +EV1 in understanding the physics of luminous AGN. However, while +the location of a given quasar spectrum on either EV1 or the C iv +blueshift–EW space must ultimately be a function of the fundamental +SMBH parameters, there is no guarantee that such a function is +linear, or even injective (i.e. one-to-one with a well-defined inverse). +For example, we cannot rule out the possibility that two objects with +different 𝑀BH and accretion rate have similar (or indeed identical) +C iv emission. +1.2 Theoretical context: AGN outflows and SEDs +Mass outflows from AGN can be launched by thermal pressure, mag- +netic forces, or radiation (Laha et al. 2021). Thermal winds can only +be launched at large radii [𝑅 ≳ 105𝑅g ≈ 5 × (𝑀BH/109𝑀⊙) parsec] +with terminal velocities of order 100–1000 km s−1 (Begelman et al. +1983; Woods et al. 1996; Mizumoto et al. 2019). Faster outflows with +speeds > 2000 km s−1, as commonly seen in broad high-ionization +ultraviolet absorption features, are most likely launched on smaller +(sub-parsec) scales. Magnetically driven winds (Blandford & Payne +1982; Emmering et al. 1992; Konigl & Kartje 1994; Fukumura et al. +2010; Yang et al. 2021a) may be important in this context, but we cur- +rently lack predictive models for how such winds would translate into +observable quantities (although see e.g. Bottorff et al. 2000; Chajet +& Hall 2013). On the other hand, radiation line driving (Castor et al. +1975; Murray et al. 1995; Murray & Chiang 1995; Elvis 2000; Proga +et al. 2000; Proga & Kallman 2004; Proga 2007; Risaliti & Elvis +2010; Elvis 2017; Nomura & Ohsuga 2017; Nomura et al. 2020; Zhu +et al. 2022) is intrinsically linked to the spectral energy distribution +(SED) of the continuum which is responsible for both ionizing the +transitions and then accelerating the flow by providing the source +of radiation pressure. By considering how the SED changes with +SMBH mass and accretion rate, authors such as Giustini & Proga +(2019) have developed unifying frameworks which make testable +predictions for luminous AGN based on the physics of radiation line +driven winds. +The ionizing continuum SED depends on the structure of the +accretion flow, which in turn is set by the SMBH mass 𝑀BH, the +mass-normalised accretion rate �𝑚 = +�𝑀BH/ �𝑀Edd and the SMBH +spin 𝑎∗. Empirically, the optical–to–X-ray SEDs of AGN are seen to +contain at least three distinct components (Elvis et al. 1994; Casebeer +et al. 2006; Leighly et al. 2007; Done et al. 2012; Jin et al. 2012). +First, any optically thick accretion disc will emit thermally, with +larger radii being cooler, giving rise to a multi-temperature blackbody +which is expected to peak in the near-ultraviolet (𝑀BH > 108𝑀⊙), +far-ultraviolet or soft X-rays (𝑀BH < 108𝑀⊙). This part of the SED +is commonly observed to peak around 1100 Å (Shang et al. 2005; +Laor & Davis 2014; Stevans et al. 2014; Vanden Berk et al. 2020), +although this peak might be expected to depend on 𝑀BH if larger +SMBHs have accretion flows which truncate at lower temperatures +(eq. 5.3.1 of Novikov & Thorne 1973). Second, a hot Comptonised +‘corona’ emits a non-thermal power law which dominates the X- +ray continuum above 1 keV (Haardt & Maraschi 1991; Titarchuk +1994). Finally, a ‘soft excess’ is seen in the X-rays below ≈1 keV, +which is usually attributed to an intermediate warm Comptonising +component (Petrucci et al. 2018). This soft excess may be a significant +contributor to the ionizing SED in the ≈100–1000 Å (≈10–100 eV) +range, where many of the ultraviolet transitions are excited, but where +direct observations of the continuum are not possible due to the high +opacity of neutral hydrogen along the line-of-sight. +For a line-driven disc-wind to emerge, the system needs strong +ultraviolet emission to produce sufficient radiation pressure, but also +a soft enough SED to avoid over-ionizing the gas (Murray et al. 1995; +Higginbottom et al. 2014). For each relevant line, the flux at the line +energy combined with the line opacity determines the line-driving +boost beyond radiation pressure from Thomson scattering. The line +opacity depends on the ionization state, which is primarily sensitive +MNRAS 000, 1–20 (2023) + +C iv and He ii emission in quasars +3 +to the flux beyond some ionization edge (64 eV for C iv). Line driving +results when this effect is summed across many lines, each with their +own energies, leading to a complex interplay between the flux of +the SED underneath all the relevant lines in the ultraviolet and the +flux of the SED beyond all the relevant ionization edges. Giustini & +Proga (2019) suggest that both 𝐿/𝐿Edd > 0.25 and 𝑀BH > 108𝑀⊙ +are required to satisfy these criteria and hence to power a strong +outflow through radiation line driving. Giustini & Proga (2019) also +expect the 𝑀BH dependence of the observed outflow properties to be +different above and below an �𝑚 of around 0.25, where they expect +the cold, optically thick accretion disc to extend down towards the +innermost stable circular orbit (ISCO) and replace the hot, optically +thin, inner accretion flow which is present at lower accretion rates. +In other words, they require 𝐿/𝐿Edd > 0.25 to ensure emission from +thermal disc emission dominates over that from the hot corona, to +accelerate a strong line-driven wind without over-ionizing the gas. +With the quantity and quality of spectroscopic data which are now +available from large surveys, these predictions from the Giustini & +Proga (2019) framework can be tested empirically. +1.3 Observational probes of quasar SEDs +From an observational viewpoint, it is relatively easy to constrain the +SED of an unobscured type-1 AGN in the rest-frame infrared, optical +and X-ray wavebands, as photometric measurements can place direct +constraints on the emission. For example, the strength of the rest- +frame 2 keV X-ray continuum relative to the near-ultraviolet contin- +uum has been shown to anti-correlate with the ultraviolet continuum +luminosity in the so-called 𝛼ox–𝐿2500 Å relation (Avni & Tananbaum +1982, 1986; Steffen et al. 2006; Just et al. 2007; Lusso & Risaliti +2016; Timlin et al. 2021), and the fractional contribution of the 2- +10 keV emission to the total bolometric power of the AGN is known +to vary as a function of the accretion rate �𝑚 (Vasudevan & Fabian +2007, 2009). However, while this X-ray waveband can make a not +insignificant contribution to the total emitted energy, it contributes a +negligible number of ionizing photons to the photoionization budget +of the BLR gas (see Appendix A). The number of ionizing photons +is instead dominated by photons at the ionization edges themselves, +which is of the order of 10-100 eV for the ultraviolet BLR (e.g. pro- +duction edges of 15 eV for Mg ii, 54 eV for He ii and 64 eV for C iv). +This extreme ultraviolet (EUV) part of the SED is not directly ob- +servable due to intervening absorption along the line-of-sight, but +plays a crucial role in the physics of the BLR. To add to the complex- +ity, the relative contribution of the warm Comptonising soft excess +to the total EUV emission is likely to be varying as a function of +𝑀BH and �𝑚, meaning that the observable 2 keV continuum may not +be a reliable proxy for the strength of the EUV SED at the ionization +edges. +C iv is a resonant doublet transition with a complicated ionic struc- +ture, so the strength of C iv emission is not necessarily a good tracer +of the ionizing SED. However, it is instead possible to probe the +EUV continuum using the He ii 𝜆1640 recombination line, which +arises from a simple hydrogenic (i.e. single electron) system. Under +the assumptions that the He ii emitting region is in equilibrium and +that the He ii continuum is optically thick, the total rate of He ii- +ionizing photons must balance the total number of recombinations +such that each He ii 𝜆1640 line photon can be associated with an ion- +izing continuum photon at or above 54 eV. This method was first used +by Zanstra (1929) to infer stellar temperatures using the strength of +Hydrogen recombination lines (section 5.10 of Osterbrock & Ferland +2006). Following previous works (Mathews & Ferland 1987; Baskin +et al. 2013; Ferland et al. 2020; Timlin et al. 2021), we will use the +strength of He ii as a proxy for the strength of the ‘unseen’ EUV +continuum which is ionizing the BLR. +1.4 This work +The first goal of this paper is to provide an up-to-date summary of +our knowledge of the ultraviolet spectral properties of type-1 quasars, +using the final data release (DR17) from the fourth iteration of SDSS. +This sample contains an order of magnitude more quasars than the +SDSS DR7 sample used by Richards et al. (2011). The large sample +size allows us to consider the emission properties as a function of +both mass and Eddington ratio simultaneously, and thus provide a +test of current SED models and disc-wind theories, which is our +second goal. +To best compare with theory and simulations, we present observed +quantities such as the C iv blueshift, He ii EW, and 𝛼ox as a function +of three physical parameters: the ultraviolet continuum luminosity, +the SMBH mass estimated from the Mg ii 𝜆2800 emission line, and +the inferred Eddington ratio. This relatively simple exercise has long +been used to gain insight into the physics of AGN (Dibai 1980), but +is subtly different from purely empirical approaches which observe +trends in emission line properties (e.g. EV1 or the C iv blueshift-EW +plane) and then try to infer which underlying physical parameters are +driving those trends. By contrast, theoretical models make predic- +tions for the SED and outflow properties as a function of the SMBH +mass and accretion rate. In this work we confront such predictions +directly with observations, showing that the ultraviolet emission lines +display different behaviour above a threshold of 𝐿bol/𝐿Edd ≈ 0.2, +consistent with predictions for radiation line-driven winds, and find- +ing good qualitative agreement between state-of-the-art SED models +and observed continuum tracers in regions of parameter space where +the models were not calibrated. +The structure of this paper is as follows. In Section 2 we present the +observational data, while in Section 3 we describe the SED models +to which we compare. We present our key results in Section 4 and +discuss their implications and limitations in Section 5. Throughout +this work, wavelengths are given in vacuum in units of Ångströms, +and we assume a flat ΛCDM cosmology with Ω𝑚 = 0.27, ΩΛ = 0.73 +and 𝐻0 = 71 km s−1 Mpc−1. Energies, frequencies and wavelengths +are given in the rest-frame unless stated otherwise. +2 OBSERVATIONAL DATA +2.1 Rest-frame ultraviolet spectra +The first aim of this paper is to quantify the behaviour of He ii 𝜆1640 +and C iv 𝜆1549 as a function of SMBH mass 𝑀BH and Eddington +ratio 𝐿/𝐿Edd. The 𝑀BH inferred from single-epoch measurements +of C iv is known to be biased as a function of the emission line +properties (Baskin & Laor 2005; Shen et al. 2008; Coatman et al. +2016, 2017; Mejía-Restrepo et al. 2018), so we will instead use the +velocity width of the Mg ii 𝜆2800 line to infer 𝑀BH. We construct +a sample of quasars from the SDSS with coverage of rest-frame +wavelengths 1450-3000 Å to include C iv, He ii and Mg ii (Fig. 1). +The original selection of the SDSS DR17 quasar sample was de- +scribed by Lyke et al. (2020) and Abdurro’uf et al. (2022). We post- +process each spectrum using a sky subtraction routine conceptually +similar to that described by Wild & Hewett (2005)1. Systemic red- +1 Measurements of spectrum properties derived from observed-frame wave- +lengths >6700 Å improve somewhat but none of the results, or conclusions, +MNRAS 000, 1–20 (2023) + +4 +M. J. Temple et al. +108 +109 +1010 +MBH [M⊙] +10−2 +10−1 +100 +L/LEdd +10 +20 +40 +100 +200 +400 +1000 +2000 +Number of quasars per bin +1500 +1550 +1600 +1650 +Rest Wavelength [˚A] +1 +2 +3 +fλ/f1700˚ +A +CIV +HeII +1400 +1600 +1800 +2000 +2200 +2400 +2600 +2800 +3000 +Rest Wavelength [˚A] +1041 +1042 +1043 +1044 +Lλ [erg s−1 ˚A−1] +10−0.2 < L/LEdd < 10−0.1 +109.4 < MBH/M⊙ < 109.5 +10−1.1 < L/LEdd < 10−1.0 +109.7 < MBH/M⊙ < 109.8 +10−0.5 < L/LEdd < 10−0.4 +108.4 < MBH/M⊙ < 108.5 +10−1.6 < L/LEdd < 10−1.5 +108.9 < MBH/M⊙ < 109.0 +MgII +CIV +HeII +Figure 1. Top left panel: the distribution of our sample of 191 391 quasars with redshifts 1.5 < 𝑧 < 2.65 in the 𝑀BH–𝐿/𝐿Edd plane. Throughout this work, we +only consider hexagonal bins where there are five or more quasars per bin. By construction, the FWHM of Mg ii 𝜆2800 increases from top-left to bottom-right +of this parameter space, while the 3000 Å continuum luminosity increases from bottom-left to top-right. Right panel: composite spectra taken from different +regions of the 𝑀BH–𝐿/𝐿Edd plane, as indicated by coloured boxes in the top left panel. A diversity of emission line properties is seen with changing SMBH +mass and Eddington ratio. Bottom left panel: comparing the composite spectra in the region around C iv 𝜆1549 and He ii 𝜆1640. Here the spectra have been +normalised at 1700 Å and plotted on a linear y-axis. The EW of He ii can be seen to correlate with the profile of C iv: the high-mass, high-Eddington composite +in blue displays weak lines and blueshifted C iv while the low-mass, high-Eddington composite in green shows much stronger line emission with no blue excess +in C iv, consistent with fig. 11 of Richards et al. (2011) and fig. A2 of Rankine et al. (2020). The difference here is that, instead of being constructed from C iv +or C iii] emission properties, objects were included based on the FWHM of Mg ii and 𝐿3000 to represent regions of the 𝑀BH–𝐿/𝐿Edd plane, and also that the +larger sample from SDSS DR17 includes fainter objects such as those contributing to the composite in red. Composite spectra spanning the full range of the +𝑀BH–𝐿/𝐿Edd space will be made available as supplemental online-only material with the journal. +shifts are calculated as described in section 3 of Rankine et al. (2020). +Our redshift estimation routine uses the rest-frame 1600-3000 Å re- +gion, deliberately excluding the C iv emission line, which is a key +difference compared to the approach employed in the SDSS quasar +catalogues. The improved redshifts and sky-subtracted spectra will be +described in a forthcoming publication by P. C. Hewett. To measure +the emission line properties, we employ the spectral reconstructions +from the Mean-Field Independent Component Analysis (ICA) car- +ried out by Rankine et al. (2020), which we have successfully used in +our previous investigations into quasar emission line physics (Tem- +ple et al. 2020, 2021a,b). The ICA-reconstructions provide a signif- +icant improvement in the measurement of emission line properties, +reducing the impact of the modest signal-to-noise ratio in the origi- +nal spectra and the effect of weak absorption lines (e.g. intervening +or outflowing C iv 𝜆𝜆1548,1550 doublets). We exclude objects with +broad low-ionization absorption features (LoBALs) and poor ICA +reconstructions. To include both C iv and Mg ii in the observed spec- +trum, we limit our sample to redshifts 1.5 < 𝑧 < 2.65. Spectra from +before the start of the BOSS survey (MJD 55000) were observed +using the original SDSS spectrograph which had a more limited +wavelength coverage; for these objects we require 1.6 < 𝑧 < 2.2 to +ensure coverage of C iv and Mg ii. The spectra of quasars are required +to possess a mean signal-to-noise ratio (per 69 km s−1SDSS pixel) +of this paper change if the original DR17 reductions of the spectra are used +instead. +≥3.0 over the rest-frame interval 1700-2200 Å. These criteria leave +a sample of 191 391 quasars. +To compute the EW of C iv emission, a power law continuum is de- +fined using the median flux in the 1445-1465 and 1700-1705 Å wave- +length windows. This continuum is then subtracted from the spec- +trum to isolate the line flux in the 1500-1600 Å wavelength region. +The He ii EW is measured in the same way across the 1620-1650 Å +wavelength region, using windows at 1610-1620 and 1700-1705 Å to +define the continuum model. The C iv emission line ‘blueshift’ is de- +fined as the Doppler shift of the wavelength bisecting the continuum- +subtracted line flux: +C iv blueshift ≡ 𝑐 × +�𝜆rest − 𝜆median +𝜆rest +� +(1) +where 𝑐 is the speed of light, 𝜆median is the rest-frame wavelength +of the observed line centroid, and 𝜆rest = 1549.48 Å is the mean +rest-frame wavelength of the C iv 𝜆𝜆1548.19,1550.77 doublet. +2.2 X-ray data +In addition to the rest-frame ultraviolet emission features, we can +use the rest-frame 2 keV X-ray continuum emission to gain further +insight into the spectral energy distributions of the quasars in our +sample. We therefore cross-match our sample of 191 391 objects +to various X-ray catalogues from the literature, in order to build a +large sample of rest-frame 2 keV measurements. 4189 objects from +our sample of 1.5 < 𝑧 < 2.65 objects with ultraviolet spectra are +MNRAS 000, 1–20 (2023) + +C iv and He ii emission in quasars +5 +included in the recent study of quasar X-ray properties by Rivera +et al. (2022), including 2820 with XMM-Newton observations from +Lusso et al. (2020), 1337 with Chandra observations from Timlin +et al. (2020), and 32 with XMM-Newton observations from Liu et al. +(2020). We make use of data from the second ROSAT All-Sky Survey +(2RXS; Boller et al. 2016) for objects included in the SDSS DR16 +SPIDERS programme (Dwelly et al. 2017; Comparat et al. 2020). The +flux limit for this survey is relatively bright so we use the Bayesian +measurements described by Coffey et al. (2019) which account for +the Eddington bias. A total of 45 objects from 2RXS are included in +our sample. Finally, we include 13, 11, and 8 objects with Chandra +observations from Timlin et al. (2021), Ni et al. (2018, 2022) and Fu +et al. (2022) respectively. These last three sub-samples were selected +to have high 𝐿UV, weak C iv and strong C iv respectively, but the +number of quasars is small and our results would be unchanged if +we were to exclude them. To improve our statistics for the number +of objects with X-ray detections, we augment our sample with 1059 +objects from the eROSITA Final Equatorial Depth Survey (eFEDS; +Liu et al. 2022) with redshifts 1.5 < 𝑧 < 2.65 and with SDSS griz +photometry (Lyke et al. 2020) or HSC griz photometry from the +eFEDS catalogue (Salvato et al. 2022). +The compilation results in a sample of 5325 quasars with measure- +ments of both their ultraviolet (2500 Å) and X-ray (2 keV) continuum +fluxes. We use directly observed X-ray fluxes with no spectral fitting, +though we have verified that the conclusions of this work would not +change if we excluded objects which may be affected by absorp- +tion. From these fluxes we compute luminosities assuming a consis- +tent cosmology (Section 1.4) across all sub-samples. We then infer +𝛼ox, the logarithm of the ratio of the rest-frame 2 keV and 2500 Å +monochromatic luminosities: +𝛼ox = log10 +� +𝜈𝐿𝜈 +� +2 keV − log10 +� +𝜈𝐿𝜈 +� +2500 Å, +(2) +as a measure of the relative strength of the X-ray emission in each +source. Objects with smaller (i.e. more negative) 𝛼ox have weaker +2 keV X-ray emission relative to the ultraviolet continuum. +2.3 Black hole masses and Eddington ratios +We estimate SMBH masses using the single-epoch virial estimator +described by Vestergaard & Osmer (2009), using the full width at +half maximum (FWHM) of the Mg ii line: +𝑀BH = 106.86 +� +FWHM(Mg ii) +1000 km s−1 +�2 � +𝐿3000 +1044 erg s−1 +�0.5 +𝑀⊙. +(3) +where 𝐿3000 is the rest-frame monochromatic continuum luminosity +𝜈𝐿𝜈 at 3000 Å. This 𝑀BH estimator assumes a relationship between +the radius of the Mg ii-emitting region and the observed 𝐿3000 which +is independent of the shape of the ionizing SED, or more generally, +independent of any changes in the accretion disc structure which +may arise with changing 𝑀BH or accretion rate. We discuss this +assumption further in Section 5.1.2. We infer the FWHM of Mg ii +from our ICA reconstructions, which provide a more robust model +of the intrinsic Mg ii profile than a conventional Gaussian fit in low +signal-to-noise (𝑆/𝑁) spectra. Using a sub-sample with 𝑆/𝑁 > 10, +we have verified that our Mg ii FWHM measurements are consistent +with those obtained from fitting a single Gaussian to Mg ii together +with an iron template (Vestergaard & Wilkes 2001) using the routine +described by Shen et al. (2011). The key results of this paper would +not change if we were to instead use such a Gaussian model for Mg ii, +but there would be significantly more scatter in lower luminosity +regions of parameter space where the spectral 𝑆/𝑁 is poorer on +average. The error budget on our resulting 𝑀BH is dominated by the +0.55 dex uncertainty on the single-epoch estimator as described by +Vestergaard & Osmer (2009). +We infer 𝐿3000 by fitting a quasar SED model (Temple et al. +2021c) to griz photometry. For sources in SDSS DR16 we use the +SDSS photometry reported by Lyke et al. (2020), and for eFEDS- +selected sources in SDSS DR17 we use the Hyper-Suprime Cam +(HSC) photometry reported by Salvato et al. (2022). Eddington lu- +minosities are calculated in the usual way, balancing the gravitational +and radiation forces in a Hydrogen-only plasma, and assuming the +dominant source of opacity is Thomson electron scattering: +𝐿Edd = 4𝜋𝐺𝑀BH𝑚p𝑐 +𝜎T += 1.26 × 1038 +� 𝑀BH +𝑀⊙ +� +erg s−1 . +(4) +The Eddington ratio 𝐿bol/𝐿Edd (hereafter 𝐿/𝐿Edd) is then estimated +assuming a constant bolometric correction of 𝐿bol = 5.15 × 𝐿3000. +We discuss this assumption further in Section 5.1.1, and show how +our key observables depend directly on FWHM(Mg ii) and 𝐿3000 in +Appendix B. +Our sample of 191 391 quasars spans 2.5 dex in luminosity, with +𝐿3000 ≈ 1044.5−47 erg s−1 and 𝐿bol ≈ 1045−47.5 erg s−1 . We infer +SMBH masses in the 108−10𝑀⊙ range and Eddington ratios from +0.01 to unity, with the distribution of our sample shown in Fig. 1. +3 MODELING THE QUASAR SED +Our second goal is to confront observational data with models for +accretion and outflow in quasars; more specifically, we aim to test +if the changes in observed emission line and continuum properties +with 𝑀BH and Eddington ratio are consistent with theoretical models +for the SED of the ionizing continuum. To this end, we make use +of qsosed2 (Kubota & Done 2018). We used the implementation of +qsosed in xspec (Arnaud 1996) to calculate SEDs, via the PyXSPEC +python wrapper (Gordon & Arnaud 2021). +In qsosed, the radiation originates from three characteristic re- +gions: an outer thermal disc, an inner hot Comptonising ‘corona’ +and an intermediate warm Comptonising component. These three +regions are assumed to be radially stratified as defined by four criti- +cal radii: 𝑅ISCO < 𝑅hot < 𝑅warm < 𝑅out. The inner and outer radii +are defined by the radius of the innermost stable circular orbit 𝑅ISCO +and the self-gravitation radius 𝑅out. The hot X-ray component origi- +nates from 𝑅ISCO < 𝑅 < 𝑅hot, and has a luminosity set by the sum +of the directly dissipated power, 𝐿diss,hot, and the seed photon lumi- +nosity, 𝐿seed. One of the key aspects of the model is the empirically +motivated assumption that the dissipated power is always 2 per cent +of the Eddington luminosity; this constraint defines the value of 𝑅hot. +The outer radius of the warm Comptonising component 𝑅warm is set +to be twice 𝑅hot. For 𝑅warm < 𝑅 < 𝑅out, the thermal disc component +is assumed to emit as described by Novikov & Thorne (1973). +qsosed has four physical input parameters: the cosine of the incli- +nation, cos𝑖, the SMBH mass, 𝑀BH, the dimensionless spin param- +eter, 𝑎∗, and the Eddington-scaled accretion rate, �𝑚 ≡ �𝑀acc/ �𝑀Edd. +We fix cos𝑖 = 0.5 and calculate grids of models in (𝑀BH, �𝑚) parame- +ter space, for non- and maximally-spinning SMBHs 𝑎∗ ∈ (0, 0.998). +We calculate models with 21 logarithmically-spaced grid points in +each direction, spanning the ranges 8 ≤ log(𝑀BH/𝑀⊙) ≤ 10 and +−1.65 ≤ log �𝑚 ≤ 0, corresponding to intervals of 0.1 and 0.0825 dex. +To compare with observations, we take the input SMBH mass and +2 https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/ +node132.html +MNRAS 000, 1–20 (2023) + +6 +M. J. Temple et al. +calculate the Eddington ratio from 𝐿3000 using the same bolometric +correction of 5.15 that we apply to the observational data (but see +Section 5.1.1 and Appendix C). Here, and in Section 4.2, we present +models for only the non-spinning case, as these are in much better +agreement with the data. We discuss the impact of SMBH spin and +system inclination in Section 5.2.3 and models with 𝑎∗ = 0.998 are +presented in Appendix D. +Although the emission line properties must depend on the ioniz- +ing SED, the exact relationship between, for example, C iv EW and +the SED is complex due to a number of confounding factors such as +BLR geometry, density and radiative transfer. The relationship to any +kinematic signatures such as C iv blueshift is even more complicated +and would require a physical model for the line formation region and +associated flow dynamics. A somewhat simpler case is the EW of +He ii 𝜆1640, which is a recombination line and therefore a reasonable +‘photon counter’. He ii has history as a tracer of the EUV continuum: +for example, in cataclysmic variables Hoare & Drew (1991) applied +a modified Zanstra (1929) method to infer boundary layer temper- +atures, and in quasars, Leighly (2004) note that a high He ii EW is +indicative of a strong X-ray continuum. Assuming Case B recombi- +nation, Mathews & Ferland (1987) give the He ii 𝜆1640 EW in terms +of the 228 Å continuum flux. Their equation can be inverted to give +the proportionality +𝐹𝜈(𝜆228) +𝐹𝜈(𝜆1640) ∝ EW(He ii 𝜆1640) Ω +4𝜋 , +(5) +where Ω/4𝜋 is the covering fraction and the proportionality constant +is dependent on the shape of the SED (Mathews & Ferland 1987 +considered a power law in 𝐹𝜈 at 228 Å). In this work we assume, +based on the above equation, that the observed He ii 𝜆1640 EW is a +reasonable proxy for the ratio of continuum luminosities 𝐿228/𝐿1640. +In Fig. 2 we present output SEDs from qsosed, in which the three +radially stratified components can be seen as separate ‘bumps’ in +the spectrum. In these plots, we show how the model SEDs change +as a function of Eddington-scaled accretion rate, �𝑚 (for fixed mass, +top panel) and SMBH mass, 𝑀BH (for fixed �𝑚, bottom panel). The +important frequencies for determining He ii EW (corresponding to +228 Å and 1640 Å) and 𝛼ox (corresponding to 2500 Å and 2 keV) are +marked. Increasing �𝑚 increases the overall luminosity of the system +and pushes the peak of the outer thermal disc component to higher +frequencies. Simultaneously, the hard X-ray slope becomes signifi- +cantly softer and 𝐿2keV only increases slowly. As a result, the higher +Eddington fraction objects are more X-ray weak relative to their ul- +traviolet flux. Increasing 𝑀BH also increases the total luminosity, but +now the peak of the thermal component moves to lower frequencies +and the hard X-ray slope stays fairly constant. In both panels of the +plot the peak of the SED can be found on either side of the low +frequency pivot points for both He ii EW and 𝛼ox, resulting in an +interesting interplay between these quantities and the fundamental +AGN parameters. +4 RESULTS +4.1 Observed properties in 𝑀BH–𝐿/𝐿Edd space +The first observational result from this work is the behaviour of +the C iv 𝜆1549 emission line morphology as a function of SMBH +mass (𝑀BH) and Eddington ratio (𝐿/𝐿Edd), shown in Fig. 3. In the +left panel, we show the C iv emission line blueshift (as defined in +Eq. 1) and in the right panel the EW of C iv. In the top panels, +consistent with previous works, we find that more luminous quasars +show weaker emission line strengths relative to the continuum and +1044 +1045 +1046 +1047 +νLν (erg s−1) +MBH = 109 M⊙ +10−1 +100 +˙m +1014 +1015 +1016 +1017 +1018 +1019 +ν (Hz) +1044 +1045 +1046 +1047 +νLν (erg s−1) +He II +αox +˙m = 0.15 +108 +109 +1010 +MBH [M⊙] +Figure 2. Output SEDs in 𝜈𝐿𝜈 units from qsosed for 𝑎∗ = 0 and cos 𝑖 = 0.5. +The vertical lines show, from left to right, the frequencies at 2500 Å, 1640 Å, +228 Å (= 54 eV) and 2 keV which together determine 𝛼ox and the EW of +He ii 𝜆1640. The He ii ionization edge at 54 eV (1.3×1016 Hz) lies in the EUV +regime where the intermediate warm Comptonising component in qsosed is +most important, but the EW of He ii can also be seen to depend on the location +of the peak of the ionizing SED. Top panel: SEDs with fixed SMBH mass of +109 𝑀⊙ and varying �𝑚 in logarithmic intervals. As �𝑚 increases the peak of +the SED moves to the blue, the luminosity increases, and the hard X-ray power +law spectral index becomes softer. Bottom panel: SEDs with fixed �𝑚 = 0.15 +and varying 𝑀BH in logarithmic intervals. As SMBH mass increases the peak +of the SED moves to the red, and the luminosity increases. A maximal spin +analogue to this plot is shown in Fig. D1. +stronger emission line blueshifts. However, when considering the +observed C iv properties as a function of both 𝑀BH and 𝐿/𝐿Edd +(bottom panel), we see a more complicated behaviour. To observe +the strongest C iv blueshifts (which are associated with the smallest +EWs), we need to look at objects with both 𝑀BH ≳ 109𝑀⊙ and +𝐿/𝐿Edd ≳ 0.2. Moreover, the contours of constant C iv blueshift +follow acute-angled ‘wedge’ shapes, which are somewhat orthogonal +to lines of constant luminosity (running diagonally top-left to bottom- +right in the 𝑀BH–𝐿/𝐿Edd space). At the same time, objects with the +strongest C iv EWs ≳ 100 Å, which have strong symmetric emission +with little or no blueshift, are found at high 𝐿/𝐿Edd ≳ 0.2 and +relatively low 𝑀BH ≲ 109𝑀⊙. +To help us to understand the physical drivers behind the trends +seen in C iv, in Figs. 4 and 5 we also show 𝛼ox and He ii EW across +the same 𝑀BH–𝐿/𝐿Edd parameter space. The 𝛼ox behaviour is as +expected from previous works (e.g. Mitchell et al. 2022), largely with +more luminous objects displaying relatively weaker X-ray emission +which is quantified by a more negative 𝛼ox. A more interesting result +MNRAS 000, 1–20 (2023) + +C iv and He ii emission in quasars +7 +1045 +1046 +1047 +νLν|3000˚ +A [erg s−1] +108 +109 +1010 +MBH [M⊙] +10−2 +10−1 +100 +L/LEdd +0 +250 +500 +750 +1000 +1250 +1500 +1750 +2000 +CIV λ1550 blueshift [km s−1] +1045 +1046 +1047 +νLν|3000˚ +A [erg s−1] +108 +109 +1010 +MBH [M⊙] +10−2 +10−1 +100 +L/LEdd +20 +40 +60 +80 +100 +120 +EW(CIV λ1550) [˚A] +Figure 3. The median observed C iv blueshift (left) and EW (right) in bins of SMBH mass, 3000 Å ultraviolet continuum luminosity (top) and Eddington ratio +(bottom). Data are shown only for bins which contain five or more objects. The C iv blueshift and EW are seen to anti-correlate: areas of parameter space with +strong blueshifts have weak EW and vice versa. 𝐿/𝐿Edd ≳ 0.2 is a necessary but not sufficient condition for observing the largest C iv blueshifts. The strongest +C iv blueshifts are observed only at large SMBH mass and large Eddington ratio, while high EW C iv emission is observed at large Eddington ratio and smaller +mass. The Baldwin effect can be observed in the sense that objects with brighter 3000 Å luminosities tend to have weaker C iv EWs on average. However, the +C iv EW behaviour as a function of 𝑀BH and 𝐿/𝐿Edd shows that the underlying drivers of the Baldwin effect are more complicated than a simple dependence +on the ultraviolet luminosity. +is seen in the EW of He ii, which is even more striking than the +behaviour seen in C iv. With the He ii EW, there is a clear transition +around 𝐿/𝐿Edd ≈ 0.2, with both the strongest and weakest line +emission only seen above this threshold. Below this Eddington limit, +there is little change in the average line properties as a function +of mass, but at 𝐿/𝐿Edd ≳ 0.2 there is a strong mass dependence +with diagonal wedge-shaped contours similar to those observed in +C iv. By contrast, the contours of constant 𝛼ox are much less closely +aligned with contours of constant C iv blueshift. +To test the robustness of these trends, we divide the 𝑀BH–𝐿/𝐿Edd +into square bins of 0.1 by 0.1 dex and compute the median absolute +deviation (MAD) in each bin. The typical MAD is 290 km s−1 in +C iv blueshift, 13 Å in C iv EW and 0.5 Å in He ii EW. The typical +scatter within each bin is therefore significantly less than the dynamic +range in the average emission line properties shown in Figs. 3 and +5, meaning that one is unlikely to find individual objects which go +against the overall trend of the population. Dividing through by the +median in each bin, the typical MAD/median in each bin is 0.29 +and 0.24 for the He ii and C iv EWs respectively, meaning that the +typical range of emission line EW within each 𝑀BH–𝐿/𝐿Edd bin is +a factor of 3.5 and 4 for He ii and C iv respectively, compared with +the dynamic range of more than a factor of six seen in the median +per-bin line properties. +4.2 Comparison with model SEDs +In the right-hand panels of Figs. 4 and 5 we show how 𝛼ox and +𝐿228/𝐿1640, respectively, vary with mass and Eddington fraction, as +modeled by qsosed. These plots can be compared to the respective +plots from the observational sample (left-hand panels), albeit with +some caveats regarding bolometric corrections (Section 5.1.1) and +𝑀BH estimates (Section 5.1.2). In a qualitative sense, the models do +a reasonably good job of reproducing the trends observed in the data. +Focusing first on 𝛼ox, we can see that the general trend of decreasing +𝛼ox with Eddington fraction is reproduced, and, in addition, the +gradient is stronger at high 𝑀BH, as observed in the data. To put +this another way, in both the data and model results, the contour of +fixed 𝛼ox curves around, from being nearly horizontal at high 𝑀BH +to being closer to vertical at low 𝑀BH. The dynamic range of model +𝛼ox values is comparable to that observed, but the models do not +produce soft enough spectra to match the data; 𝛼ox ≈ −1.9 can be +found in some bins in the quasar sample but the minimum value of +𝛼ox in the models is −1.79. +The comparison of the model 𝐿228/𝐿1640 ratio and the observed +He ii 𝜆1640 EW is also broadly encouraging, at least at relatively +high Eddington fractions. This finding is perhaps more interesting as +the He ii EW is probing a portion of the SED that is not accessible +directly. The basic behaviour, of decreasing He ii EW with 𝑀BH at +MNRAS 000, 1–20 (2023) + +8 +M. J. Temple et al. +1045 +1046 +1047 +νLν|3000˚ +A [erg s−1] +108 +109 +1010 +MBH [M⊙] +10−2 +10−1 +100 +L/LEdd +−1.9 +−1.8 +−1.7 +−1.6 +−1.5 +−1.4 +−1.3 +−1.2 +αox +1045 +1046 +1047 +ν Lν|3000˚ +A [erg s−1] +108 +109 +1010 +MBH [M⊙] +10−2 +10−1 +100 +L/LEdd +−1.9 +−1.8 +−1.7 +−1.6 +−1.5 +−1.4 +−1.3 +−1.2 +αox +Figure 4. Left panel: The median observed 𝛼ox in bins of SMBH mass, 3000 Å ultraviolet continuum luminosity (top) and Eddington ratio (bottom) for the +5325 objects from our sample with 2 keV X-ray measurements. Data are shown only for bins which contain five or more objects. Right panel: the predicted 𝛼ox +from low spin qsosed models in the same parameter space. The observations show good agreement with the models, with 𝛼ox more negative (i.e. more X-ray +weak) in objects with brighter ultraviolet luminosities. In Fig. D1 we show equivalent models but with high spin, which do not show such agreement with the +observations, suggesting that the 𝑧 ≈ 2 SDSS quasar population may be more consistent with low SMBH spins on average. +high Eddington fractions, is well matched by the models. The models +also capture the diagonal contours of constant He ii EW, in which +the transition to low He ii EWs occurs at higher masses for higher +Eddington fractions. As discussed above, at low Eddington fractions +(𝐿/𝐿Edd ≲ 0.2), something fundamentally switches in the data, with +gradients generally being shallower and along a different direction in +the parameter space. This relatively sharp change is not reproduced +by the models, and may be telling us something fundamental about +the quasar accretion process (see Section 5.2.1 for a discussion). +5 DISCUSSION +We have quantified the average behaviour of C iv 𝜆1549, He ii 𝜆1640 +and 𝛼ox as a function of both 𝑀BH and 𝐿/𝐿Edd, and compared our +observations with predictions from qsosed models. In this section +we now discuss these results. We first outline the key caveats in +our findings (Sections 5.1.1 and 5.1.2), before discussing possible +interpretations of our results within the context of AGN accretion and +outflow theories (Sections 5.2.1 and 5.2.2). Finally, we discuss some +wider implications and possible future applications (Section 5.3), +before summarizing our key conclusions in Section 6. +5.1 Key assumptions and limitations +5.1.1 Bolometric corrections +A large part of this work has attempted to quantify the ‘unseen’ ex- +treme ultraviolet (EUV) portion of the SED which is not directly +observable, but which can instead be probed via the He ii emission +line. This portion of the SED contributes a significant amount to the +bolometric luminosity of a quasar. To estimate bolometric luminosi- +ties (and Eddington ratios 𝐿bol/𝐿Edd), we have assumed a constant +bolometric correction 𝑓bol ≡ 𝐿bol/𝐿3000 of 5.15, consistent with +previous works in the literature (e.g. Richards et al. 2006; Krawczyk +et al. 2013). However, we have also shown that the He ii strength +is changing as a function of 𝑀BH and 𝐿/𝐿Edd, so we expect the +strength of the EUV continuum and hence the bolometric correction +to be varying with 𝑀BH and 𝐿/𝐿Edd. Using our qsosed models, +we attempt to quantify this effect in Fig. 6. While our chosen value +of 𝑓bol = 5.15 lies within the range of values spanned by our grid +of model SEDs, there is variation of around a factor of two in 𝑓bol +depending on the values of 𝑀BH and 𝐿/𝐿Edd we consider. While +this could in principle lead to systematic biases in our estimation of +𝐿/𝐿Edd, we show in Appendix C that these biases are likely to be +small compared to the magnitude of the trends we observe. +We can however, briefly describe what might happen if we were to +adopt a non-constant bolometric correction when inferring 𝐿/𝐿Edd +from our observations. For two objects, both at �𝑚 = 0.2, the 𝑓bol +MNRAS 000, 1–20 (2023) + +C iv and He ii emission in quasars +9 +1045 +1046 +1047 +νLν|3000˚ +A [erg s−1] +108 +109 +1010 +MBH [M⊙] +10−2 +10−1 +100 +L/LEdd +0.5 +1 +2 +4 +EW(HeII λ1640) [˚A] +1045 +1046 +1047 +ν Lν|3000˚ +A [erg s−1] +108 +109 +1010 +MBH [M⊙] +10−2 +10−1 +100 +L/LEdd +0.03 +0.04 +0.06 +0.1 +L228/L1640 +Figure 5. Left panel: The median observed He ii EW in bins of SMBH mass, 3000 Å ultraviolet continuum luminosity (top) and Eddington ratio (bottom). +Data are shown only for bins which contain five or more objects. Right panel: the predicted strength of He ii ionizing photons at 228 Å relative to the 1640 Å +continuum from qsosed models. Above an Eddington ratio of ≈0.2, there is a strong trend as a function of SMBH mass, with high mass objects showing the +weakest He ii emission and low mass objects showing the strongest He ii emission. The model predictions show qualitatively similar behaviour in this region of +parameter space, explaining the diagonal contours in constant He ii. Below 𝐿/𝐿Edd ≲ 0.2, the observed He ii displays much weaker trends, and does not agree +with the model predictions, suggesting that in this regime either the SED models are less accurate or the structure of the BLR is changing. +inferred from the qsosed models would be ≈6 and ≈3 for 𝑀BH = +108𝑀⊙ and 1010𝑀⊙ respectively. This would skew the observations +in Fig. 5, moving the location of the strongest He ii EW (at low +𝑀BH) to larger 𝐿/𝐿Edd, more in line with the 𝐿/𝐿Edd threshold at +high 𝑀BH above which we see the weakest He ii and largest C iv +blueshifts. +5.1.2 Black hole mass estimates +As well as the assumption of a constant bolometric correction, we +have used a single-epoch virial estimator to estimate SMBH masses +throughout this work. The caveats associated with such estimates +are numerous and have been reviewed by Shen (2013). Here we +discuss some of the issues which are most relevant to our method +and results. Most notably, the BLR radius–luminosity relation (as +encoded through the virial 𝑓 factor) may depend on the shape of the +SED. Other uncertainties arising from (for example) orientation are +likely to be random, in the sense that they will add scatter to our +𝑀BH estimates but should not bias our results. While it is possible +that our observed distribution of quasars in the 𝑀BH–𝐿/𝐿Edd plane +is not the same as the intrinsic distribution, the fact that we do still +observe such striking behaviour in the He ii and C iv emission line +properties as a function of our inferred 𝑀BH and 𝐿/𝐿Edd is telling +us that any random scatter or noise in our 𝑀BH estimates is small +enough not to ‘wash out’ the observed trends. +We used the FWHM of the Mg ii line to estimate 𝑀BH. Shen +et al. (2008) showed that such Mg ii-derived 𝑀BH estimates corre- +late tightly with those derived from H𝛽 across the full 108−10 𝑀⊙ +mass range, with the distribution of log +� +𝑀H𝛽 +BH +�𝑀Mg ii +BH +� +following +a Gaussian with mean 0.034 and dispersion 0.22 dex. Shen & Liu +(2012) extended this analysis to higher redshifts and higher lumi- +nosities, more appropriate for the objects in this work, and again +found that the Mg ii properties remained well correlated with those +of H𝛽. The Mg ii-derived 𝑀BH estimates we use in this work are +therefore unlikely to be biased compared to those which we would +have derived from a single-epoch H𝛽 measurement. The possibility +remains, however, that such estimates are biased as a function of the +SED, or equivalently, as a function of 𝑀BH and 𝐿/𝐿Edd. +Early concerns about the universality of the BLR radius– +luminosity relation were discussed by Kaspi et al. (2005) and Collin +et al. (2006). More recently, various authors have tried to account +for possible SED-dependent biases in single-epoch 𝑀BH estimates +(Du & Wang 2019; Dalla Bontà et al. 2020; Fonseca Alvarez et al. +2020; Martínez-Aldama et al. 2020), either using the accretion rate +directly or by using the strength of optical iron emission 𝑅Fe ii as a +proxy. However, Khadka et al. (2022a,b) and Yu et al. (2022a) find +MNRAS 000, 1–20 (2023) + +10 +M. J. Temple et al. +1044 +1045 +1046 +1047 +νLν|3000˚ +A [erg s−1] +4 +6 +8 +10 +12 +fbol +fbol = 5.15 +108 +109 +1010 +MBH [M⊙] +Figure 6. The predicted bolometric correction, 𝑓bol ≡ 𝐿bol/𝐿3000, as a func- +tion of 𝐿3000, from qsosed models. The points are colour-coded by 𝑀BH with +a logarithmic normalisation, and points of constant mass are joined with solid +lines so that the trends with Eddington ratio can be understood by following +individual lines from left to right. The adopted bolometric correction in this +work, 𝑓bol = 5.15, is shown as a horizontal dashed line. 𝑓bol ranges from +≈ 3 − 10, and our adopted 𝑓bol is bounded by this range; however 𝑓bol does +have a clear dependence on mass and luminosity in the model SEDs. Our +assumption of a fixed 𝑓bol could lead to an artificially reduced dynamic range +in the inferred 𝐿bol at 𝑀BH = 108𝑀⊙ and an artificially increased range of +𝐿bol at 𝑀BH = 1010𝑀⊙. +the opposite result, with the inclusion of 𝑅Fe ii having no effect on +the scatter in either the Mg ii or H𝛽 radius–luminosity relations. +While the literature remains divided, we argue it is still true that +any SED-dependent bias in our single-epoch 𝑀BH estimates must be +contained within the scatter on the BLR radius–luminosity relation, +i.e. within 0.3-0.5 dex. This scatter is smaller than the range spanned +by our sample by a factor of ≈3, meaning that SED-dependent bi- +ases in our 𝑀BH cannot explain the observed trends presented in +Section 4.1. +5.2 Quasar physics +5.2.1 AGN accretion models +In Section 4.2, we compared our observations with predictions from +the qsosed models of Kubota & Done (2018). The predictions for 𝛼ox +made by these models have recently been tested over a much broader +parameter space (𝑀BH ≈ 107−10𝑀⊙ and 𝐿3000 ≈ 1043.5−47 erg s−1 ) +by Mitchell et al. (2022), who find that the qsosed model predicts +the optical and X-ray SED fairly well for 𝑀BH ≲ 109𝑀⊙, but that +at higher masses the outer accretion disc spectra are predicted to be +too cool to match the observed data, especially at lower Eddington +ratios. This finding is consistent with our result (in Fig. 4) that the +2 keV emission is predicted to be slightly stronger (relative to the +2500 Å emission) than observed at 𝑀BH ≈ 109.5𝑀⊙. +In this work we have also quantified the He ii emission, which +provides a new, complementary probe of the ionizing SED across +the 𝑀BH–𝐿/𝐿Edd space. In Section 4.2 we found that, for 𝐿/𝐿Edd ≳ +0.2, the observed He ii EW is qualitatively similar to the behaviour +of the 54 eV ionizing SED predicted by the qsosed models. The +observations are consistent with a scenario in which (at least for +𝐿/𝐿Edd ≳ 0.2) the strength of He ii emission is set directly by the +ionizing photon luminosity at 54 eV, and thus that He ii is providing +a probe of the EUV which is not directly observable. Moreover, the +observed He ii EW behaviour provides further evidence for the soft +excess to be an intermediate, warm Comptonising component which +behaves in the way in which the qsosed models predict. The strongest +and weakest 228 Å emission (relative to the 1640 Å continuum) are +both produced at high Eddington ratios, at low (≈ 108𝑀⊙) and high +(≈ 1010𝑀⊙) SMBH mass respectively. +However, the match between the observed He ii and the predicted +strength of the 54 eV ionizing luminosity is not perfect, especially +in the 𝐿/𝐿Edd ≲ 0.2 regime. This mismatch might suggest a decou- +pling between the He ii EW and the 228 Å continuum flux at these +Eddington ratios, perhaps if changes in the BLR covering factor lead +to differences in the fraction of the continuum source which is re- +processed into emission lines. Another possibility is that the He ii +continuum becomes optically thin, for instance if the density of the +BLR were to decrease (which could indicate the absence of a dense +outflow). Alternatively, the mismatch could suggest that the SED +models are inaccurate in this Eddington ratio regime. Intriguingly, +this regime is similar to the region of the 𝑀BH–𝐿2500 Å space where +Mitchell et al. (2022) find a mismatch between the observed and pre- +dicted 𝛼ox. Either way, the observed switch in He ii behaviour above +and below 𝐿/𝐿Edd ≈ 0.2 (which is not reflected in the qsosed mod- +els) suggests that something fundamental is changing in the structure +of either the BLR or the accretion flow. +C iv is a resonant doublet with a more complicated ionic structure +than He ii. However, the close correspondence of the C iv blueshift +and EW with the He ii EW, allied with the fact that the He ii behaviour +can be consistently explained with trends in the SED, suggests that the +C iv morphology is governed by accretion physics - specifically the +shape of the SED in the near and extreme ultraviolet regions. Given +the proximity of the C iv and He ii ionization edges, at 64 eV and +54 eV respectively, this result is perhaps unsurprising. More notable +is the fact that the observed 𝛼ox behaviour does not correspond so +well to the C iv morphology, as demonstrated by the differing trends +in Figs. 3 and 4. This shows how the 2 keV X-ray emission is a weaker +proxy than He ii for the physical mechanisms which control the C iv +emission. +Finally, the fact that our 𝐿/𝐿Edd ≳ 0.2 observations of He ii and +𝛼ox can be well explained by changes in the predicted model SEDs is +significant. The qsosed models were calibrated using observations +of just three objects with 𝑀BH = 108𝑀⊙ (Kubota & Done 2018), and +had not been critically assessed at higher SMBH masses prior to the +recent work of Mitchell et al. (2022). The fact that the models agree +with our observations across a relatively wide range of parameter +space was not necessarily to be expected. Moreover, the theoretical +models show that the different phenomenological behaviours ob- +served in 𝛼ox and He ii (Figs. 4 and 5) have natural origins in the +SED, and that the observations presented in this work are broadly in +agreement with our current understanding of the accretion physics in +AGN. +5.2.2 AGN outflow models +In this subsection we test the predictions made by Giustini & Proga +(2019), who summarize current understanding of AGN accretion and +outflow mechanisms with a particular focus on the physical condi- +tions required to drive powerful winds from the accretion disc through +radiation line driving. We note again that the picture described by +Giustini & Proga (2019) might not be the only plausible model for +AGN outflows, but we choose to compare with their picture as it pro- +MNRAS 000, 1–20 (2023) + +C iv and He ii emission in quasars +11 +vides clear testable predictions within a well-defined framework. In +particular, Giustini & Proga (2019) suggest that both 𝐿/𝐿Edd > 0.25 +and 𝑀BH > 108𝑀⊙ are required to power strong outflows from AGN +through radiation line driving: below these thresholds the X-ray flux +is strong enough to over-ionize material and the ultraviolet flux will +be too low to accelerate a line-driven wind. +For the purposes of this comparison, we assume that any blue- +wing excess in the C iv emission line profile is tracing an outflow +along the line-of-sight from the accretion disc, and hence that the +blueshift presented in Fig. 3 is a measure of the strength of emission +from the outflowing wind (Leighly & Moore 2004; Richards et al. +2011). The origin of the C iv emission line blueshift is still debated +(see Gaskell & Goosmann 2013, 2016, for an alternative view), but a +growing body of work is connecting the C iv emission morphology +with more unambiguous tracers of line driven winds. For example, +the strengths and velocities of broad C iv absorption troughs have +been shown to correlate with the C iv emission blueshift (Rankine +et al. 2020; Rodríguez Hidalgo & Rankine 2022), and the velocities of +narrow C iv line-locked ‘triplet’ absorption features are also strongly +correlated with the emission blueshift (Rankine et al. in preparation). +For the discussion in this subsection, we therefore assume that +objects with larger C iv blueshifts have stronger disc winds. To be +more precise, the C iv blueshift is taken as a measure of the strength +of emission from outflowing gas relative to the strength of emission +from virialized gas at the systemic redshift. In this paradigm, the +trends seen in Fig. 3 are in good agreement with the picture proposed +by Giustini & Proga (2019). We see large (≳ 1000 km s−1) median +C iv blueshifts only in bins with high SMBH masses and high Ed- +dington ratios. Furthermore, we do indeed see a more complicated +mass dependence above 𝐿/𝐿Edd ≈ 0.2. High 𝐿/𝐿Edd is therefore +a necessary, but not sufficient, condition for observing large C iv +blueshifts, consistent with the results of Baskin & Laor (2005). +In detail, we only observe strong outflow signatures in objects +with 𝑀BH ≳ 109𝑀⊙, which is somewhat higher than the criterion +of 𝑀BH ≳ 108𝑀⊙ proposed by Giustini & Proga (2019). Requiring +𝑀BH ≳ 109𝑀⊙ and 𝐿/𝐿Edd ≳ 0.2 together ensures that the crite- +rion 𝐿bol ≳ 1045.5 erg s−1 is satisfied. Above this 𝐿bol threshold, +Zakamska & Greene (2014) suggest that quasar winds are capable of +driving ionized gas (as traced by [O iii] 𝜆5008 emission) beyond the +escape velocity of the host galaxy. The kinematics of C iv and [O iii] +are known to correlate (Coatman et al. 2019), and our observed C iv +blueshift behaviour is therefore consistent with the conclusion of Za- +kamska & Greene (2014) that 𝐿bol ≳ 1045.5 erg s−1 is required for +quasar feedback to operate. +For 108𝑀⊙ ≲ 𝑀BH ≲ 109𝑀⊙ and 𝐿/𝐿Edd ≳ 0.2, we see the +strongest He ii and strongest non-outflowing C iv line emission. One +possible explanation for this behaviour would be that this emission +represents ionized material which has been launched from the ac- +cretion disc, but lacks the ultraviolet luminosity to accelerate the +outflow, meaning that such material falls back and virializes instead +of escaping. In such a scenario the strong symmetric emission from +high-ionization ultraviolet lines would represent a failed line-driven +wind, analogous to models of the low-ionization BLR which repre- +sent a failed dust-driven wind (Czerny & Hryniewicz 2011; Baskin +& Laor 2018, see also Elvis 2017). +While we observe a reasonably good qualitative agreement be- +tween the C iv blueshift behaviour and the Giustini & Proga (2019) +predictions for line-driven winds, the reality is likely more compli- +cated. In particular, Giustini & Proga (2019) do not consider any +emission from a ‘soft excess’. Instead they assume that the ionizing +SED consists of just two components, emitted from a thermal disc +and a hot corona. Such a simple model is unlikely to explain our +observational results: the different behaviour of He ii EW and 𝛼ox +as a function of 𝑀BH and 𝐿/𝐿Edd points to the presence of a third +spectral component in the EUV which can vary separately from the +disc and corona. +Other physical effects could also be at play. In particular, as the +accretion rate increases above �𝑚 ≳ 0.3, we expect the disc structure +to transition between geometries akin to slim discs and thin discs +(Abramowicz et al. 1988; Abramowicz & Fragile 2013). At low ac- +cretion rates, slim discs are well approximated by the Novikov & +Thorne (1973) thin disc solution, as used in qsosed, but we expect +this to be less accurate as �𝑚 increases. In other words, the regime +in which qsosed appears to best match our data is also the regime +in which we might expect it to be least accurate. The origin of the +apparent transition around 𝐿/𝐿Edd ≈ 0.2 in Figs. 3 and 5 is there- +fore still uncertain and further work is required to fully understand +the interplay between AGN accretion flows, the ionizing SEDs they +produce, and the outflows they drive. +Line-driven winds from high Eddington ratio AGN are often cited +as a potentially important component of radiative-mode (quasar- +mode) feedback (Zubovas & King 2012). While difficult to observe +directly, such feedback modes are required to regulate galaxy growth +and explain the tight SMBH-galaxy correlations observed in the local +universe (see Fabian 2012, for a review). However, most SMBHs in +the local universe do not have masses above 109𝑀⊙, so our results +might suggest that the line-driven winds traced by C iv cannot have +a significant effect on their host galaxies’ growth and co-evolution +as they never reach the SMBH masses required to launch strong +winds. There are at least two solutions to this apparent problem. +First is that radiative-mode feedback could still be operating through +ionized gas outflows, but that the gas is in a different ionization phase +and is not seen in C iv, but instead in other bands such as the X-ray +‘ultra-fast outflows’ (Laha et al. 2021). Second could be that quasar- +mode feedback is only effective when coupled to dusty gas (Fabian +et al. 2008; Ishibashi et al. 2018; Ricci et al. 2022), thus having +most impact when the AGN is obscured by dust (Temple et al. 2019; +Lansbury et al. 2020; Jun et al. 2021; Assef et al. 2022). +5.2.3 SMBH spin and system inclination +In our qsosed modeling, we kept inclination fixed at cos𝑖 = 0.5 +and only presented the non-spinning SMBH case, 𝑎∗ = 0. However, +both of these parameters have an impact on the predicted SEDs. The +impact of SMBH spin is particularly pronounced; plots matching +those in the right-hand panels of Figs. 4 and 5 are presented in +Appendix D. The basic finding from the maximal spin models is +that the observed trends of 𝛼ox with mass and Eddington fraction +are not reproduced, for reasons that are explained in Appendix D. In +fact, all of the maximal spin models have 𝛼ox ≳ −1.5, meaning that +the X-ray luminosity is always quite high compared to the optical +and ultraviolet, and the observed soft spectra at high mass and high +Eddington fraction are not reproduced for 𝑎∗ = 1. If there are a +significant proportion of maximally spinning SMBHs in our quasar +sample, this would imply that the model predictions are not valid +for high spin objects, potentially undermining many of the results +discussed in Section 5.2.1. Alternatively, if the qsosed models are +correct, the good agreement at low spin and poor agreement at high +spin would imply that most SDSS quasars at 𝑧 ≈ 2 typically have low +or moderate SMBH spins. +SMBH spin is most commonly estimated from broad iron line +emission in the X-ray band (Reynolds 2019). Spin measurements +tend to be rather high, with the majority of X-ray measurements in +AGN consistent with maximally spinning SMBHs. This apparent +MNRAS 000, 1–20 (2023) + +12 +M. J. Temple et al. +preference might initially appear to be inconsistent with our results. +However, there are a number of factors at work. First, discs around +maximally spinning SMBHs have higher radiative efficiencies and +are thus more luminous. As shown in figure 3 of Reynolds (2019), +this might lead to high spins being over-represented in a sample. +One could also imagine further selection effects if spins are eas- +ier to measure when they are close to maximal and the iron line +is broader. Second, the majority of spin measurements are at lower +masses (𝑀BH ≲ 108 𝑀⊙) than in our sample, with only a handful of +spin measurements in our considered mass regime. In fact, there is +some tentative evidence for a decrease of SMBH spin with increasing +mass (Sisk-Reynés et al. 2022), behaviour that is expected if accre- +tion is coherent at low masses and more incoherent at higher masses, +as predicted by both semi-analytic models and hydrodynamic sim- +ulations (King et al. 2008; Sesana et al. 2014; Zhang & Lu 2019; +Bustamante & Springel 2019). If our results do indeed favour low or +moderate SMBH spins in luminous quasars, then they are consistent +with this proposed trend, especially since it is the high-mass quasars +which cannot be matched by the maximally spinning qsosed models. +Any conclusions drawn here should be treated with caution, given (i) +the difficulties in obtaining reliable and unbiased spin measurements, +and (ii) the lack of knowledge about the impact of spin on the EUV +and X-ray regions of the quasar SED. +We fixed inclination in the qsosed models, adopting cos𝑖 = 0.5. +Following Copernican reasoning, we expect AGN to have a random, +isotropic distribution of viewing angles, in which case the mean +viewing angle of all AGN is cos𝑖 = 0.5. Factoring in obscuration +by a putative ‘torus’ beyond some maximum inclination, and fore- +shortening/limb darkening of the disc continuum might be expected +to bias this mean viewing angle to lower (more face-on) inclinations +(e.g. Krolik & Voit 1998; Matthews et al. 2017). If we were to adopt +a different inclination in qsosed, we can think about how the model +predictions would change. +The hard X-ray source in qsosed is isotropic, whereas the warm +and thermal components have a disc-like geometry and thus pro- +duced an observed luminosity ∝ cos𝑖, such that lower inclinations +have higher luminosities. The impact of inclination on the outputs +from qsosed can thus be straightforwardly understood. Changing +inclination from 𝑖1 → 𝑖2 results in a fractional change in 𝐿3000 of +(cos𝑖2/cos𝑖1). Since, for a given input �𝑚, we calculate 𝐿/𝐿Edd from +𝐿3000, we obtain a linear scaling of the 𝑦-axis of the right-hand +panels of Figs 5 and 4 by the same factor. The change in 𝛼ox is +Δ𝛼ox = −0.3838 log10(cos𝑖2/cos𝑖1); adopting a lower inclination +with cos𝑖 = 0.75 would result in a more negative 𝛼ox in all simula- +tion bins by ≈ 0.07. Finally, the ratio 𝐿228/𝐿1640 undergoes small +changes with cos𝑖, but these are fairly uniform across the simulation +grid and thus unimportant, given that the proportionality constant +between 𝐿228/𝐿1640 and He ii EW is not known. Furthermore, the +He ii EW depends on the 𝐿228 seen by the He ii gas, rather than +the 𝐿228 seen by a distant observer, meaning that the true inclination +dependence of He ii EW would depend on the BLR geometry. We +have explicitly checked that the anticipated changes in 𝐿3000 and 𝛼ox +are indeed reproduced in qsosed, except for small departures in the +𝛼ox due to contamination of the 2 keV flux by the warm component. +5.3 Wider implications and future work +5.3.1 Comparison with other populations +Recent work has attempted to compare the ultraviolet emission prop- +erties in high redshift (𝑧 ≳ 6) quasars with their lower redshift ana- +logues (Mazzucchelli et al. 2017; Meyer et al. 2019; Shen et al. 2019; +Schindler et al. 2020; Yang et al. 2021b; Lai et al. 2022; Wang et al. +2022). Such quasars are (by selection) very luminous, and generally +display large C iv blueshifts. From our results, we would argue that +comparative studies should match AGN samples not just in luminos- +ity, but in two independent parameters which trace 𝐿/𝐿Edd and 𝑀BH. +Stepney et al. (in preparation) will discuss this further in a study of +C iv and He ii emission in SDSS quasars with redshifts 𝑧 > 3.5. +Current samples of 𝑧 ≳ 6 quasars include a significant number of +objects with inferred 𝐿/𝐿Edd > 1, which lie outside the parameter +space explored in this work. We have verified that the sample of 37 +quasar spectra with redshifts 𝑧 > 6.3 presented by Yang et al. (2021b) +typically show narrower Mg ii profiles than 𝑧 ≈ 2 SDSS objects with +the same 3000 Å luminosities, suggesting smaller 𝑀BH and larger +𝐿/𝐿Edd (for a given 𝐿3000) than the quasars characterized in this +work. For such objects it is therefore not surprising that their typical +ultraviolet emission properties are different from the majority of the +SDSS population at 𝑧 ≈ 2. +5.3.2 BLR metallicities +The relative strengths of many ultraviolet emission lines are known +to correlate with the C iv and He ii emission properties (Richards +et al. 2011; Temple et al. 2020). In particular, the flux ratios of high +ionization ultraviolet lines such as C iv, N v 𝜆1240 and Si iv 𝜆1400 +are tightly correlated with the C iv blueshift and He ii EW (Temple +et al. 2021b). Assuming no changes in density or ionization structure +or geometry of the BLR, changes in these line ratios are sometimes +taken to reflect changes in the metal content of the BLR (Nagao et al. +2006). Such an interpretation, combined with the results in this work, +gives rise to a paradigm where the metal content of quasar BLRs is +largest in objects with the largest 𝑀BH and 𝐿/𝐿Edd, as noted by Xu +et al. (2018) and Śniegowska et al. (2021) respectively. +However, as shown in Temple et al. (2021b), the variation in these +line ratios can be explained with changes in the density of the emitting +gas, and need not involve changes in metallicity (see also appendix +A4 of Casebeer et al. 2006). In particular, the ultraviolet emission +line ratios seen in objects with large C iv blueshifts can be explained +by emission from relatively dense gas which is located closer to +the ionizing source, while the line ratios in objects with high EW, +symmetric C iv emission are consistent with emission from less dense +gas at larger radii. Given the trends seen in Figs. 3 and 5, this is a +much more natural explanation: objects with different SMBH masses +and accretion rates have different accretion flows, which give rise to +different EUV SEDs (as traced by He ii) and different kinematic and +density structures in the BLR (traced by the C iv blueshift and high +ionization line ratios respectively). Under this alternative paradigm +the BLR metallicity would be free to vary independently of 𝑀BH and +𝐿/𝐿Edd, and need not be super-solar in the early universe (cf. Lai +et al. 2022; Wang et al. 2022). +5.3.3 Quasar cosmology +Quasars are visible out to large cosmological distances, and display +remarkably homogeneous behaviour across cosmic time. A grow- +ing body of work has proposed the non-linear scaling between 𝐿UV +and 𝐿2 keV (i.e. the 𝛼ox–𝐿2500 Å relation) as a way to use quasars +as standardizable candles for cosmological measurements (Risaliti +& Lusso 2015, 2017, 2019; Lusso & Risaliti 2017; Salvestrini et al. +2019; Lusso et al. 2020; Sacchi et al. 2022; Khadka & Ratra 2022). +More recently, however, Petrosian et al. (2022) have shown that an +independent determination of the cosmological distance–redshift re- +lation cannot be constructed simply by using the empirical correlation +MNRAS 000, 1–20 (2023) + +C iv and He ii emission in quasars +13 +between two continuum luminosities, as such luminosities must as- +sume a cosmological model to be inferred from fluxes and redshifts, +leading to circular reasoning. +In this work we have shown that it should be possible to break this +circularity by including information from the emission line prop- +erties. With knowledge of the Mg ii velocity width, and either the +He ii strength or the C iv properties, one should be able to locate +an object in the 𝑀BH–𝐿/𝐿Edd plane, and hence infer the intrinsic +luminosity in a cosmology-independent way. By comparing to the +observed fluxes one could then (in principle) infer a constraint on +the Hubble parameter 𝐻(𝑧). However, further work is still required. +In particular, we need to build a sample of quasars with ultraviolet +emission line measurements which have independent measurements +of the luminosity distance, in order to calibrate our 𝑀BH–𝐿/𝐿Edd +space in a cosmology-independent manner, in an analogous way to +the use of the ‘inverse distance ladder’ to calibrate type Ia supernovae +as standard candles (e.g. Freedman et al. 2001; Riess et al. 2021). +5.3.4 Time variability and upcoming surveys +Changes in 𝐿/𝐿Edd for a quasar with fixed 𝑀BH will lead to changes +in the emitted spectrum, but such changes in SMBH fueling are +expected to generally occur on the viscous time-scale, which is on the +of order of tens to thousands of years. However, SMBH accretion is +inherently stochastic and the emitted flux varies by a factor of a few on +shorter time-scales of just years. The time-scale and amplitude of this +intrinsic ‘flickering’ are now known to depend on the SMBH mass +and accretion rate (Yu et al. 2022b), and this stochastic flickering will +contribute to the scatter within each binned region of our parameter +space (Section 4.1). +In terms of spectroscopic variability, Rivera et al. (2020) showed +that individual SDSS-RM quasars with multiple epochs of spec- +troscopy (i.e. with fixed 𝑀BH) can vary in essentially every direction +in the C iv blueshift–EW space, although objects with large blueshifts +tend to show a change in blueshift and objects with strong EW show +a change in EW. In the near future, SDSS-V (Kollmeier et al. 2017) +will provide multi-epoch spectroscopic data for tens of thousands of +luminous quasars, providing new insights into AGN variability. +At the same time, surveys such as DESI (Alexander et al. 2022; +Chaussidon et al. 2022) and 4MOST (Merloni et al. 2019; Eltvedt +et al. 2022) will probe fainter, yielding spectra of lower luminosity +quasars than the sample investigated in this work, and future data +releases from the eROSITA all-sky survey will include X-ray flux +measurements for millions of AGN. Together these surveys will pro- +vide new constraints on the spectroscopic properties and ionizing +SEDs of luminous AGN across the 𝑀BH–𝐿/𝐿Edd parameter space. +6 CONCLUSIONS +We have investigated the rest-frame ultraviolet emission line proper- +ties in 191 391 SDSS quasars with redshifts 1.5 < 𝑧 < 2.65. We can +infer 𝛼ox, the logarithmic ratio of the rest-frame 2 keV and 2500 Å +luminosities, for 5325 quasars in our sample. Using the FWHM of +Mg ii 𝜆2800 as a proxy for the virial velocity, we quantify the av- +erage properties of the C iv 𝜆1549 and He ii 𝜆1640 emission lines +across the two-dimensional space spanned by 𝑀BH and 𝐿/𝐿Edd, and +use these observations to confront qualitative predictions of when +radiation-driven outflows should dominate kinetic feedback mecha- +nisms (Giustini & Proga 2019) and theoretical SEDs arising from +models of AGN accretion flows (Kubota & Done 2018). Our main +conclusions are: +(i) As shown in previous works (Richards et al. 2011; Rankine +et al. 2020), the blueshift and EW of C iv correlate with the EW of +He ii. Objects with strong He ii have high EW C iv with little or no +blue excess, while objects with weaker He ii show smaller EW C iv +with larger C iv blueshifts. +(ii) We recover a Baldwin effect, but instead of simply correlating +with the ultraviolet luminosity, we find that the C iv and He ii prop- +erties display more complicated trends in the 𝑀BH–𝐿/𝐿Edd plane. +The dynamic range in He ii EW is greatest at Eddington ratios ≳0.2 +(Fig. 5). The largest C iv blueshifts are only observed at high 𝐿/𝐿Edd +and high 𝑀BH, while the highest EWs are seen only at high 𝐿/𝐿Edd +and relatively low 𝑀BH (Fig. 3). Composite spectra from these two +extrema are shown in blue and green in Fig. 1. +(iii) In contrast to the ultraviolet emission line properties, but +consistent with previous work in the literature, 𝛼ox displays a much +simpler behaviour across the 𝑀BH–𝐿/𝐿Edd plane (Fig. 4). 𝛼ox cor- +relates with the ultraviolet continuum luminosity in a more direct +manner than the emission lines, albeit in a much smaller sample. Fu- +ture data releases from eROSITA, SDSS-V and 4MOST will increase +the number of known quasars with X-ray data. +(iv) Under the assumption that blueshifted C iv emission is trac- +ing a disc wind accelerated by radiation line driving, we find our +results are consistent with the global scheme for accretion and out- +flow mechanisms proposed by Giustini & Proga (2019). In particular, +an Eddington-scaled mass accretion rate �𝑚 ≳ 0.25 is required for the +formation of the strongest line-driven winds. Giustini & Proga (2019) +suggest that 𝑀BH > 108𝑀⊙ is also required to launch strong line- +driven winds, however we only observe the largest C iv blueshifts in +objects with Mg ii-inferred 𝑀BH ≳ 109𝑀⊙. Strong line emission at +𝑀BH ≲ 109𝑀⊙ could perhaps indicate a ‘failed’ line-driven wind. +(v) Absent large changes in the density or geometry of the broad +line region, the strength of He ii is probing the strength of 54 eV ion- +izing radiation in the ‘unseen’ portion of the ultraviolet SED. Above +𝐿/𝐿Edd ≈ 0.2, we find that the EW of He ii is broadly consistent +with the qsosed model (Kubota & Done 2018). In other words, the +relative strength of the 54 eV flux (which is photoionizing the broad +line region) compared to the 1640 Å continuum is consistent with a +relatively simple model where the peak temperature of the accretion +disc blackbody changes as a function of SMBH mass and accre- +tion rate and the strength of the ‘soft excess’ is adjusted to give the +correct bolometric luminosity while keeping the strength of the hot +coronal emission fixed at two per cent of the Eddington luminosity +(as proposed by Kubota & Done 2018). +(vi) Below 𝐿/𝐿Edd ≈ 0.2, something appears to change in the +physics of the broad line region, with no strong C iv blueshifts ob- +served and much weaker trends in He ii. The simple SED models do +not provide as good a match to the observed He ii trends, consistent +with the results of Mitchell et al. (2022) who find a discrepancy +between the observed and predicted 𝛼ox in the same region of the +𝑀BH–𝐿/𝐿Edd parameter space. +(vii) Similar to Mitchell et al. (2022), we also find no strong +evidence for high SMBH spins in our quasar sample: the zero-spin +qsosed models provide an acceptable match to the SED probes across +a significant portion of our observed parameter space while the max- +imally spinning models do not. If a significant fraction of our quasar +sample have maximally spinning SMBHs, this would suggest that +the qsosed model assumptions are not valid for high spin objects. +Alternatively, taking the model results at face value would suggest +low or moderate spins in typical SDSS quasars at 𝑧 ≈ 2. +MNRAS 000, 1–20 (2023) + +14 +M. J. Temple et al. +ACKNOWLEDGEMENTS +We gratefully acknowledge useful discussions with Chris Done and +Jake Mitchell. MJT thanks Chiara Mazzucchelli, Claudio Ricci and +Roberto Assef for insightful comments, and Jinyi Yang for sharing +the sample of 𝑧 > 6 quasar spectra from Yang et al. (2021b). +MJT acknowledges support from a FONDECYT postdoctoral fel- +lowship (3220516). JHM acknowledges funding from the Royal So- +ciety. ALR acknowledges support from UKRI (MR/T020989/1). +Funding for the Sloan Digital Sky Survey IV has been provided by +the Alfred P. Sloan Foundation, the U.S. Department of Energy Office +of Science, and the Participating Institutions. SDSS-IV acknowledges +support and resources from the Center for High-Performance Com- +puting at the University of Utah. The SDSS web site is www.sdss.org. +SDSS-IV is managed by the Astrophysical Research Consortium +for the Participating Institutions of the SDSS Collaboration including +the Brazilian Participation Group, the Carnegie Institution for Sci- +ence, Carnegie Mellon University, the Chilean Participation Group, +the French Participation Group, Harvard-Smithsonian Center for As- +trophysics, Instituto de Astrofísica de Canarias, The Johns Hopkins +University, Kavli Institute for the Physics and Mathematics of the Uni- +verse (IPMU) / University of Tokyo, the Korean Participation Group, +Lawrence Berkeley National Laboratory, Leibniz Institut für Astro- +physik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA +Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), +Max-Planck-Institut für Extraterrestrische Physik (MPE), National +Astronomical Observatories of China, New Mexico State Univer- +sity, New York University, University of Notre Dame, Observatário +Nacional / MCTI, The Ohio State University, Pennsylvania State +University, Shanghai Astronomical Observatory, United Kingdom +Participation Group, Universidad Nacional Autónoma de México, +University of Arizona, University of Colorado Boulder, University +of Oxford, University of Portsmouth, University of Utah, Univer- +sity of Virginia, University of Washington, University of Wisconsin, +Vanderbilt University, and Yale University. +For the purpose of open access, the authors will apply a Cre- +ative Commons Attribution (CC BY) licence to any Author Accepted +Manuscript version arising from this submission. +DATA AVAILABILITY +The spectroscopic data underlying this article are available from +SDSS.3 The X-ray data sets underlying this article are available from +the references given in Section 2.2. The composite spectra in Fig. 1 +are available as online supplementary material. +REFERENCES +Abdurro’uf et al., 2022, ApJS, 259, 35 +Abramowicz M. A., Fragile P. C., 2013, Living Reviews in Relativity, 16, 1 +Abramowicz M. A., Czerny B., Lasota J. P., Szuszkiewicz E., 1988, ApJ, 332, +646 +Alexander D. M., et al., 2022, arXiv e-prints, p. arXiv:2208.08517 +Arnaud K. A., 1996, in Jacoby G. H., Barnes J., eds, Astronomical Society +of the Pacific Conference Series Vol. 101, Astronomical Data Analysis +Software and Systems V. p. 17 +Assef R. J., et al., 2022, ApJ, 934, 101 +Avni Y., Tananbaum H., 1982, ApJ, 262, L17 +Avni Y., Tananbaum H., 1986, ApJ, 305, 83 +3 https://www.sdss4.org/dr17/ +Bachev R., Marziani P., Sulentic J. W., Zamanov R., Calvani M., Dultzin- +Hacyan D., 2004, ApJ, 617, 171 +Baldwin J. A., 1977, ApJ, 214, 679 +Baldwin J. A., Netzer H., 1978, ApJ, 226, 1 +Baskin A., Laor A., 2004, MNRAS, 350, L31 +Baskin A., Laor A., 2005, MNRAS, 356, 1029 +Baskin A., Laor A., 2018, MNRAS, 474, 1970 +Baskin A., Laor A., Hamann F., 2013, MNRAS, 432, 1525 +Begelman M. C., McKee C. F., Shields G. A., 1983, ApJ, 271, 70 +Blandford R. D., Payne D. G., 1982, MNRAS, 199, 883 +Boller T., Freyberg M. J., Trümper J., Haberl F., Voges W., Nandra K., 2016, +A&A, 588, A103 +Boroson T. A., Green R. F., 1992, ApJS, 80, 109 +Bottorff M. C., Korista K. T., Shlosman I., 2000, ApJ, 537, 134 +Brodzeller A., Dawson K., 2022, AJ, 163, 110 +Bustamante S., Springel V., 2019, MNRAS, 490, 4133 +Calistro Rivera G., et al., 2021, A&A, 649, A102 +Casebeer D. A., Leighly K. M., Baron E., 2006, ApJ, 637, 157 +Castor J. I., Abbott D. C., Klein R. I., 1975, ApJ, 195, 157 +Chajet L. S., Hall P. B., 2013, MNRAS, 429, 3214 +Chaussidon E., et al., 2022, arXiv e-prints, p. arXiv:2208.08511 +Coatman L., Hewett P. C., Banerji M., Richards G. T., 2016, MNRAS, 461, +647 +Coatman L., Hewett P. C., Banerji M., Richards G. T., Hennawi J. F., +Prochaska J. X., 2017, MNRAS, 465, 2120 +Coatman L., Hewett P. C., Banerji M., Richards G. T., Hennawi J. F., +Prochaska J. X., 2019, MNRAS, 486, 5335 +Coffey D., et al., 2019, A&A, 625, A123 +Collin S., Kawaguchi T., Peterson B. M., Vestergaard M., 2006, A&A, 456, +75 +Comparat J., et al., 2020, A&A, 636, A97 +Croom S. M., et al., 2002, MNRAS, 337, 275 +Czerny B., Hryniewicz K., 2011, A&A, 525, L8 +Dalla Bontà E., et al., 2020, ApJ, 903, 112 +Davidson K., Netzer H., 1979, Reviews of Modern Physics, 51, 715 +Dibai E. A., 1980, Soviet Ast., 24, 389 +Done C., Davis S. W., Jin C., Blaes O., Ward M., 2012, MNRAS, 420, 1848 +Du P., Wang J.-M., 2019, ApJ, 886, 42 +Dwelly T., et al., 2017, MNRAS, 469, 1065 +Eltvedt A., et al., 2022, arXiv e-prints, p. arXiv:2211.07324 +Elvis M., 2000, ApJ, 545, 63 +Elvis M., 2017, ApJ, 847, 56 +Elvis M., et al., 1994, ApJS, 95, 1 +Emmering R. T., Blandford R. D., Shlosman I., 1992, ApJ, 385, 460 +Fabian A. C., 2012, ARA&A, 50, 455 +Fabian A. C., Vasudevan R. V., Gandhi P., 2008, MNRAS, 385, L43 +Fawcett V. A., Alexander D. M., Rosario D. J., Klindt L., Lusso E., Morabito +L. K., Calistro Rivera G., 2022, MNRAS, 513, 1254 +Ferland G. J., et al., 2013, Rev. Mex. Astron. Astrofis., 49, 137 +Ferland G. J., et al., 2017, Rev. Mex. Astron. Astrofis., 53, 385 +Ferland G. J., Done C., Jin C., Landt H., Ward M. J., 2020, MNRAS, 494, +5917 +Fonseca Alvarez G., et al., 2020, ApJ, 899, 73 +Freedman W. L., et al., 2001, ApJ, 553, 47 +Fu S., Brandt W. N., Zou F., Laor A., Garmire G. P., Ni Q., Timlin John D. +I., Xue Y., 2022, ApJ, 934, 97 +Fukumura K., Kazanas D., Contopoulos I., Behar E., 2010, ApJ, 715, 636 +Gaskell C. M., 1982, ApJ, 263, 79 +Gaskell C. M., Goosmann R. W., 2013, ApJ, 769, 30 +Gaskell C. M., Goosmann R. W., 2016, Ap&SS, 361, 67 +Giustini M., Proga D., 2019, A&A, 630, A94 +Gordon C., Arnaud K., 2021, PyXspec: Python interface to XSPEC spectral- +fitting program, Astrophysics Source Code Library, record ascl:2101.014 +(ascl:2101.014) +Haardt F., Maraschi L., 1991, ApJ, 380, L51 +Hewett P. C., Wild V., 2010, MNRAS, 405, 2302 +Higginbottom N., Proga D., Knigge C., Long K. S., Matthews J. H., Sim +S. A., 2014, ApJ, 789, 19 +MNRAS 000, 1–20 (2023) + +C iv and He ii emission in quasars +15 +Hoare M. G., Drew J. E., 1991, MNRAS, 249, 452 +Ishibashi W., Fabian A. C., Ricci C., Celotti A., 2018, MNRAS, 479, 3335 +Jensen T. W., et al., 2016, ApJ, 833, 199 +Jin C., Ward M., Done C., Gelbord J., 2012, MNRAS, 420, 1825 +Jun H. D., Assef R. J., Carroll C. M., Hickox R. C., Kim Y., Lee J., Ricci C., +Stern D., 2021, ApJ, 906, 21 +Just D. W., Brandt W. N., Shemmer O., Steffen A. T., Schneider D. P., Chartas +G., Garmire G. P., 2007, ApJ, 665, 1004 +Kaspi S., Maoz D., Netzer H., Peterson B. M., Vestergaard M., Jannuzi B. T., +2005, ApJ, 629, 61 +Khadka N., Ratra B., 2022, MNRAS, 510, 2753 +Khadka N., Martínez-Aldama M. L., Zajaček M., Czerny B., Ratra B., 2022a, +MNRAS, 513, 1985 +Khadka N., Zajaček M., Panda S., Martínez-Aldama M. L., Ratra B., 2022b, +MNRAS, 515, 3729 +King A. R., Pringle J. E., Hofmann J. A., 2008, MNRAS, 385, 1621 +Kollmeier J. A., et al., 2017, arXiv e-prints, p. arXiv:1711.03234 +Konigl A., Kartje J. F., 1994, ApJ, 434, 446 +Krawczyk C. M., Richards G. T., Mehta S. S., Vogeley M. S., Gallagher S. C., +Leighly K. M., Ross N. P., Schneider D. P., 2013, ApJS, 206, 4 +Krolik J. H., Kallman T. R., 1988, ApJ, 324, 714 +Krolik J. H., Voit G. M., 1998, ApJ, 497, L5 +Kruczek N. E., et al., 2011, AJ, 142, 130 +Kubota A., Done C., 2018, MNRAS, 480, 1247 +Kwan J., Krolik J. H., 1981, ApJ, 250, 478 +Laha S., Reynolds C. S., Reeves J., Kriss G., Guainazzi M., Smith R., Veilleux +S., Proga D., 2021, Nature Astronomy, 5, 13 +Lai S., et al., 2022, MNRAS, 513, 1801 +Lansbury G. B., Banerji M., Fabian A. C., Temple M. J., 2020, MNRAS, 495, +2652 +Laor A., Davis S. W., 2014, MNRAS, 438, 3024 +Leighly K. M., 2004, ApJ, 611, 125 +Leighly K. M., Moore J. R., 2004, ApJ, 611, 107 +Leighly K. M., Halpern J. P., Jenkins E. B., Casebeer D., 2007, ApJS, 173, 1 +Liu T., et al., 2020, ApJS, 250, 32 +Liu T., et al., 2022, A&A, 661, A5 +Lusso E., Risaliti G., 2016, ApJ, 819, 154 +Lusso E., Risaliti G., 2017, A&A, 602, A79 +Lusso E., et al., 2020, A&A, 642, A150 +Lusso E., et al., 2021, A&A, 653, A158 +Lyke B. W., et al., 2020, ApJS, 250, 8 +Lynden-Bell D., 1969, Nature, 223, 690 +Maddox N., et al., 2017, MNRAS, 470, 2314 +Marlar A., Shemmer O., Brotherton M. S., Richards G. T., Dix C., 2022, ApJ, +931, 41 +Martínez-Aldama M. L., Zajaček M., Czerny B., Panda S., 2020, ApJ, 903, +86 +Mathews W. G., Ferland G. J., 1987, ApJ, 323, 456 +Matthews J. H., Knigge C., Long K. S., 2017, MNRAS, 467, 2571 +Mazzucchelli C., et al., 2017, ApJ, 849, 91 +Mejía-Restrepo J. E., Trakhtenbrot B., Lira P., Netzer H., 2018, MNRAS, +478, 1929 +Merloni A., et al., 2019, The Messenger, 175, 42 +Meyer R. A., Bosman S. E. I., Ellis R. S., 2019, MNRAS, 487, 3305 +Mitchell J. A. J., Done C., Ward M. J., Kynoch D., Hagen S., Lusso E., Landt +H., 2022, arXiv e-prints, p. arXiv:2210.11977 +Mizumoto M., Done C., Tomaru R., Edwards I., 2019, MNRAS, 489, 1152 +Murray N., Chiang J., 1995, ApJ, 454, L105 +Murray N., Chiang J., Grossman S. A., Voit G. M., 1995, ApJ, 451, 498 +Nagao T., Marconi A., Maiolino R., 2006, A&A, 447, 157 +Ni Q., et al., 2018, MNRAS, 480, 5184 +Ni Q., et al., 2022, MNRAS, 511, 5251 +Nomura M., Ohsuga K., 2017, MNRAS, 465, 2873 +Nomura M., Ohsuga K., Done C., 2020, MNRAS, 494, 3616 +Novikov I. D., Thorne K. S., 1973, in Black Holes (Les Astres Occlus). pp +343–450 +Osterbrock D. E., Ferland G. J., 2006, Astrophysics of gaseous nebulae and +active galactic nuclei +Petrosian V., Singal J., Mutchnick S., 2022, ApJ, 935, L19 +Petrucci P. O., Ursini F., De Rosa A., Bianchi S., Cappi M., Matt G., Dadina +M., Malzac J., 2018, A&A, 611, A59 +Proga D., 2007, ApJ, 661, 693 +Proga D., Kallman T. R., 2004, ApJ, 616, 688 +Proga D., Stone J. M., Kallman T. R., 2000, ApJ, 543, 686 +Rankine A. L., Hewett P. C., Banerji M., Richards G. T., 2020, MNRAS, 492, +4553 +Rankine A. L., Matthews J. H., Hewett P. C., Banerji M., Morabito L. K., +Richards G. T., 2021, MNRAS, 502, 4154 +Reynolds C. S., 2019, Nature Astronomy, 3, 41 +Ricci C., et al., 2022, ApJ, 938, 67 +Richards G. T., Vanden Berk D. E., Reichard T. A., Hall P. B., Schneider +D. P., SubbaRao M., Thakar A. R., York D. G., 2002, AJ, 124, 1 +Richards G. T., et al., 2006, ApJS, 166, 470 +Richards G. T., et al., 2011, AJ, 141, 167 +Riess A. G., Casertano S., Yuan W., Bowers J. B., Macri L., Zinn J. C., Scolnic +D., 2021, ApJ, 908, L6 +Risaliti G., Elvis M., 2010, A&A, 516, A89 +Risaliti G., Lusso E., 2015, ApJ, 815, 33 +Risaliti G., Lusso E., 2017, Astronomische Nachrichten, 338, 329 +Risaliti G., Lusso E., 2019, Nature Astronomy, 3, 272 +Rivera A. B., Richards G. T., Hewett P. C., Rankine A. L., 2020, ApJ, 899, 96 +Rivera A. B., Richards G. T., Gallagher S. C., McCaffrey T. V., Rankine A. L., +Hewett P. C., Shemmer O., 2022, ApJ, 931, 154 +Rodríguez Hidalgo P., Rankine A. L., 2022, ApJ, 939, L24 +Sacchi A., et al., 2022, A&A, 663, L7 +Salvato M., et al., 2022, A&A, 661, A3 +Salvestrini F., Risaliti G., Bisogni S., Lusso E., Vignali C., 2019, A&A, 631, +A120 +Schindler J.-T., et al., 2020, ApJ, 905, 51 +Schneider D. P., et al., 2010, AJ, 139, 2360 +Sesana A., Barausse E., Dotti M., Rossi E. M., 2014, ApJ, 794, 104 +Shang Z., Wills B. J., Robinson E. L., Wills D., Laor A., Xie B., Yuan J., +2003, ApJ, 586, 52 +Shang Z., et al., 2005, ApJ, 619, 41 +Shen Y., 2013, Bulletin of the Astronomical Society of India, 41, 61 +Shen Y., Ho L. C., 2014, Nature, 513, 210 +Shen Y., Liu X., 2012, ApJ, 753, 125 +Shen Y., Greene J. E., Strauss M. A., Richards G. T., Schneider D. P., 2008, +ApJ, 680, 169 +Shen Y., et al., 2011, ApJS, 194, 45 +Shen Y., et al., 2019, ApJ, 873, 35 +Sisk-Reynés J., Reynolds C. S., Matthews J. H., Smith R. N., 2022, MNRAS, +514, 2568 +Śniegowska M., Marziani P., Czerny B., Panda S., Martínez-Aldama M. L., +del Olmo A., D’Onofrio M., 2021, ApJ, 910, 115 +Steffen A. T., Strateva I., Brandt W. N., Alexander D. M., Koekemoer A. M., +Lehmer B. D., Schneider D. P., Vignali C., 2006, AJ, 131, 2826 +Stevans M. L., Shull J. M., Danforth C. W., Tilton E. M., 2014, ApJ, 794, 75 +Sulentic J., Marziani P., 2015, Frontiers in Astronomy and Space Sciences, +2, 6 +Sulentic J. W., Zwitter T., Marziani P., Dultzin-Hacyan D., 2000, ApJ, 536, +L5 +Sulentic J. W., Bachev R., Marziani P., Negrete C. A., Dultzin D., 2007, ApJ, +666, 757 +Sun J., Shen Y., 2015, ApJ, 804, L15 +Temple M. J., Banerji M., Hewett P. C., Coatman L., Maddox N., Peroux C., +2019, MNRAS, 487, 2594 +Temple M. J., Ferland G. J., Rankine A. L., Hewett P. C., Badnell N. R., +Ballance C. P., Del Zanna G., Dufresne R. P., 2020, MNRAS, 496, 2565 +Temple M. J., Banerji M., Hewett P. C., Rankine A. L., Richards G. T., 2021a, +MNRAS, 501, 3061 +Temple M. J., Ferland G. J., Rankine A. L., Chatzikos M., Hewett P. C., +2021b, MNRAS, 505, 3247 +Temple M. J., Hewett P. C., Banerji M., 2021c, MNRAS, 508, 737 +Timlin J. D., Brandt W. N., Ni Q., Luo B., Pu X., Schneider D. P., Vivek M., +Yi W., 2020, MNRAS, 492, 719 +MNRAS 000, 1–20 (2023) + +16 +M. J. Temple et al. +Timlin John D. I., Brandt W. N., Laor A., 2021, MNRAS, 504, 5556 +Titarchuk L., 1994, ApJ, 434, 570 +Vanden Berk D. E., Wesolowski S. C., Yeckley M. J., Marcinik J. M., Quash- +nock J. M., Machia L. M., Wu J., 2020, MNRAS, 493, 2745 +Vasudevan R. V., Fabian A. C., 2007, MNRAS, 381, 1235 +Vasudevan R. V., Fabian A. C., 2009, MNRAS, 392, 1124 +Verner D. A., Ferland G. J., Korista K. T., Yakovlev D. G., 1996, ApJ, 465, +487 +Vestergaard M., Osmer P. S., 2009, ApJ, 699, 800 +Vestergaard M., Wilkes B. J., 2001, ApJS, 134, 1 +Vietri G., et al., 2018, A&A, 617, A81 +Vietri G., et al., 2020, A&A, 644, A175 +Wang S., et al., 2022, ApJ, 925, 121 +Wild V., Hewett P. C., 2005, MNRAS, 358, 1083 +Wilkes B. J., 1984, MNRAS, 207, 73 +Wills B. J., Laor A., Brotherton M. S., Wills D., Wilkes B. J., Ferland G. J., +Shang Z., 1999, ApJ, 515, L53 +Wolf J., et al., 2020, MNRAS, 492, 3580 +Wolfire M. G., Vallini L., Chevance M., 2022, ARA&A, 60, 247 +Woods D. T., Klein R. I., Castor J. I., McKee C. F., Bell J. B., 1996, ApJ, +461, 767 +Xu F., Bian F., Shen Y., Zuo W., Fan X., Zhu Z., 2018, MNRAS, 480, 345 +Yang X.-H., Ablimit K., Li Q.-X., 2021a, ApJ, 914, 31 +Yang J., et al., 2021b, ApJ, 923, 262 +York D. G., et al., 2000, AJ, 120, 1579 +Yu Z., et al., 2022a, arXiv e-prints, p. arXiv:2208.05491 +Yu W., Richards G. T., Vogeley M. S., Moreno J., Graham M. J., 2022b, ApJ, +936, 132 +Zakamska N. L., Greene J. E., 2014, MNRAS, 442, 784 +Zanstra H., 1929, Publs Dom. Ap. Obs. Victoria, 4, 209 +Zappacosta L., et al., 2020, A&A, 635, L5 +Zhang X., Lu Y., 2019, ApJ, 873, 101 +Zhu Y., Bu D.-F., Yang X.-H., Yuan F., Lin W.-B., 2022, MNRAS, 513, 1141 +Zubovas K., King A., 2012, ApJ, 745, L34 +APPENDIX A: THE ROLE OF X-RAYS IN +PHOTOIONIZATION +This appendix shows that the X-ray portion of the SED is an insignif- +icant source of ionization for typical AGN emission line regions. +This is surprising since the SEDs shown in Fig. 2 have a signifi- +cant fraction of their power at high energies. The ultraviolet–X-ray +hardness ratio (𝛼ox) does correlate with emission line properties, as +summarized above, but this is not likely a direct relation with the +observed X-rays. +To illustrate this point, we use the hardest and softest SEDs from +our model grid (Section 3), corresponding to �𝑚 = 0.027 and �𝑚 = +1.000 at 𝑀BH = 1010𝑀⊙. These SEDs are shown in the top panel of +Figure A1, and have 𝛼ox = −0.05 in the hardest and 𝛼ox = −2.05 in +the softest cases. +The photoionization rate for a given shell 𝑛 is +Γ𝑛 = +∫ ∞ +𝜈0 +𝜎𝜈𝜙𝜈 𝑑𝜈[s−1] +(A1) +where 𝜈0, 𝜎𝜈, and 𝜙𝜈 are the photoionization threshold of shell 𝑛, its +energy-dependent photoionization cross section [cm2] , and the flux +of ionizing photons [cm−2 s−1 Hz−1] (Osterbrock & Ferland 2006). +The total photoionization rate is the sum over all shells, +Γtotal = +∑︁ +𝑛 +Γ𝑛[s−1] +(A2) +The flux of ionizing photons 𝜙𝜈 enters in the photoionization +rate (Eq. A1). This is the ratio 𝜙𝜈 = 4𝜋𝜈𝐽𝜈/(ℎ𝜈2) and is shown in +the lower panel of Fig. A1. The photon flux near 2 keV is typically +C K +He, C L +H +2 keV +φν Softest SED (αox=-2.05) +φν Hardest SED (αox=-0.05) +φν +φν [cm-2 Hz-1 s-1] +10−6 +10−3 +1 +1000 +106 +Hz +1014 +1015 +1016 +1017 +1018 +1019 +4πν Jν +C K +He, C L +H +2 keV +ν Fν Softest SED (αox=-2.05) +ν Fν Hardest SED (αox=-0.05) +4π ν Jν [erg cm-2 s-1] +105 +106 +107 +108 +109 +1014 +1015 +1016 +1017 +1018 +1019 +Figure A1. The upper panel shows the hardest and softest SEDs presented in +this paper. The lower panel shows the flux of photons 𝜙𝜈 for the same models. +Some important energies are indicated by the hashed lines near the bottom +of each panel. These show the ionization potentials of ground-state H0 and +He2+, the L and K shells of C2+, and 2 keV. The flux of ionizing photons is +orders of magnitude smaller at 2 keV compared to the flux at the He ii and +C iv photoionization edges even for the hardest SED. +∼6 dex fainter than the value near the peak. Photoionization is photon- +counting and there are relatively few high-energy photons despite +their significant energy. +The photon flux is multiplied by the photoionization cross section +to derive the photoionization rate (Eq. A1). We concentrate on C2+ +since photoionization of that ion produces C3+ and C iv 𝜆1549 emis- +sion. The shell-dependent cross sections for photoionization of C2+, +taken from Verner et al. (1996), are shown in Fig. A2. Both the 1s2 K +shell in the X-ray and the lower energy 2s2 L shell are shown. Both +shells have two electrons and, as expected, the peak photoionization +cross sections are similar. +MNRAS 000, 1–20 (2023) + +C iv and He ii emission in quasars +17 +C2+ cross section +K +L +σphoto [cm2] +10−21 +10−20 +10−19 +10−18 +Hz +1014 +1015 +1016 +1017 +1018 +1019 +Figure A2. The K and L shell photoionization cross sections for C2+ are +shown as a function of energy. The cross section are from the calculations by +Verner et al. (1996). These, together with the flux of photons (lower panel of +Figure A1), determine the photoionization rate (Equation A1) and the effects +of light upon matter. +Table A1. Photoionization rates for H0 and the K and L shells of C2+. +Shell +ΓSoftest [s−1] +ΓHardest [s−1] +H0 K +8.24e+00 +5.52e+00 +C2+ L +2.66e-01 +4.66e-01 +C2+ K +4.49e-03 +2.71e-02 +The photoionization rates for H0 and the two shells of C2+ are +listed in Table A1. The C2+ K-shell rate is 17 to 60 times smaller +then the L-shell rate. Both are ∼ 10 − 30 times smaller than the H0 +photoionization rate. From this comparison we expect that the effects +of the EUV and XUV will be more important than the X-ray, mainly +due to the larger number of softer photons. +Figure A3 shows the rate at which photons interact with matter for +our two reference SEDs and a solar composition. Calculations are +done with Cloudy version 22.01, as last described by Ferland et al. +(2017). Cloud parameters are typical of the C iv emitting region of +an AGN. The vertical axis is the total light-matter interaction rate at +a particular frequency and is the product of the photon flux and the +total gas opacity, evaluated for the appropriate chemical composition +and degree of ionization. The 50-912 Å ultraviolet region is ∼7 dex +more interactive than 2 keV X-rays. +The result that the X-rays hardly matter at all is surprising given the +shape of the SEDs in the upper panel of Figure A1. This is because +photoionization is photon counting, and the paucity of X-ray photons +cannot make up for their great energy. High-energy photons would +dominate the physics if softer parts of the SED were extinguished +so only X-rays strike the gas. Indeed, this is the ‘XDR’ model of +atomic and molecular regions of clouds exposed to ionizing radiation +(Wolfire et al. 2022). It would be difficult to detect this XDR emission +C K +He, C L +H +2 keV +Light-matter interaction rate +2 keV +φν σν Softest SED (αox=-2.05) +φν σν Hardest SED (αox=-0.05) +φν σν [s-1] +10−3 +1 +106 +109 +Hz +1014 +1015 +1016 +1017 +1018 +1019 +Figure A3. This shows the rate that photons interact with matter, the product +of the flux of photons and the total gas opacity. X-rays interact with matter +with a rate about ∼ 7 dex slower than the rate near the peak rate around +50 − 912Å. +since emission from the gas which absorbed the softer radiation +would be far stronger. This is discussed in Section 4.1 of Ferland +et al. (2013). The ultraviolet luminosity of a realistic SED has more +power than the relatively hard X-ray portion that drives an XDR. +The full SED striking a cloud produces successive H+/H0/H2 layers, +which are brighter than the deep X-ray heated regions. Emission +from regions powered by lower-energy light would dominate over +the XDR. +Correlations between the X-ray portion of the SED and emission- +line properties are observed. These are likely due to other correlations +with the softer portions of the SED, as evident in Figure 2, and not +the X-rays themselves. +APPENDIX B: ADDITIONAL OBSERVATIONS +In this Appendix we present additional observational results. First, +in Fig. B1 we present the He ii EW, the C iv blueshift and 𝛼ox as a +function of the FWHM of Mg ii 𝜆2800 and the 3000 Å continuum +luminosity. These two parameters are measured directly from the +SDSS spectroscopy and photometry respectively. By contrast, the +plots in the main text show observed properties as a function of +𝑀BH ∝ 𝐿0.5 +3000FWHM2 +Mg ii +(B1) +and +𝐿/𝐿Edd ∝ 𝐿3000/𝑀BH ∝ 𝐿0.5 +3000FWHM−2 +Mg ii. +(B2) +Given that both 𝑀BH and 𝐿/𝐿Edd depend on the observed parameters +FWHMMg ii and 𝐿3000, this might lead to induced correlations in the +𝑀BH–𝐿/𝐿Edd space. However, in practice our inferred 𝑀BH–𝐿/𝐿Edd +space is simply a rotation and reflection of the FWHMMg ii–𝐿3000 +space, where we (reassuringly) still see clear trends. Furthermore, we +see the same He ii behaviour as a function of FWHMMg ii and 𝐿3000 +in the X-ray detected sub-sample as in our full sample, meaning that +MNRAS 000, 1–20 (2023) + +18 +M. J. Temple et al. +1045 +1046 +1047 +νLν|3000˚ +A [erg s−1] +All sources +10000 +2500 +5000 +7500 +FWHM(MgII) [km s−1] +1045 +1046 +1047 +νLν|3000˚ +A [erg s−1] +0.5 +1 +2 +4 +EW(HeII λ1640) [˚A] +0 +250 +500 +750 +1000 +1250 +1500 +1750 +2000 +CIV λ1550 blueshift [km s−1] +1045 +1046 +1047 +νLν|3000˚ +A [erg s−1] +X-ray detected sources +10000 +2500 +5000 +7500 +FWHM(MgII) [km s−1] +1045 +1046 +1047 +νLν|3000˚ +A [erg s−1] +0.5 +1 +2 +4 +EW(HeII λ1640) [˚A] +−1.9 +−1.8 +−1.7 +−1.6 +−1.5 +−1.4 +−1.3 +−1.2 +αox +Figure B1. Observed quasar properties as a function of the FWHM of Mg ii 𝜆2800 and the 3000 Å continuum luminosity. Left panel: The median He ii EW +(top) and C iv blueshift (bottom) in our full sample of 191 391 objects. Right panel: The median He ii EW (top) and 𝛼ox (bottom) in our sub-sample of 5325 +X-ray detected sources. The He ii behaviour is identical in both panels (modulo the sample size), suggesting that our X-ray detected sub-sample is not biased +in terms of its ultraviolet emission properties. Moreover, clear differences are seen in the behaviour of He ii and 𝛼ox within the X-ray subsample: the strongest +He ii emission is seen only at low Mg ii FWHM while the strongest 2 keV X-ray emission is seen only at the lowest 3000 Å luminosities. +our X-ray detected objects are not obviously biased compared to our +full sample. +Second, we show the 2 keV X-ray continuum and He ii 𝜆1640 line +luminosities in Fig. B2. Assuming no changes in the BLR covering +factor, and that the He ii continuum is optically thick, 𝐿He ii can +be taken as a proxy for the continuum luminosity at 54 eV. These +two observables show qualitatively different behaviour: contours of +constant 𝐿2 keV are largely aligned with lines of constant 𝑀BH, which +is consistent with the assumption in qsosed that the hard X-ray power +law component emits a constant fraction of the Eddington luminosity. +He ii behaves in a much more complex manner, with the gradient +vector of increasing 𝐿He ii changing depending on the location in the +𝑀BH–𝐿/𝐿Edd space. +APPENDIX C: BOLOMETRIC CORRECTIONS +In the data, we apply a fixed bolometric correction of 5.15 to estimate +𝐿bol from 𝜈𝐿𝜈 at 𝜈 = 3000Å. In reality, the bolometric correction +will vary as a function of 𝑀BH and 𝐿/𝐿Edd. We discussed the vari- +ation of the bolometric correction from the qsosed model grid in +Section 5.1.1, showing a range in 𝑓bol by a factor of ≈ 2 − 3, where +𝑓bol ≡ 𝜈𝐿𝜈|3000Å/𝐿bol is calculated from each individual qsosed +model. Although a true ‘Apples versus Apples’ comparison is only +really possible with full knowledge of the intrinsic SED, in com- +paring our qsosed models with data we tried to match scalings and +biases in the data introduced by the fixed bolometric correction by +applying appropriate transformations to the qsosed outputs. For the +SMBH mass, the estimate in the data from the Mg ii line width but +also requires an estimate of the line formation radius, for which we +follow the usual method and assume that the BLR radius scales as +𝑅BLR ∝ 𝐿1/2. The 𝐿 in this expression should really be some appro- +priate ionizing luminosity, but 𝐿bol is normally used and we follow +this convention. As a result, the bolometric correction enters into the +SMBH mass estimate and implies a bias in the SMBH mass estimates +with respect to the true SMBH mass by factor of ( 𝑓bol/5.15)−1/2. +As a result, when plotting 𝑀BH along the 𝑥-axis of Figs. 5 and 4, we +apply the scaling +𝑀BH = ( 𝑓bol/5.15)−1/2𝑀BH, q, +(C1) +where 𝑀BH, q denotes the input qsosed grid value (the ‘true’ SMBH +mass). For 𝐿/𝐿Edd, correction factors appear in both the numer- +ator and denominator. 𝐿Edd ∝ 𝑀BH, introducing a bias factor +( 𝑓bol/5.15)−1/2 into the Eddington ratio estimate, while the numera- +tor is 𝐿bol and so contains a straightforward bias factor of 𝑓bol/5.15. +As a result, the relationship between the 𝐿/𝐿Edd plotted in Figs. 5 +and 4, and the dimensionless, Eddington-scaled accretion rate used +as input to qsosed is given by +𝐿/𝐿Edd = �𝑚 × ( 𝑓bol/5.15) × ( 𝑓bol/5.15)−1/2. +(C2) +MNRAS 000, 1–20 (2023) + +C iv and He ii emission in quasars +19 +1045 +1046 +1047 +νLν|3000˚ +A [erg s−1] +108 +109 +1010 +MBH [M⊙] +10−2 +10−1 +100 +L/LEdd +1042 +1043 +1044 +LHeII λ1640 [erg s−1] +1045 +1046 +1047 +νLν|3000˚ +A [erg s−1] +108 +109 +1010 +MBH [M⊙] +10−2 +10−1 +100 +L/LEdd +1044 +1045 +νLν|2 keV [erg s−1] +Figure B2. He ii line luminosity (left) and 2 keV X-ray continuum luminosity (right) as a function of SMBH mass, luminosity and Eddington ratio. Contours of +constant 𝐿2 keV are largely aligned with lines of constant 𝑀BH, consistent with the assumption in qsosed that the hard X-ray emission is equal to 2 per cent of +the Eddington limit. +108 +109 +1010 +MBH,q [M⊙] +10−2 +10−1 +100 +˙m ≡ ˙M/ ˙MEdd +108 +109 +1010 +MBH,q [M⊙] +10−2 +10−1 +100 +(ν Lν|3000˚ +A,q/LEdd) × 5.15 +108 +109 +1010 +MBH ≡ MBH,q × (fbol/5.15)−1/2 [M⊙] +10−2 +10−1 +100 +L/LEdd +0.03 +0.04 +0.06 +0.1 +L228/L1640 +Figure C1. An illustration of how the qsosed predictions change with differing treatments of the bolometric correction, focusing on the bottom-right panel of +Fig. 4. Left panel: the model outputs plotted as a function of the intrinsic, input values of �𝑚 and 𝑀BH, q. Centre panel: as in the left panel, but with the 𝑦-axis +is replaced with 𝐿bol/𝐿Edd where 𝐿bol is calculated from 𝐿3000 using a constant bolometric correction of 5.15. Right panel: as in the centre panel, but with the +𝑥-axis scaled by ( 𝑓bol/5.15)−1/2 and the 𝑦-axis scaled by ( 𝑓bol/5.15)1/2 to capture the impact of the bolometric correction on SMBH mass estimates in the +observed data. See the main Appendix text for details. +The effect of introducing these scaling factors as transformations +from the initial qsosed grid is shown in Fig. C1, to show how the +right-hand panels of Figs. 4 and 5 would change if we had made a +different presentation choice. The scale factors twist and distort the +simulation grid slightly from the original uniform parameter space, +but, overall, the effects are fairly modest because only square-root +terms distinguish the rightmost panel from the original input grid. +APPENDIX D: BLACK HOLE SPIN +In Fig. D1 we show how the predictions of 𝛼ox and our He ii 𝜆1640 +EW proxy (𝐿228/𝐿1640) from qsosed change if we instead con- +sider a maximally spinning SMBH. While the qualitative trends +in the 𝐿228/𝐿1640 are broadly in line with the low spin case, the +𝑎∗ = 1 models fail to reproduce the observed low values of 𝛼ox at +high Eddington fractions and SMBH masses (see discussion in Sec- +MNRAS 000, 1–20 (2023) + +20 +M. J. Temple et al. +1044 +1045 +1046 +1047 +νLν (erg s−1) +MBH = 109 M⊙ +10−1 +100 +˙m +1014 +1015 +1016 +1017 +1018 +1019 +ν (Hz) +1044 +1045 +1046 +1047 +νLν (erg s−1) +He II +αox +˙m = 0.15 +108 +109 +1010 +MBH [M⊙] +1045 +1046 +1047 +ν Lν|3000˚ +A [erg s−1] +108 +109 +1010 +MBH [M⊙] +10−2 +10−1 +100 +L/LEdd +−1.9 +−1.8 +−1.7 +−1.6 +−1.5 +−1.4 +−1.3 +−1.2 +αox +1045 +1046 +1047 +ν Lν|3000˚ +A [erg s−1] +108 +109 +1010 +MBH [M⊙] +10−2 +10−1 +100 +L/LEdd +0.06 +0.1 +L228/L1640 +Figure D1. Left panel: As Fig. 2, but for the maximally spinning case, 𝑎∗ = 0.998. The grey shaded area shows the range of the SEDs shown in Fig. 2 for the +non-spinning case. Centre and right panels: qsosed predictions for 𝛼ox and 𝐿228/𝐿1640 for a maximally spinning SMBH with 𝑎∗ = 0.998 (cf. Figs. 4 and 5). +tion 5.2.3). The reason for this can be understood from the left-hand +panel of Fig. D1, where we show the qsosed broadband spectrum +(the analogue to Fig. 2) for the maximally spinning case. Inspection +of the high �𝑚 models in the top-panel reveals that the 𝛼ox behaviour +is driven by a combination of stronger X-rays and the movement of +the peak of the thermal component. At high spin, the thermal peak +moves blueward to higher energies, such that the lower frequency +pivot point falls further from the peak and has lower flux compared +to the low spin model. In qsosed, this behaviour comes about in +a slightly convoluted way, but is driven by the decrease of the ra- +dius 𝑅warm (and corresponding temperature increase). This decrease +happens because 𝑅warm = 2𝑅hot, and 𝑅hot must move inwards as +spin increases, because 𝑅ISCO moves closer to the SMBH so 𝑅hot +must also decrease from eq. 2 of Kubota & Done (2018) to maintain +the model assumption that the dissipated power is 2 per cent of the +Eddington luminosity. One could clearly construct other models in +which the critical radii change in different ways when the spin is +changed, which is partly why we caution against over-interpreting +the fact that maximally spins appear difficult to reconcile with the +data. +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–20 (2023) + diff --git a/K9E0T4oBgHgl3EQfzwIE/content/tmp_files/load_file.txt b/K9E0T4oBgHgl3EQfzwIE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..39fe2054fccaac25b5e89ae54a99334f64f6a6c0 --- /dev/null +++ b/K9E0T4oBgHgl3EQfzwIE/content/tmp_files/load_file.txt @@ -0,0 +1,2140 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf,len=2139 +page_content='MNRAS 000, 1–20 (2023) Preprint 10 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='0 Testing AGN outflow and accretion models with C iv and He ii emission line demographics in 𝑧 ≈ 2 quasars Matthew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Temple ,1★ James H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Matthews ,2,3 Paul C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Hewett ,3 Amy L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Rankine ,4 Gordon T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Richards ,5 Manda Banerji ,6 Gary J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Ferland ,7 Christian Knigge 6 and Matthew Stepney 6 1Instituto de Estudios Astrofísicos, Universidad Diego Portales, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Ejército Libertador 441, Santiago 8370191, Chile 2Department of Physics, Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK 3Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK 4Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK 5Department of Physics, Drexel University, 32 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 32nd Street, Philadelphia, PA 19104, USA 6School of Physics & Astronomy, University of Southampton, Southampton SO17 1BJ, UK 7Department of Physics and Astronomy, The University of Kentucky, Lexington, KY 40506, USA Submitted to MNRAS 2022 December 22 ABSTRACT Using ≈190,000 spectra from the seventeenth data release of the Sloan Digital Sky Survey, we investigate the ultraviolet emission line properties in 𝑧 ≈ 2 quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Specifically, we quantify how the shape of C iv 𝜆1549 and the equivalent width (EW) of He ii 𝜆1640 depend on the black hole mass and Eddington ratio inferred from Mg ii 𝜆2800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Above 𝐿/𝐿Edd ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2, there is a strong mass dependence in both C iv blueshift and He ii EW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Large C iv blueshifts are observed only in regions with both high mass and high accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Including X-ray measurements for a subsample of 5,300 objects, we interpret our observations in the context of AGN accretion and outflow mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The observed trends in He ii and 2 keV strength are broadly consistent with theoretical qsosed models of AGN spectral energy distributions (SEDs) for low spin black holes, where the ionizing SED depends on the accretion disc temperature and the strength of the soft excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' High spin models are not consistent with observations, suggesting SDSS quasars at 𝑧 ≈ 2 may in general have low spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We find a dramatic switch in behaviour at 𝐿/𝐿Edd ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2: the ultraviolet emission properties show much weaker trends, and no longer agree with qsosed predictions, hinting at changes in the structure of the broad line region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Overall the observed emission line trends are generally consistent with predictions for radiation line driving where quasar outflows are governed by the SED, which itself results from the accretion flow and hence depends on both the SMBH mass and accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Key words: quasars: emission lines 1 INTRODUCTION 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1 Observational context: spectroscopic properties of quasars The spectroscopic properties of type-1 quasars have long been appre- ciated for their potential to provide insight into the physical processes responsible for luminous active galactic nuclei (AGN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Baldwin & Netzer 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Davidson & Netzer 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Kwan & Krolik 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Krolik & Kallman 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Elvis 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' These processes include the excitation of various line- and continuum-emitting regions, and mechanisms for launching outflows which might ‘feed back’ energy to their host galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Such processes are ultimately powered by accretion onto supermassive black holes (SMBHs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Lynden-Bell 1969), and thus de- pend primarily on the mass of the SMBH and the accretion rate, with potential second-order drivers including the spin of the SMBH and the metal content of the accreting material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The search for insight has gained much from identifying and ex- ★ E-mail: Matthew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='Temple@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='udp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='cl ploring the properties which are observed to vary the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Such diversity in the observed quantities must ultimately be driven by some of the physics which we would like to use to better constrain both the growth of SMBHs and their effect on the galactic ecosystems in which they reside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The most famous result of these investigations is arguably the identification of the so-called ‘eigenvector 1’ (EV1), which accounts for the largest amount of correlated variance in the optical spectra of low-redshift (𝑧 < 1) type-1 AGN spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Most authors now agree that the EV1 is driven by the mass-normalised accretion rate (the Eddington ratio), possibly convolved with some orientation effect (Boroson & Green 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Wills et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Sulentic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Shen & Ho 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Sun & Shen 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Sulentic & Marziani 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Similarly, the ultraviolet emission features in quasar spectra also show a rich phenomenology (Croom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Jensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Brodzeller & Dawson 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Early work by Baldwin (1977) showed that the equivalent widths (EWs) of various ultraviolet lines, most notably C iv 𝜆1549, were anti-correlated with the ultraviolet contin- uum luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Shang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2003) showed that this ‘Baldwin effect’ © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='02675v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='GA] 6 Jan 2023 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Temple et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' was independent of EV1 in 22 quasars with 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='4, implying dif- ferent physical drivers for these correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Early observations also demonstrated that the centroid of the C iv emission line is commonly shifted to the blue (Gaskell 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Wilkes 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Richards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Within a sample of 87 Palomar-Green quasars, Baskin & Laor (2004, 2005) found that large C iv blueshifts were only seen in objects with high Eddington ratios, although not all quasars with high Eddington ratios had large C iv blueshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The EV1 formalism was extended by Bachev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2004) and Sulentic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2007) to include the velocity shift of C iv, again finding that large C iv blueshifts are seen only in so-called ‘Population A’ quasars with high Eddington ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' With the start of the Sloan Digital Sky Survey (SDSS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' York et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2000), large samples of rest-frame ultraviolet quasar spectra became available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Equally as important were methods to accurately charac- terize the systemic redshift of each quasar (Hewett & Wild 2010), which is necessary to infer the velocity shift of any emission features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A notable work by Richards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2011) summarized the state of the field a decade ago at the time of the seventh data release (DR7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2010) from SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Using ≈35 000 quasar spectra, Richards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2011) confirmed the Baldwin effect and showed that the EW of C iv line also anti-correlates with the magnitude of the C iv blueshift: quasars with higher luminosities show, on aver- age, weaker C iv emission which is more strongly blueshifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C iv blueshifts could be a signature of emission from ionized gas being driven away from the accretion disc along the line-of-sight to the ob- server (Leighly & Moore 2004), in which case the results of Richards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2011) can be interpreted as brighter objects showing stronger emission from outflowing gas and weaker emission from the virial- ized broad line region (BLR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We discuss this interpretation further in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2, but do not assume anything about the origin of C iv line shifts when presenting our observational results in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Richards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2011) also demonstrated that the C iv properties are strongly correlated with the EW of the nearby He ii 𝜆1640 emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' More recent work by Rankine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2020) has shown that the correlations between the EW of He ii and both the EW and blueshift of C iv are also present in quasars with broad absorption features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We now know that the C iv and He ii properties are strongly correlated with the properties of other ultraviolet emission features such as Ly 𝛼, N v, Si iv and O iv] (Temple et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2021b), Fe iii, Al iii, Si iii] and C iii] (Temple et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2020), as well as the optical [O iii] emission (Vietri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Coatman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Vietri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2020), the strength of near infrared emission from dust at the sublimation temperature (Temple et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2021a), the strength of the far infrared emission (Maddox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2017), the radio detection fraction (Rankine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2021) and the strength of the 2 keV X-ray continuum (Kruczek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Zappacosta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Timlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Lusso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Marlar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Rivera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Tentative links have also been found between the C iv blueshift and the amount of continuum reddening ascribed to nuclear dust (Calistro Rivera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Fawcett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The existence of such correlations - between parameters which trace emission at different wavelengths and from different physical regions - suggests that they are driven (either directly or indirectly) by changes in some of the fundamental physical parameters which govern the properties of a SMBH and its surrounding regions, such as the SMBH mass, spin, and accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The space spanned by C iv blueshift and C iv EW therefore appears to be just as important as EV1 in understanding the physics of luminous AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' However, while the location of a given quasar spectrum on either EV1 or the C iv blueshift–EW space must ultimately be a function of the fundamental SMBH parameters, there is no guarantee that such a function is linear, or even injective (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' one-to-one with a well-defined inverse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' For example, we cannot rule out the possibility that two objects with different 𝑀BH and accretion rate have similar (or indeed identical) C iv emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 Theoretical context: AGN outflows and SEDs Mass outflows from AGN can be launched by thermal pressure, mag- netic forces, or radiation (Laha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Thermal winds can only be launched at large radii [𝑅 ≳ 105𝑅g ≈ 5 × (𝑀BH/109𝑀⊙) parsec] with terminal velocities of order 100–1000 km s−1 (Begelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Woods et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Mizumoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Faster outflows with speeds > 2000 km s−1, as commonly seen in broad high-ionization ultraviolet absorption features, are most likely launched on smaller (sub-parsec) scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Magnetically driven winds (Blandford & Payne 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Emmering et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Konigl & Kartje 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Fukumura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2021a) may be important in this context, but we cur- rently lack predictive models for how such winds would translate into observable quantities (although see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Bottorff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Chajet & Hall 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' On the other hand, radiation line driving (Castor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Murray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Murray & Chiang 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Elvis 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Proga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Proga & Kallman 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Proga 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Risaliti & Elvis 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Elvis 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Nomura & Ohsuga 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Nomura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2022) is intrinsically linked to the spectral energy distribution (SED) of the continuum which is responsible for both ionizing the transitions and then accelerating the flow by providing the source of radiation pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' By considering how the SED changes with SMBH mass and accretion rate, authors such as Giustini & Proga (2019) have developed unifying frameworks which make testable predictions for luminous AGN based on the physics of radiation line driven winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The ionizing continuum SED depends on the structure of the accretion flow, which in turn is set by the SMBH mass 𝑀BH, the mass-normalised accretion rate �𝑚 = �𝑀BH/ �𝑀Edd and the SMBH spin 𝑎∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Empirically, the optical–to–X-ray SEDs of AGN are seen to contain at least three distinct components (Elvis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Casebeer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Leighly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Done et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' First, any optically thick accretion disc will emit thermally, with larger radii being cooler, giving rise to a multi-temperature blackbody which is expected to peak in the near-ultraviolet (𝑀BH > 108𝑀⊙), far-ultraviolet or soft X-rays (𝑀BH < 108𝑀⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This part of the SED is commonly observed to peak around 1100 Å (Shang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Laor & Davis 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Stevans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Vanden Berk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2020), although this peak might be expected to depend on 𝑀BH if larger SMBHs have accretion flows which truncate at lower temperatures (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1 of Novikov & Thorne 1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Second, a hot Comptonised ‘corona’ emits a non-thermal power law which dominates the X- ray continuum above 1 keV (Haardt & Maraschi 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Titarchuk 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Finally, a ‘soft excess’ is seen in the X-rays below ≈1 keV, which is usually attributed to an intermediate warm Comptonising component (Petrucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This soft excess may be a significant contributor to the ionizing SED in the ≈100–1000 Å (≈10–100 eV) range, where many of the ultraviolet transitions are excited, but where direct observations of the continuum are not possible due to the high opacity of neutral hydrogen along the line-of-sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' For a line-driven disc-wind to emerge, the system needs strong ultraviolet emission to produce sufficient radiation pressure, but also a soft enough SED to avoid over-ionizing the gas (Murray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Higginbottom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' For each relevant line, the flux at the line energy combined with the line opacity determines the line-driving boost beyond radiation pressure from Thomson scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The line opacity depends on the ionization state, which is primarily sensitive MNRAS 000, 1–20 (2023) C iv and He ii emission in quasars 3 to the flux beyond some ionization edge (64 eV for C iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Line driving results when this effect is summed across many lines, each with their own energies, leading to a complex interplay between the flux of the SED underneath all the relevant lines in the ultraviolet and the flux of the SED beyond all the relevant ionization edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Giustini & Proga (2019) suggest that both 𝐿/𝐿Edd > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='25 and 𝑀BH > 108𝑀⊙ are required to satisfy these criteria and hence to power a strong outflow through radiation line driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Giustini & Proga (2019) also expect the 𝑀BH dependence of the observed outflow properties to be different above and below an �𝑚 of around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='25, where they expect the cold, optically thick accretion disc to extend down towards the innermost stable circular orbit (ISCO) and replace the hot, optically thin, inner accretion flow which is present at lower accretion rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In other words, they require 𝐿/𝐿Edd > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='25 to ensure emission from thermal disc emission dominates over that from the hot corona, to accelerate a strong line-driven wind without over-ionizing the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' With the quantity and quality of spectroscopic data which are now available from large surveys, these predictions from the Giustini & Proga (2019) framework can be tested empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3 Observational probes of quasar SEDs From an observational viewpoint, it is relatively easy to constrain the SED of an unobscured type-1 AGN in the rest-frame infrared, optical and X-ray wavebands, as photometric measurements can place direct constraints on the emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' For example, the strength of the rest- frame 2 keV X-ray continuum relative to the near-ultraviolet contin- uum has been shown to anti-correlate with the ultraviolet continuum luminosity in the so-called 𝛼ox–𝐿2500 Å relation (Avni & Tananbaum 1982, 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Steffen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Just et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Lusso & Risaliti 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Timlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2021), and the fractional contribution of the 2- 10 keV emission to the total bolometric power of the AGN is known to vary as a function of the accretion rate �𝑚 (Vasudevan & Fabian 2007, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' However, while this X-ray waveband can make a not insignificant contribution to the total emitted energy, it contributes a negligible number of ionizing photons to the photoionization budget of the BLR gas (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The number of ionizing photons is instead dominated by photons at the ionization edges themselves, which is of the order of 10-100 eV for the ultraviolet BLR (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' pro- duction edges of 15 eV for Mg ii, 54 eV for He ii and 64 eV for C iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This extreme ultraviolet (EUV) part of the SED is not directly ob- servable due to intervening absorption along the line-of-sight, but plays a crucial role in the physics of the BLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' To add to the complex- ity, the relative contribution of the warm Comptonising soft excess to the total EUV emission is likely to be varying as a function of 𝑀BH and �𝑚, meaning that the observable 2 keV continuum may not be a reliable proxy for the strength of the EUV SED at the ionization edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C iv is a resonant doublet transition with a complicated ionic struc- ture, so the strength of C iv emission is not necessarily a good tracer of the ionizing SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' However, it is instead possible to probe the EUV continuum using the He ii 𝜆1640 recombination line, which arises from a simple hydrogenic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' single electron) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Under the assumptions that the He ii emitting region is in equilibrium and that the He ii continuum is optically thick, the total rate of He ii- ionizing photons must balance the total number of recombinations such that each He ii 𝜆1640 line photon can be associated with an ion- izing continuum photon at or above 54 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This method was first used by Zanstra (1929) to infer stellar temperatures using the strength of Hydrogen recombination lines (section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='10 of Osterbrock & Ferland 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Following previous works (Mathews & Ferland 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Baskin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Ferland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Timlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2021), we will use the strength of He ii as a proxy for the strength of the ‘unseen’ EUV continuum which is ionizing the BLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='4 This work The first goal of this paper is to provide an up-to-date summary of our knowledge of the ultraviolet spectral properties of type-1 quasars, using the final data release (DR17) from the fourth iteration of SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This sample contains an order of magnitude more quasars than the SDSS DR7 sample used by Richards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The large sample size allows us to consider the emission properties as a function of both mass and Eddington ratio simultaneously, and thus provide a test of current SED models and disc-wind theories, which is our second goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' To best compare with theory and simulations, we present observed quantities such as the C iv blueshift, He ii EW, and 𝛼ox as a function of three physical parameters: the ultraviolet continuum luminosity, the SMBH mass estimated from the Mg ii 𝜆2800 emission line, and the inferred Eddington ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This relatively simple exercise has long been used to gain insight into the physics of AGN (Dibai 1980), but is subtly different from purely empirical approaches which observe trends in emission line properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' EV1 or the C iv blueshift-EW plane) and then try to infer which underlying physical parameters are driving those trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' By contrast, theoretical models make predic- tions for the SED and outflow properties as a function of the SMBH mass and accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In this work we confront such predictions directly with observations, showing that the ultraviolet emission lines display different behaviour above a threshold of 𝐿bol/𝐿Edd ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2, consistent with predictions for radiation line-driven winds, and find- ing good qualitative agreement between state-of-the-art SED models and observed continuum tracers in regions of parameter space where the models were not calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The structure of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In Section 2 we present the observational data, while in Section 3 we describe the SED models to which we compare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We present our key results in Section 4 and discuss their implications and limitations in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Throughout this work, wavelengths are given in vacuum in units of Ångströms, and we assume a flat ΛCDM cosmology with Ω𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='27, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='73 and 𝐻0 = 71 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Energies, frequencies and wavelengths are given in the rest-frame unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2 OBSERVATIONAL DATA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1 Rest-frame ultraviolet spectra The first aim of this paper is to quantify the behaviour of He ii 𝜆1640 and C iv 𝜆1549 as a function of SMBH mass 𝑀BH and Eddington ratio 𝐿/𝐿Edd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The 𝑀BH inferred from single-epoch measurements of C iv is known to be biased as a function of the emission line properties (Baskin & Laor 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Coatman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2016, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Mejía-Restrepo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2018), so we will instead use the velocity width of the Mg ii 𝜆2800 line to infer 𝑀BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We construct a sample of quasars from the SDSS with coverage of rest-frame wavelengths 1450-3000 Å to include C iv, He ii and Mg ii (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The original selection of the SDSS DR17 quasar sample was de- scribed by Lyke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2020) and Abdurro’uf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We post- process each spectrum using a sky subtraction routine conceptually similar to that described by Wild & Hewett (2005)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Systemic red- 1 Measurements of spectrum properties derived from observed-frame wave- lengths >6700 Å improve somewhat but none of the results, or conclusions, MNRAS 000, 1–20 (2023) 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Temple et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 108 109 1010 MBH [M⊙] 10−2 10−1 100 L/LEdd 10 20 40 100 200 400 1000 2000 Number of quasars per bin 1500 1550 1600 1650 Rest Wavelength [˚A] 1 2 3 fλ/f1700˚ A CIV HeII 1400 1600 1800 2000 2200 2400 2600 2800 3000 Rest Wavelength [˚A] 1041 1042 1043 1044 Lλ [erg s−1 ˚A−1] 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 < L/LEdd < 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='4 < MBH/M⊙ < 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1 < L/LEdd < 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='0 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='7 < MBH/M⊙ < 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='8 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 < L/LEdd < 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='4 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='4 < MBH/M⊙ < 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='6 < L/LEdd < 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='9 < MBH/M⊙ < 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='0 MgII CIV HeII Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Top left panel: the distribution of our sample of 191 391 quasars with redshifts 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 < 𝑧 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='65 in the 𝑀BH–𝐿/𝐿Edd plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Throughout this work, we only consider hexagonal bins where there are five or more quasars per bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' By construction, the FWHM of Mg ii 𝜆2800 increases from top-left to bottom-right of this parameter space, while the 3000 Å continuum luminosity increases from bottom-left to top-right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Right panel: composite spectra taken from different regions of the 𝑀BH–𝐿/𝐿Edd plane, as indicated by coloured boxes in the top left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A diversity of emission line properties is seen with changing SMBH mass and Eddington ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Bottom left panel: comparing the composite spectra in the region around C iv 𝜆1549 and He ii 𝜆1640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Here the spectra have been normalised at 1700 Å and plotted on a linear y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The EW of He ii can be seen to correlate with the profile of C iv: the high-mass, high-Eddington composite in blue displays weak lines and blueshifted C iv while the low-mass, high-Eddington composite in green shows much stronger line emission with no blue excess in C iv, consistent with fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 11 of Richards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2011) and fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A2 of Rankine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The difference here is that, instead of being constructed from C iv or C iii] emission properties, objects were included based on the FWHM of Mg ii and 𝐿3000 to represent regions of the 𝑀BH–𝐿/𝐿Edd plane, and also that the larger sample from SDSS DR17 includes fainter objects such as those contributing to the composite in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Composite spectra spanning the full range of the 𝑀BH–𝐿/𝐿Edd space will be made available as supplemental online-only material with the journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' shifts are calculated as described in section 3 of Rankine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Our redshift estimation routine uses the rest-frame 1600-3000 Å re- gion, deliberately excluding the C iv emission line, which is a key difference compared to the approach employed in the SDSS quasar catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The improved redshifts and sky-subtracted spectra will be described in a forthcoming publication by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Hewett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' To measure the emission line properties, we employ the spectral reconstructions from the Mean-Field Independent Component Analysis (ICA) car- ried out by Rankine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2020), which we have successfully used in our previous investigations into quasar emission line physics (Tem- ple et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2020, 2021a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The ICA-reconstructions provide a signif- icant improvement in the measurement of emission line properties, reducing the impact of the modest signal-to-noise ratio in the origi- nal spectra and the effect of weak absorption lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' intervening or outflowing C iv 𝜆𝜆1548,1550 doublets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We exclude objects with broad low-ionization absorption features (LoBALs) and poor ICA reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' To include both C iv and Mg ii in the observed spec- trum, we limit our sample to redshifts 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 < 𝑧 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Spectra from before the start of the BOSS survey (MJD 55000) were observed using the original SDSS spectrograph which had a more limited wavelength coverage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' for these objects we require 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='6 < 𝑧 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 to ensure coverage of C iv and Mg ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The spectra of quasars are required to possess a mean signal-to-noise ratio (per 69 km s−1SDSS pixel) of this paper change if the original DR17 reductions of the spectra are used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' ≥3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='0 over the rest-frame interval 1700-2200 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' These criteria leave a sample of 191 391 quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' To compute the EW of C iv emission, a power law continuum is de- fined using the median flux in the 1445-1465 and 1700-1705 Å wave- length windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This continuum is then subtracted from the spec- trum to isolate the line flux in the 1500-1600 Å wavelength region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The He ii EW is measured in the same way across the 1620-1650 Å wavelength region, using windows at 1610-1620 and 1700-1705 Å to define the continuum model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The C iv emission line ‘blueshift’ is de- fined as the Doppler shift of the wavelength bisecting the continuum- subtracted line flux: C iv blueshift ≡ 𝑐 × �𝜆rest − 𝜆median 𝜆rest � (1) where 𝑐 is the speed of light, 𝜆median is the rest-frame wavelength of the observed line centroid, and 𝜆rest = 1549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='48 Å is the mean rest-frame wavelength of the C iv 𝜆𝜆1548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='19,1550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='77 doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 X-ray data In addition to the rest-frame ultraviolet emission features, we can use the rest-frame 2 keV X-ray continuum emission to gain further insight into the spectral energy distributions of the quasars in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We therefore cross-match our sample of 191 391 objects to various X-ray catalogues from the literature, in order to build a large sample of rest-frame 2 keV measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 4189 objects from our sample of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 < 𝑧 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='65 objects with ultraviolet spectra are MNRAS 000, 1–20 (2023) C iv and He ii emission in quasars 5 included in the recent study of quasar X-ray properties by Rivera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2022), including 2820 with XMM-Newton observations from Lusso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2020), 1337 with Chandra observations from Timlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2020), and 32 with XMM-Newton observations from Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We make use of data from the second ROSAT All-Sky Survey (2RXS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Boller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2016) for objects included in the SDSS DR16 SPIDERS programme (Dwelly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Comparat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The flux limit for this survey is relatively bright so we use the Bayesian measurements described by Coffey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2019) which account for the Eddington bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A total of 45 objects from 2RXS are included in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Finally, we include 13, 11, and 8 objects with Chandra observations from Timlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2021), Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2018, 2022) and Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2022) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' These last three sub-samples were selected to have high 𝐿UV, weak C iv and strong C iv respectively, but the number of quasars is small and our results would be unchanged if we were to exclude them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' To improve our statistics for the number of objects with X-ray detections, we augment our sample with 1059 objects from the eROSITA Final Equatorial Depth Survey (eFEDS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2022) with redshifts 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 < 𝑧 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='65 and with SDSS griz photometry (Lyke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2020) or HSC griz photometry from the eFEDS catalogue (Salvato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The compilation results in a sample of 5325 quasars with measure- ments of both their ultraviolet (2500 Å) and X-ray (2 keV) continuum fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We use directly observed X-ray fluxes with no spectral fitting, though we have verified that the conclusions of this work would not change if we excluded objects which may be affected by absorp- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' From these fluxes we compute luminosities assuming a consis- tent cosmology (Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='4) across all sub-samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We then infer 𝛼ox, the logarithm of the ratio of the rest-frame 2 keV and 2500 Å monochromatic luminosities: 𝛼ox = log10 � 𝜈𝐿𝜈 � 2 keV − log10 � 𝜈𝐿𝜈 � 2500 Å, (2) as a measure of the relative strength of the X-ray emission in each source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Objects with smaller (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' more negative) 𝛼ox have weaker 2 keV X-ray emission relative to the ultraviolet continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3 Black hole masses and Eddington ratios We estimate SMBH masses using the single-epoch virial estimator described by Vestergaard & Osmer (2009), using the full width at half maximum (FWHM) of the Mg ii line: 𝑀BH = 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='86 � FWHM(Mg ii) 1000 km s−1 �2 � 𝐿3000 1044 erg s−1 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (3) where 𝐿3000 is the rest-frame monochromatic continuum luminosity 𝜈𝐿𝜈 at 3000 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This 𝑀BH estimator assumes a relationship between the radius of the Mg ii-emitting region and the observed 𝐿3000 which is independent of the shape of the ionizing SED, or more generally, independent of any changes in the accretion disc structure which may arise with changing 𝑀BH or accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We discuss this assumption further in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We infer the FWHM of Mg ii from our ICA reconstructions, which provide a more robust model of the intrinsic Mg ii profile than a conventional Gaussian fit in low signal-to-noise (𝑆/𝑁) spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Using a sub-sample with 𝑆/𝑁 > 10, we have verified that our Mg ii FWHM measurements are consistent with those obtained from fitting a single Gaussian to Mg ii together with an iron template (Vestergaard & Wilkes 2001) using the routine described by Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The key results of this paper would not change if we were to instead use such a Gaussian model for Mg ii, but there would be significantly more scatter in lower luminosity regions of parameter space where the spectral 𝑆/𝑁 is poorer on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The error budget on our resulting 𝑀BH is dominated by the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='55 dex uncertainty on the single-epoch estimator as described by Vestergaard & Osmer (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We infer 𝐿3000 by fitting a quasar SED model (Temple et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2021c) to griz photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' For sources in SDSS DR16 we use the SDSS photometry reported by Lyke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2020), and for eFEDS- selected sources in SDSS DR17 we use the Hyper-Suprime Cam (HSC) photometry reported by Salvato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Eddington lu- minosities are calculated in the usual way, balancing the gravitational and radiation forces in a Hydrogen-only plasma, and assuming the dominant source of opacity is Thomson electron scattering: 𝐿Edd = 4𝜋𝐺𝑀BH𝑚p𝑐 𝜎T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='26 × 1038 � 𝑀BH 𝑀⊙ � erg s−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (4) The Eddington ratio 𝐿bol/𝐿Edd (hereafter 𝐿/𝐿Edd) is then estimated assuming a constant bolometric correction of 𝐿bol = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='15 × 𝐿3000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We discuss this assumption further in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1, and show how our key observables depend directly on FWHM(Mg ii) and 𝐿3000 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Our sample of 191 391 quasars spans 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 dex in luminosity, with 𝐿3000 ≈ 1044.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5−47 erg s−1 and 𝐿bol ≈ 1045−47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 erg s−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We infer SMBH masses in the 108−10𝑀⊙ range and Eddington ratios from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='01 to unity, with the distribution of our sample shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 3 MODELING THE QUASAR SED Our second goal is to confront observational data with models for accretion and outflow in quasars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' more specifically, we aim to test if the changes in observed emission line and continuum properties with 𝑀BH and Eddington ratio are consistent with theoretical models for the SED of the ionizing continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' To this end, we make use of qsosed2 (Kubota & Done 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We used the implementation of qsosed in xspec (Arnaud 1996) to calculate SEDs, via the PyXSPEC python wrapper (Gordon & Arnaud 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In qsosed, the radiation originates from three characteristic re- gions: an outer thermal disc, an inner hot Comptonising ‘corona’ and an intermediate warm Comptonising component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' These three regions are assumed to be radially stratified as defined by four criti- cal radii: 𝑅ISCO < 𝑅hot < 𝑅warm < 𝑅out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The inner and outer radii are defined by the radius of the innermost stable circular orbit 𝑅ISCO and the self-gravitation radius 𝑅out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The hot X-ray component origi- nates from 𝑅ISCO < 𝑅 < 𝑅hot, and has a luminosity set by the sum of the directly dissipated power, 𝐿diss,hot, and the seed photon lumi- nosity, 𝐿seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' One of the key aspects of the model is the empirically motivated assumption that the dissipated power is always 2 per cent of the Eddington luminosity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' this constraint defines the value of 𝑅hot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The outer radius of the warm Comptonising component 𝑅warm is set to be twice 𝑅hot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' For 𝑅warm < 𝑅 < 𝑅out, the thermal disc component is assumed to emit as described by Novikov & Thorne (1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' qsosed has four physical input parameters: the cosine of the incli- nation, cos𝑖, the SMBH mass, 𝑀BH, the dimensionless spin param- eter, 𝑎∗, and the Eddington-scaled accretion rate, �𝑚 ≡ �𝑀acc/ �𝑀Edd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We fix cos𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 and calculate grids of models in (𝑀BH, �𝑚) parame- ter space, for non- and maximally-spinning SMBHs 𝑎∗ ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We calculate models with 21 logarithmically-spaced grid points in each direction, spanning the ranges 8 ≤ log(𝑀BH/𝑀⊙) ≤ 10 and −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='65 ≤ log �𝑚 ≤ 0, corresponding to intervals of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='0825 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' To compare with observations, we take the input SMBH mass and 2 https://heasarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='gov/xanadu/xspec/manual/ node132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='html MNRAS 000, 1–20 (2023) 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Temple et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' calculate the Eddington ratio from 𝐿3000 using the same bolometric correction of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='15 that we apply to the observational data (but see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1 and Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Here, and in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2, we present models for only the non-spinning case, as these are in much better agreement with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We discuss the impact of SMBH spin and system inclination in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3 and models with 𝑎∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='998 are presented in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Although the emission line properties must depend on the ioniz- ing SED, the exact relationship between, for example, C iv EW and the SED is complex due to a number of confounding factors such as BLR geometry, density and radiative transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The relationship to any kinematic signatures such as C iv blueshift is even more complicated and would require a physical model for the line formation region and associated flow dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A somewhat simpler case is the EW of He ii 𝜆1640, which is a recombination line and therefore a reasonable ‘photon counter’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' He ii has history as a tracer of the EUV continuum: for example, in cataclysmic variables Hoare & Drew (1991) applied a modified Zanstra (1929) method to infer boundary layer temper- atures, and in quasars, Leighly (2004) note that a high He ii EW is indicative of a strong X-ray continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Assuming Case B recombi- nation, Mathews & Ferland (1987) give the He ii 𝜆1640 EW in terms of the 228 Å continuum flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Their equation can be inverted to give the proportionality 𝐹𝜈(𝜆228) 𝐹𝜈(𝜆1640) ∝ EW(He ii 𝜆1640) Ω 4𝜋 , (5) where Ω/4𝜋 is the covering fraction and the proportionality constant is dependent on the shape of the SED (Mathews & Ferland 1987 considered a power law in 𝐹𝜈 at 228 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In this work we assume, based on the above equation, that the observed He ii 𝜆1640 EW is a reasonable proxy for the ratio of continuum luminosities 𝐿228/𝐿1640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2 we present output SEDs from qsosed, in which the three radially stratified components can be seen as separate ‘bumps’ in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In these plots, we show how the model SEDs change as a function of Eddington-scaled accretion rate, �𝑚 (for fixed mass, top panel) and SMBH mass, 𝑀BH (for fixed �𝑚, bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The important frequencies for determining He ii EW (corresponding to 228 Å and 1640 Å) and 𝛼ox (corresponding to 2500 Å and 2 keV) are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Increasing �𝑚 increases the overall luminosity of the system and pushes the peak of the outer thermal disc component to higher frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Simultaneously, the hard X-ray slope becomes signifi- cantly softer and 𝐿2keV only increases slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' As a result, the higher Eddington fraction objects are more X-ray weak relative to their ul- traviolet flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Increasing 𝑀BH also increases the total luminosity, but now the peak of the thermal component moves to lower frequencies and the hard X-ray slope stays fairly constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In both panels of the plot the peak of the SED can be found on either side of the low frequency pivot points for both He ii EW and 𝛼ox, resulting in an interesting interplay between these quantities and the fundamental AGN parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 4 RESULTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1 Observed properties in 𝑀BH–𝐿/𝐿Edd space The first observational result from this work is the behaviour of the C iv 𝜆1549 emission line morphology as a function of SMBH mass (𝑀BH) and Eddington ratio (𝐿/𝐿Edd), shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In the left panel, we show the C iv emission line blueshift (as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 1) and in the right panel the EW of C iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In the top panels, consistent with previous works, we find that more luminous quasars show weaker emission line strengths relative to the continuum and 1044 1045 1046 1047 νLν (erg s−1) MBH = 109 M⊙ 10−1 100 ˙m 1014 1015 1016 1017 1018 1019 ν (Hz) 1044 1045 1046 1047 νLν (erg s−1) He II αox ˙m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='15 108 109 1010 MBH [M⊙] Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Output SEDs in 𝜈𝐿𝜈 units from qsosed for 𝑎∗ = 0 and cos 𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The vertical lines show, from left to right, the frequencies at 2500 Å, 1640 Å, 228 Å (= 54 eV) and 2 keV which together determine 𝛼ox and the EW of He ii 𝜆1640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The He ii ionization edge at 54 eV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3×1016 Hz) lies in the EUV regime where the intermediate warm Comptonising component in qsosed is most important, but the EW of He ii can also be seen to depend on the location of the peak of the ionizing SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Top panel: SEDs with fixed SMBH mass of 109 𝑀⊙ and varying �𝑚 in logarithmic intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' As �𝑚 increases the peak of the SED moves to the blue, the luminosity increases, and the hard X-ray power law spectral index becomes softer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Bottom panel: SEDs with fixed �𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='15 and varying 𝑀BH in logarithmic intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' As SMBH mass increases the peak of the SED moves to the red, and the luminosity increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A maximal spin analogue to this plot is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' stronger emission line blueshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' However, when considering the observed C iv properties as a function of both 𝑀BH and 𝐿/𝐿Edd (bottom panel), we see a more complicated behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' To observe the strongest C iv blueshifts (which are associated with the smallest EWs), we need to look at objects with both 𝑀BH ≳ 109𝑀⊙ and 𝐿/𝐿Edd ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Moreover, the contours of constant C iv blueshift follow acute-angled ‘wedge’ shapes, which are somewhat orthogonal to lines of constant luminosity (running diagonally top-left to bottom- right in the 𝑀BH–𝐿/𝐿Edd space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' At the same time, objects with the strongest C iv EWs ≳ 100 Å, which have strong symmetric emission with little or no blueshift, are found at high 𝐿/𝐿Edd ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 and relatively low 𝑀BH ≲ 109𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' To help us to understand the physical drivers behind the trends seen in C iv, in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 4 and 5 we also show 𝛼ox and He ii EW across the same 𝑀BH–𝐿/𝐿Edd parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The 𝛼ox behaviour is as expected from previous works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2022), largely with more luminous objects displaying relatively weaker X-ray emission which is quantified by a more negative 𝛼ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A more interesting result MNRAS 000, 1–20 (2023) C iv and He ii emission in quasars 7 1045 1046 1047 νLν|3000˚ A [erg s−1] 108 109 1010 MBH [M⊙] 10−2 10−1 100 L/LEdd 0 250 500 750 1000 1250 1500 1750 2000 CIV λ1550 blueshift [km s−1] 1045 1046 1047 νLν|3000˚ A [erg s−1] 108 109 1010 MBH [M⊙] 10−2 10−1 100 L/LEdd 20 40 60 80 100 120 EW(CIV λ1550) [˚A] Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The median observed C iv blueshift (left) and EW (right) in bins of SMBH mass, 3000 Å ultraviolet continuum luminosity (top) and Eddington ratio (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Data are shown only for bins which contain five or more objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The C iv blueshift and EW are seen to anti-correlate: areas of parameter space with strong blueshifts have weak EW and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 𝐿/𝐿Edd ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 is a necessary but not sufficient condition for observing the largest C iv blueshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The strongest C iv blueshifts are observed only at large SMBH mass and large Eddington ratio, while high EW C iv emission is observed at large Eddington ratio and smaller mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The Baldwin effect can be observed in the sense that objects with brighter 3000 Å luminosities tend to have weaker C iv EWs on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' However, the C iv EW behaviour as a function of 𝑀BH and 𝐿/𝐿Edd shows that the underlying drivers of the Baldwin effect are more complicated than a simple dependence on the ultraviolet luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' is seen in the EW of He ii, which is even more striking than the behaviour seen in C iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' With the He ii EW, there is a clear transition around 𝐿/𝐿Edd ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2, with both the strongest and weakest line emission only seen above this threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Below this Eddington limit, there is little change in the average line properties as a function of mass, but at 𝐿/𝐿Edd ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 there is a strong mass dependence with diagonal wedge-shaped contours similar to those observed in C iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' By contrast, the contours of constant 𝛼ox are much less closely aligned with contours of constant C iv blueshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' To test the robustness of these trends, we divide the 𝑀BH–𝐿/𝐿Edd into square bins of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1 by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1 dex and compute the median absolute deviation (MAD) in each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The typical MAD is 290 km s−1 in C iv blueshift, 13 Å in C iv EW and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 Å in He ii EW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The typical scatter within each bin is therefore significantly less than the dynamic range in the average emission line properties shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 3 and 5, meaning that one is unlikely to find individual objects which go against the overall trend of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Dividing through by the median in each bin, the typical MAD/median in each bin is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='29 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='24 for the He ii and C iv EWs respectively, meaning that the typical range of emission line EW within each 𝑀BH–𝐿/𝐿Edd bin is a factor of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 and 4 for He ii and C iv respectively, compared with the dynamic range of more than a factor of six seen in the median per-bin line properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 Comparison with model SEDs In the right-hand panels of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 4 and 5 we show how 𝛼ox and 𝐿228/𝐿1640, respectively, vary with mass and Eddington fraction, as modeled by qsosed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' These plots can be compared to the respective plots from the observational sample (left-hand panels), albeit with some caveats regarding bolometric corrections (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1) and 𝑀BH estimates (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In a qualitative sense, the models do a reasonably good job of reproducing the trends observed in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Focusing first on 𝛼ox, we can see that the general trend of decreasing 𝛼ox with Eddington fraction is reproduced, and, in addition, the gradient is stronger at high 𝑀BH, as observed in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' To put this another way, in both the data and model results, the contour of fixed 𝛼ox curves around, from being nearly horizontal at high 𝑀BH to being closer to vertical at low 𝑀BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The dynamic range of model 𝛼ox values is comparable to that observed, but the models do not produce soft enough spectra to match the data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 𝛼ox ≈ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='9 can be found in some bins in the quasar sample but the minimum value of 𝛼ox in the models is −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The comparison of the model 𝐿228/𝐿1640 ratio and the observed He ii 𝜆1640 EW is also broadly encouraging, at least at relatively high Eddington fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This finding is perhaps more interesting as the He ii EW is probing a portion of the SED that is not accessible directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The basic behaviour, of decreasing He ii EW with 𝑀BH at MNRAS 000, 1–20 (2023) 8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Temple et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 1045 1046 1047 νLν|3000˚ A [erg s−1] 108 109 1010 MBH [M⊙] 10−2 10−1 100 L/LEdd −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='9 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='7 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='4 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 αox 1045 1046 1047 ν Lν|3000˚ A [erg s−1] 108 109 1010 MBH [M⊙] 10−2 10−1 100 L/LEdd −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='9 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='7 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='4 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 αox Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Left panel: The median observed 𝛼ox in bins of SMBH mass, 3000 Å ultraviolet continuum luminosity (top) and Eddington ratio (bottom) for the 5325 objects from our sample with 2 keV X-ray measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Data are shown only for bins which contain five or more objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Right panel: the predicted 𝛼ox from low spin qsosed models in the same parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The observations show good agreement with the models, with 𝛼ox more negative (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' more X-ray weak) in objects with brighter ultraviolet luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' D1 we show equivalent models but with high spin, which do not show such agreement with the observations, suggesting that the 𝑧 ≈ 2 SDSS quasar population may be more consistent with low SMBH spins on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' high Eddington fractions, is well matched by the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The models also capture the diagonal contours of constant He ii EW, in which the transition to low He ii EWs occurs at higher masses for higher Eddington fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' As discussed above, at low Eddington fractions (𝐿/𝐿Edd ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2), something fundamentally switches in the data, with gradients generally being shallower and along a different direction in the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This relatively sharp change is not reproduced by the models, and may be telling us something fundamental about the quasar accretion process (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1 for a discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 5 DISCUSSION We have quantified the average behaviour of C iv 𝜆1549, He ii 𝜆1640 and 𝛼ox as a function of both 𝑀BH and 𝐿/𝐿Edd, and compared our observations with predictions from qsosed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In this section we now discuss these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We first outline the key caveats in our findings (Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2), before discussing possible interpretations of our results within the context of AGN accretion and outflow theories (Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Finally, we discuss some wider implications and possible future applications (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3), before summarizing our key conclusions in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1 Key assumptions and limitations 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1 Bolometric corrections A large part of this work has attempted to quantify the ‘unseen’ ex- treme ultraviolet (EUV) portion of the SED which is not directly observable, but which can instead be probed via the He ii emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This portion of the SED contributes a significant amount to the bolometric luminosity of a quasar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' To estimate bolometric luminosi- ties (and Eddington ratios 𝐿bol/𝐿Edd), we have assumed a constant bolometric correction 𝑓bol ≡ 𝐿bol/𝐿3000 of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='15, consistent with previous works in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Richards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Krawczyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' However, we have also shown that the He ii strength is changing as a function of 𝑀BH and 𝐿/𝐿Edd, so we expect the strength of the EUV continuum and hence the bolometric correction to be varying with 𝑀BH and 𝐿/𝐿Edd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Using our qsosed models, we attempt to quantify this effect in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' While our chosen value of 𝑓bol = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='15 lies within the range of values spanned by our grid of model SEDs, there is variation of around a factor of two in 𝑓bol depending on the values of 𝑀BH and 𝐿/𝐿Edd we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' While this could in principle lead to systematic biases in our estimation of 𝐿/𝐿Edd, we show in Appendix C that these biases are likely to be small compared to the magnitude of the trends we observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We can however, briefly describe what might happen if we were to adopt a non-constant bolometric correction when inferring 𝐿/𝐿Edd from our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' For two objects, both at �𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2, the 𝑓bol MNRAS 000, 1–20 (2023) C iv and He ii emission in quasars 9 1045 1046 1047 νLν|3000˚ A [erg s−1] 108 109 1010 MBH [M⊙] 10−2 10−1 100 L/LEdd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 1 2 4 EW(HeII λ1640) [˚A] 1045 1046 1047 ν Lν|3000˚ A [erg s−1] 108 109 1010 MBH [M⊙] 10−2 10−1 100 L/LEdd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1 L228/L1640 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Left panel: The median observed He ii EW in bins of SMBH mass, 3000 Å ultraviolet continuum luminosity (top) and Eddington ratio (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Data are shown only for bins which contain five or more objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Right panel: the predicted strength of He ii ionizing photons at 228 Å relative to the 1640 Å continuum from qsosed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Above an Eddington ratio of ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2, there is a strong trend as a function of SMBH mass, with high mass objects showing the weakest He ii emission and low mass objects showing the strongest He ii emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The model predictions show qualitatively similar behaviour in this region of parameter space, explaining the diagonal contours in constant He ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Below 𝐿/𝐿Edd ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2, the observed He ii displays much weaker trends, and does not agree with the model predictions, suggesting that in this regime either the SED models are less accurate or the structure of the BLR is changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' inferred from the qsosed models would be ≈6 and ≈3 for 𝑀BH = 108𝑀⊙ and 1010𝑀⊙ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This would skew the observations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 5, moving the location of the strongest He ii EW (at low 𝑀BH) to larger 𝐿/𝐿Edd, more in line with the 𝐿/𝐿Edd threshold at high 𝑀BH above which we see the weakest He ii and largest C iv blueshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 Black hole mass estimates As well as the assumption of a constant bolometric correction, we have used a single-epoch virial estimator to estimate SMBH masses throughout this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The caveats associated with such estimates are numerous and have been reviewed by Shen (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Here we discuss some of the issues which are most relevant to our method and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Most notably, the BLR radius–luminosity relation (as encoded through the virial 𝑓 factor) may depend on the shape of the SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Other uncertainties arising from (for example) orientation are likely to be random, in the sense that they will add scatter to our 𝑀BH estimates but should not bias our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' While it is possible that our observed distribution of quasars in the 𝑀BH–𝐿/𝐿Edd plane is not the same as the intrinsic distribution, the fact that we do still observe such striking behaviour in the He ii and C iv emission line properties as a function of our inferred 𝑀BH and 𝐿/𝐿Edd is telling us that any random scatter or noise in our 𝑀BH estimates is small enough not to ‘wash out’ the observed trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We used the FWHM of the Mg ii line to estimate 𝑀BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2008) showed that such Mg ii-derived 𝑀BH estimates corre- late tightly with those derived from H𝛽 across the full 108−10 𝑀⊙ mass range, with the distribution of log � 𝑀H𝛽 BH �𝑀Mg ii BH � following a Gaussian with mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='034 and dispersion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='22 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Shen & Liu (2012) extended this analysis to higher redshifts and higher lumi- nosities, more appropriate for the objects in this work, and again found that the Mg ii properties remained well correlated with those of H𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The Mg ii-derived 𝑀BH estimates we use in this work are therefore unlikely to be biased compared to those which we would have derived from a single-epoch H𝛽 measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The possibility remains, however, that such estimates are biased as a function of the SED, or equivalently, as a function of 𝑀BH and 𝐿/𝐿Edd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Early concerns about the universality of the BLR radius– luminosity relation were discussed by Kaspi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2005) and Collin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' More recently, various authors have tried to account for possible SED-dependent biases in single-epoch 𝑀BH estimates (Du & Wang 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Dalla Bontà et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Fonseca Alvarez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Martínez-Aldama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2020), either using the accretion rate directly or by using the strength of optical iron emission 𝑅Fe ii as a proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' However, Khadka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2022a,b) and Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2022a) find MNRAS 000, 1–20 (2023) 10 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Temple et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 1044 1045 1046 1047 νLν|3000˚ A [erg s−1] 4 6 8 10 12 fbol fbol = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='15 108 109 1010 MBH [M⊙] Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The predicted bolometric correction, 𝑓bol ≡ 𝐿bol/𝐿3000, as a func- tion of 𝐿3000, from qsosed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The points are colour-coded by 𝑀BH with a logarithmic normalisation, and points of constant mass are joined with solid lines so that the trends with Eddington ratio can be understood by following individual lines from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The adopted bolometric correction in this work, 𝑓bol = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='15, is shown as a horizontal dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 𝑓bol ranges from ≈ 3 − 10, and our adopted 𝑓bol is bounded by this range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' however 𝑓bol does have a clear dependence on mass and luminosity in the model SEDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Our assumption of a fixed 𝑓bol could lead to an artificially reduced dynamic range in the inferred 𝐿bol at 𝑀BH = 108𝑀⊙ and an artificially increased range of 𝐿bol at 𝑀BH = 1010𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' the opposite result, with the inclusion of 𝑅Fe ii having no effect on the scatter in either the Mg ii or H𝛽 radius–luminosity relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' While the literature remains divided, we argue it is still true that any SED-dependent bias in our single-epoch 𝑀BH estimates must be contained within the scatter on the BLR radius–luminosity relation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This scatter is smaller than the range spanned by our sample by a factor of ≈3, meaning that SED-dependent bi- ases in our 𝑀BH cannot explain the observed trends presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 Quasar physics 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1 AGN accretion models In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2, we compared our observations with predictions from the qsosed models of Kubota & Done (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The predictions for 𝛼ox made by these models have recently been tested over a much broader parameter space (𝑀BH ≈ 107−10𝑀⊙ and 𝐿3000 ≈ 1043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5−47 erg s−1 ) by Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2022), who find that the qsosed model predicts the optical and X-ray SED fairly well for 𝑀BH ≲ 109𝑀⊙, but that at higher masses the outer accretion disc spectra are predicted to be too cool to match the observed data, especially at lower Eddington ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This finding is consistent with our result (in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 4) that the 2 keV emission is predicted to be slightly stronger (relative to the 2500 Å emission) than observed at 𝑀BH ≈ 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In this work we have also quantified the He ii emission, which provides a new, complementary probe of the ionizing SED across the 𝑀BH–𝐿/𝐿Edd space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 we found that, for 𝐿/𝐿Edd ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2, the observed He ii EW is qualitatively similar to the behaviour of the 54 eV ionizing SED predicted by the qsosed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The observations are consistent with a scenario in which (at least for 𝐿/𝐿Edd ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2) the strength of He ii emission is set directly by the ionizing photon luminosity at 54 eV, and thus that He ii is providing a probe of the EUV which is not directly observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Moreover, the observed He ii EW behaviour provides further evidence for the soft excess to be an intermediate, warm Comptonising component which behaves in the way in which the qsosed models predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The strongest and weakest 228 Å emission (relative to the 1640 Å continuum) are both produced at high Eddington ratios, at low (≈ 108𝑀⊙) and high (≈ 1010𝑀⊙) SMBH mass respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' However, the match between the observed He ii and the predicted strength of the 54 eV ionizing luminosity is not perfect, especially in the 𝐿/𝐿Edd ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This mismatch might suggest a decou- pling between the He ii EW and the 228 Å continuum flux at these Eddington ratios, perhaps if changes in the BLR covering factor lead to differences in the fraction of the continuum source which is re- processed into emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Another possibility is that the He ii continuum becomes optically thin, for instance if the density of the BLR were to decrease (which could indicate the absence of a dense outflow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Alternatively, the mismatch could suggest that the SED models are inaccurate in this Eddington ratio regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Intriguingly, this regime is similar to the region of the 𝑀BH–𝐿2500 Å space where Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2022) find a mismatch between the observed and pre- dicted 𝛼ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Either way, the observed switch in He ii behaviour above and below 𝐿/𝐿Edd ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 (which is not reflected in the qsosed mod- els) suggests that something fundamental is changing in the structure of either the BLR or the accretion flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C iv is a resonant doublet with a more complicated ionic structure than He ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' However, the close correspondence of the C iv blueshift and EW with the He ii EW, allied with the fact that the He ii behaviour can be consistently explained with trends in the SED, suggests that the C iv morphology is governed by accretion physics - specifically the shape of the SED in the near and extreme ultraviolet regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Given the proximity of the C iv and He ii ionization edges, at 64 eV and 54 eV respectively, this result is perhaps unsurprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' More notable is the fact that the observed 𝛼ox behaviour does not correspond so well to the C iv morphology, as demonstrated by the differing trends in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This shows how the 2 keV X-ray emission is a weaker proxy than He ii for the physical mechanisms which control the C iv emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Finally, the fact that our 𝐿/𝐿Edd ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 observations of He ii and 𝛼ox can be well explained by changes in the predicted model SEDs is significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The qsosed models were calibrated using observations of just three objects with 𝑀BH = 108𝑀⊙ (Kubota & Done 2018), and had not been critically assessed at higher SMBH masses prior to the recent work of Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The fact that the models agree with our observations across a relatively wide range of parameter space was not necessarily to be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Moreover, the theoretical models show that the different phenomenological behaviours ob- served in 𝛼ox and He ii (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 4 and 5) have natural origins in the SED, and that the observations presented in this work are broadly in agreement with our current understanding of the accretion physics in AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 AGN outflow models In this subsection we test the predictions made by Giustini & Proga (2019), who summarize current understanding of AGN accretion and outflow mechanisms with a particular focus on the physical condi- tions required to drive powerful winds from the accretion disc through radiation line driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We note again that the picture described by Giustini & Proga (2019) might not be the only plausible model for AGN outflows, but we choose to compare with their picture as it pro- MNRAS 000, 1–20 (2023) C iv and He ii emission in quasars 11 vides clear testable predictions within a well-defined framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In particular, Giustini & Proga (2019) suggest that both 𝐿/𝐿Edd > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='25 and 𝑀BH > 108𝑀⊙ are required to power strong outflows from AGN through radiation line driving: below these thresholds the X-ray flux is strong enough to over-ionize material and the ultraviolet flux will be too low to accelerate a line-driven wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' For the purposes of this comparison, we assume that any blue- wing excess in the C iv emission line profile is tracing an outflow along the line-of-sight from the accretion disc, and hence that the blueshift presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 3 is a measure of the strength of emission from the outflowing wind (Leighly & Moore 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Richards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The origin of the C iv emission line blueshift is still debated (see Gaskell & Goosmann 2013, 2016, for an alternative view), but a growing body of work is connecting the C iv emission morphology with more unambiguous tracers of line driven winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' For example, the strengths and velocities of broad C iv absorption troughs have been shown to correlate with the C iv emission blueshift (Rankine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Rodríguez Hidalgo & Rankine 2022), and the velocities of narrow C iv line-locked ‘triplet’ absorption features are also strongly correlated with the emission blueshift (Rankine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' For the discussion in this subsection, we therefore assume that objects with larger C iv blueshifts have stronger disc winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' To be more precise, the C iv blueshift is taken as a measure of the strength of emission from outflowing gas relative to the strength of emission from virialized gas at the systemic redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In this paradigm, the trends seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 3 are in good agreement with the picture proposed by Giustini & Proga (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We see large (≳ 1000 km s−1) median C iv blueshifts only in bins with high SMBH masses and high Ed- dington ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Furthermore, we do indeed see a more complicated mass dependence above 𝐿/𝐿Edd ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' High 𝐿/𝐿Edd is therefore a necessary, but not sufficient, condition for observing large C iv blueshifts, consistent with the results of Baskin & Laor (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In detail, we only observe strong outflow signatures in objects with 𝑀BH ≳ 109𝑀⊙, which is somewhat higher than the criterion of 𝑀BH ≳ 108𝑀⊙ proposed by Giustini & Proga (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Requiring 𝑀BH ≳ 109𝑀⊙ and 𝐿/𝐿Edd ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 together ensures that the crite- rion 𝐿bol ≳ 1045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 erg s−1 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Above this 𝐿bol threshold, Zakamska & Greene (2014) suggest that quasar winds are capable of driving ionized gas (as traced by [O iii] 𝜆5008 emission) beyond the escape velocity of the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The kinematics of C iv and [O iii] are known to correlate (Coatman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2019), and our observed C iv blueshift behaviour is therefore consistent with the conclusion of Za- kamska & Greene (2014) that 𝐿bol ≳ 1045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 erg s−1 is required for quasar feedback to operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' For 108𝑀⊙ ≲ 𝑀BH ≲ 109𝑀⊙ and 𝐿/𝐿Edd ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2, we see the strongest He ii and strongest non-outflowing C iv line emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' One possible explanation for this behaviour would be that this emission represents ionized material which has been launched from the ac- cretion disc, but lacks the ultraviolet luminosity to accelerate the outflow, meaning that such material falls back and virializes instead of escaping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In such a scenario the strong symmetric emission from high-ionization ultraviolet lines would represent a failed line-driven wind, analogous to models of the low-ionization BLR which repre- sent a failed dust-driven wind (Czerny & Hryniewicz 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Baskin & Laor 2018, see also Elvis 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' While we observe a reasonably good qualitative agreement be- tween the C iv blueshift behaviour and the Giustini & Proga (2019) predictions for line-driven winds, the reality is likely more compli- cated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In particular, Giustini & Proga (2019) do not consider any emission from a ‘soft excess’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Instead they assume that the ionizing SED consists of just two components, emitted from a thermal disc and a hot corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Such a simple model is unlikely to explain our observational results: the different behaviour of He ii EW and 𝛼ox as a function of 𝑀BH and 𝐿/𝐿Edd points to the presence of a third spectral component in the EUV which can vary separately from the disc and corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Other physical effects could also be at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In particular, as the accretion rate increases above �𝑚 ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3, we expect the disc structure to transition between geometries akin to slim discs and thin discs (Abramowicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Abramowicz & Fragile 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' At low ac- cretion rates, slim discs are well approximated by the Novikov & Thorne (1973) thin disc solution, as used in qsosed, but we expect this to be less accurate as �𝑚 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In other words, the regime in which qsosed appears to best match our data is also the regime in which we might expect it to be least accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The origin of the apparent transition around 𝐿/𝐿Edd ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 3 and 5 is there- fore still uncertain and further work is required to fully understand the interplay between AGN accretion flows, the ionizing SEDs they produce, and the outflows they drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Line-driven winds from high Eddington ratio AGN are often cited as a potentially important component of radiative-mode (quasar- mode) feedback (Zubovas & King 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' While difficult to observe directly, such feedback modes are required to regulate galaxy growth and explain the tight SMBH-galaxy correlations observed in the local universe (see Fabian 2012, for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' However, most SMBHs in the local universe do not have masses above 109𝑀⊙, so our results might suggest that the line-driven winds traced by C iv cannot have a significant effect on their host galaxies’ growth and co-evolution as they never reach the SMBH masses required to launch strong winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' There are at least two solutions to this apparent problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' First is that radiative-mode feedback could still be operating through ionized gas outflows, but that the gas is in a different ionization phase and is not seen in C iv, but instead in other bands such as the X-ray ‘ultra-fast outflows’ (Laha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Second could be that quasar- mode feedback is only effective when coupled to dusty gas (Fabian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Ishibashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Ricci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2022), thus having most impact when the AGN is obscured by dust (Temple et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Lansbury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Jun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Assef et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3 SMBH spin and system inclination In our qsosed modeling, we kept inclination fixed at cos𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 and only presented the non-spinning SMBH case, 𝑎∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' However, both of these parameters have an impact on the predicted SEDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The impact of SMBH spin is particularly pronounced;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' plots matching those in the right-hand panels of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 4 and 5 are presented in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The basic finding from the maximal spin models is that the observed trends of 𝛼ox with mass and Eddington fraction are not reproduced, for reasons that are explained in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In fact, all of the maximal spin models have 𝛼ox ≳ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5, meaning that the X-ray luminosity is always quite high compared to the optical and ultraviolet, and the observed soft spectra at high mass and high Eddington fraction are not reproduced for 𝑎∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' If there are a significant proportion of maximally spinning SMBHs in our quasar sample, this would imply that the model predictions are not valid for high spin objects, potentially undermining many of the results discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Alternatively, if the qsosed models are correct, the good agreement at low spin and poor agreement at high spin would imply that most SDSS quasars at 𝑧 ≈ 2 typically have low or moderate SMBH spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' SMBH spin is most commonly estimated from broad iron line emission in the X-ray band (Reynolds 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Spin measurements tend to be rather high, with the majority of X-ray measurements in AGN consistent with maximally spinning SMBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This apparent MNRAS 000, 1–20 (2023) 12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Temple et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' preference might initially appear to be inconsistent with our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' However, there are a number of factors at work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' First, discs around maximally spinning SMBHs have higher radiative efficiencies and are thus more luminous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' As shown in figure 3 of Reynolds (2019), this might lead to high spins being over-represented in a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' One could also imagine further selection effects if spins are eas- ier to measure when they are close to maximal and the iron line is broader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Second, the majority of spin measurements are at lower masses (𝑀BH ≲ 108 𝑀⊙) than in our sample, with only a handful of spin measurements in our considered mass regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In fact, there is some tentative evidence for a decrease of SMBH spin with increasing mass (Sisk-Reynés et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2022), behaviour that is expected if accre- tion is coherent at low masses and more incoherent at higher masses, as predicted by both semi-analytic models and hydrodynamic sim- ulations (King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Sesana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Zhang & Lu 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Bustamante & Springel 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' If our results do indeed favour low or moderate SMBH spins in luminous quasars, then they are consistent with this proposed trend, especially since it is the high-mass quasars which cannot be matched by the maximally spinning qsosed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Any conclusions drawn here should be treated with caution, given (i) the difficulties in obtaining reliable and unbiased spin measurements, and (ii) the lack of knowledge about the impact of spin on the EUV and X-ray regions of the quasar SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We fixed inclination in the qsosed models, adopting cos𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Following Copernican reasoning, we expect AGN to have a random, isotropic distribution of viewing angles, in which case the mean viewing angle of all AGN is cos𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Factoring in obscuration by a putative ‘torus’ beyond some maximum inclination, and fore- shortening/limb darkening of the disc continuum might be expected to bias this mean viewing angle to lower (more face-on) inclinations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Krolik & Voit 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Matthews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' If we were to adopt a different inclination in qsosed, we can think about how the model predictions would change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The hard X-ray source in qsosed is isotropic, whereas the warm and thermal components have a disc-like geometry and thus pro- duced an observed luminosity ∝ cos𝑖, such that lower inclinations have higher luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The impact of inclination on the outputs from qsosed can thus be straightforwardly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Changing inclination from 𝑖1 → 𝑖2 results in a fractional change in 𝐿3000 of (cos𝑖2/cos𝑖1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Since, for a given input �𝑚, we calculate 𝐿/𝐿Edd from 𝐿3000, we obtain a linear scaling of the 𝑦-axis of the right-hand panels of Figs 5 and 4 by the same factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The change in 𝛼ox is Δ𝛼ox = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3838 log10(cos𝑖2/cos𝑖1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' adopting a lower inclination with cos𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='75 would result in a more negative 𝛼ox in all simula- tion bins by ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Finally, the ratio 𝐿228/𝐿1640 undergoes small changes with cos𝑖, but these are fairly uniform across the simulation grid and thus unimportant, given that the proportionality constant between 𝐿228/𝐿1640 and He ii EW is not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Furthermore, the He ii EW depends on the 𝐿228 seen by the He ii gas, rather than the 𝐿228 seen by a distant observer, meaning that the true inclination dependence of He ii EW would depend on the BLR geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We have explicitly checked that the anticipated changes in 𝐿3000 and 𝛼ox are indeed reproduced in qsosed, except for small departures in the 𝛼ox due to contamination of the 2 keV flux by the warm component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3 Wider implications and future work 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1 Comparison with other populations Recent work has attempted to compare the ultraviolet emission prop- erties in high redshift (𝑧 ≳ 6) quasars with their lower redshift ana- logues (Mazzucchelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Meyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Schindler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Such quasars are (by selection) very luminous, and generally display large C iv blueshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' From our results, we would argue that comparative studies should match AGN samples not just in luminos- ity, but in two independent parameters which trace 𝐿/𝐿Edd and 𝑀BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Stepney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (in preparation) will discuss this further in a study of C iv and He ii emission in SDSS quasars with redshifts 𝑧 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Current samples of 𝑧 ≳ 6 quasars include a significant number of objects with inferred 𝐿/𝐿Edd > 1, which lie outside the parameter space explored in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We have verified that the sample of 37 quasar spectra with redshifts 𝑧 > 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3 presented by Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2021b) typically show narrower Mg ii profiles than 𝑧 ≈ 2 SDSS objects with the same 3000 Å luminosities, suggesting smaller 𝑀BH and larger 𝐿/𝐿Edd (for a given 𝐿3000) than the quasars characterized in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' For such objects it is therefore not surprising that their typical ultraviolet emission properties are different from the majority of the SDSS population at 𝑧 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 BLR metallicities The relative strengths of many ultraviolet emission lines are known to correlate with the C iv and He ii emission properties (Richards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Temple et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In particular, the flux ratios of high ionization ultraviolet lines such as C iv, N v 𝜆1240 and Si iv 𝜆1400 are tightly correlated with the C iv blueshift and He ii EW (Temple et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Assuming no changes in density or ionization structure or geometry of the BLR, changes in these line ratios are sometimes taken to reflect changes in the metal content of the BLR (Nagao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Such an interpretation, combined with the results in this work, gives rise to a paradigm where the metal content of quasar BLRs is largest in objects with the largest 𝑀BH and 𝐿/𝐿Edd, as noted by Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2018) and Śniegowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2021) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' However, as shown in Temple et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2021b), the variation in these line ratios can be explained with changes in the density of the emitting gas, and need not involve changes in metallicity (see also appendix A4 of Casebeer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In particular, the ultraviolet emission line ratios seen in objects with large C iv blueshifts can be explained by emission from relatively dense gas which is located closer to the ionizing source, while the line ratios in objects with high EW, symmetric C iv emission are consistent with emission from less dense gas at larger radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Given the trends seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 3 and 5, this is a much more natural explanation: objects with different SMBH masses and accretion rates have different accretion flows, which give rise to different EUV SEDs (as traced by He ii) and different kinematic and density structures in the BLR (traced by the C iv blueshift and high ionization line ratios respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Under this alternative paradigm the BLR metallicity would be free to vary independently of 𝑀BH and 𝐿/𝐿Edd, and need not be super-solar in the early universe (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3 Quasar cosmology Quasars are visible out to large cosmological distances, and display remarkably homogeneous behaviour across cosmic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A grow- ing body of work has proposed the non-linear scaling between 𝐿UV and 𝐿2 keV (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' the 𝛼ox–𝐿2500 Å relation) as a way to use quasars as standardizable candles for cosmological measurements (Risaliti & Lusso 2015, 2017, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Lusso & Risaliti 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Salvestrini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Lusso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Sacchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Khadka & Ratra 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' More recently, however, Petrosian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2022) have shown that an independent determination of the cosmological distance–redshift re- lation cannot be constructed simply by using the empirical correlation MNRAS 000, 1–20 (2023) C iv and He ii emission in quasars 13 between two continuum luminosities, as such luminosities must as- sume a cosmological model to be inferred from fluxes and redshifts, leading to circular reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In this work we have shown that it should be possible to break this circularity by including information from the emission line prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' With knowledge of the Mg ii velocity width, and either the He ii strength or the C iv properties, one should be able to locate an object in the 𝑀BH–𝐿/𝐿Edd plane, and hence infer the intrinsic luminosity in a cosmology-independent way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' By comparing to the observed fluxes one could then (in principle) infer a constraint on the Hubble parameter 𝐻(𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' However, further work is still required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In particular, we need to build a sample of quasars with ultraviolet emission line measurements which have independent measurements of the luminosity distance, in order to calibrate our 𝑀BH–𝐿/𝐿Edd space in a cosmology-independent manner, in an analogous way to the use of the ‘inverse distance ladder’ to calibrate type Ia supernovae as standard candles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Freedman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Riess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='4 Time variability and upcoming surveys Changes in 𝐿/𝐿Edd for a quasar with fixed 𝑀BH will lead to changes in the emitted spectrum, but such changes in SMBH fueling are expected to generally occur on the viscous time-scale, which is on the of order of tens to thousands of years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' However, SMBH accretion is inherently stochastic and the emitted flux varies by a factor of a few on shorter time-scales of just years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The time-scale and amplitude of this intrinsic ‘flickering’ are now known to depend on the SMBH mass and accretion rate (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2022b), and this stochastic flickering will contribute to the scatter within each binned region of our parameter space (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In terms of spectroscopic variability, Rivera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2020) showed that individual SDSS-RM quasars with multiple epochs of spec- troscopy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' with fixed 𝑀BH) can vary in essentially every direction in the C iv blueshift–EW space, although objects with large blueshifts tend to show a change in blueshift and objects with strong EW show a change in EW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In the near future, SDSS-V (Kollmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2017) will provide multi-epoch spectroscopic data for tens of thousands of luminous quasars, providing new insights into AGN variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' At the same time, surveys such as DESI (Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Chaussidon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2022) and 4MOST (Merloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Eltvedt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2022) will probe fainter, yielding spectra of lower luminosity quasars than the sample investigated in this work, and future data releases from the eROSITA all-sky survey will include X-ray flux measurements for millions of AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Together these surveys will pro- vide new constraints on the spectroscopic properties and ionizing SEDs of luminous AGN across the 𝑀BH–𝐿/𝐿Edd parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 6 CONCLUSIONS We have investigated the rest-frame ultraviolet emission line proper- ties in 191 391 SDSS quasars with redshifts 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 < 𝑧 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We can infer 𝛼ox, the logarithmic ratio of the rest-frame 2 keV and 2500 Å luminosities, for 5325 quasars in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Using the FWHM of Mg ii 𝜆2800 as a proxy for the virial velocity, we quantify the av- erage properties of the C iv 𝜆1549 and He ii 𝜆1640 emission lines across the two-dimensional space spanned by 𝑀BH and 𝐿/𝐿Edd, and use these observations to confront qualitative predictions of when radiation-driven outflows should dominate kinetic feedback mecha- nisms (Giustini & Proga 2019) and theoretical SEDs arising from models of AGN accretion flows (Kubota & Done 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Our main conclusions are: (i) As shown in previous works (Richards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Rankine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2020), the blueshift and EW of C iv correlate with the EW of He ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Objects with strong He ii have high EW C iv with little or no blue excess, while objects with weaker He ii show smaller EW C iv with larger C iv blueshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (ii) We recover a Baldwin effect, but instead of simply correlating with the ultraviolet luminosity, we find that the C iv and He ii prop- erties display more complicated trends in the 𝑀BH–𝐿/𝐿Edd plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The dynamic range in He ii EW is greatest at Eddington ratios ≳0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The largest C iv blueshifts are only observed at high 𝐿/𝐿Edd and high 𝑀BH, while the highest EWs are seen only at high 𝐿/𝐿Edd and relatively low 𝑀BH (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Composite spectra from these two extrema are shown in blue and green in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (iii) In contrast to the ultraviolet emission line properties, but consistent with previous work in the literature, 𝛼ox displays a much simpler behaviour across the 𝑀BH–𝐿/𝐿Edd plane (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 𝛼ox cor- relates with the ultraviolet continuum luminosity in a more direct manner than the emission lines, albeit in a much smaller sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Fu- ture data releases from eROSITA, SDSS-V and 4MOST will increase the number of known quasars with X-ray data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (iv) Under the assumption that blueshifted C iv emission is trac- ing a disc wind accelerated by radiation line driving, we find our results are consistent with the global scheme for accretion and out- flow mechanisms proposed by Giustini & Proga (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In particular, an Eddington-scaled mass accretion rate �𝑚 ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='25 is required for the formation of the strongest line-driven winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Giustini & Proga (2019) suggest that 𝑀BH > 108𝑀⊙ is also required to launch strong line- driven winds, however we only observe the largest C iv blueshifts in objects with Mg ii-inferred 𝑀BH ≳ 109𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Strong line emission at 𝑀BH ≲ 109𝑀⊙ could perhaps indicate a ‘failed’ line-driven wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (v) Absent large changes in the density or geometry of the broad line region, the strength of He ii is probing the strength of 54 eV ion- izing radiation in the ‘unseen’ portion of the ultraviolet SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Above 𝐿/𝐿Edd ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2, we find that the EW of He ii is broadly consistent with the qsosed model (Kubota & Done 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In other words,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='relative strength of the 54 eV flux (which is photoionizing the broad ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='line region) compared to the 1640 Å continuum is consistent with a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='relatively simple model where the peak temperature of the accretion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='disc blackbody changes as a function of SMBH mass and accre- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='tion rate and the strength of the ‘soft excess’ is adjusted to give the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='correct bolometric luminosity while keeping the strength of the hot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='coronal emission fixed at two per cent of the Eddington luminosity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='(as proposed by Kubota & Done 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (vi) Below 𝐿/𝐿Edd ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2, something appears to change in the physics of the broad line region, with no strong C iv blueshifts ob- served and much weaker trends in He ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The simple SED models do not provide as good a match to the observed He ii trends, consistent with the results of Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2022) who find a discrepancy between the observed and predicted 𝛼ox in the same region of the 𝑀BH–𝐿/𝐿Edd parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (vii) Similar to Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2022), we also find no strong evidence for high SMBH spins in our quasar sample: the zero-spin qsosed models provide an acceptable match to the SED probes across a significant portion of our observed parameter space while the max- imally spinning models do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' If a significant fraction of our quasar sample have maximally spinning SMBHs, this would suggest that the qsosed model assumptions are not valid for high spin objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Alternatively, taking the model results at face value would suggest low or moderate spins in typical SDSS quasars at 𝑧 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' MNRAS 000, 1–20 (2023) 14 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Temple et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We gratefully acknowledge useful discussions with Chris Done and Jake Mitchell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' MJT thanks Chiara Mazzucchelli, Claudio Ricci and Roberto Assef for insightful comments, and Jinyi Yang for sharing the sample of 𝑧 > 6 quasar spectra from Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' MJT acknowledges support from a FONDECYT postdoctoral fel- lowship (3220516).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' JHM acknowledges funding from the Royal So- ciety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' ALR acknowledges support from UKRI (MR/T020989/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Sloan Foundation, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Department of Energy Office of Science, and the Participating Institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' SDSS-IV acknowledges support and resources from the Center for High-Performance Com- puting at the University of Utah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The SDSS web site is www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='sdss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} 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Harvard-Smithsonian Center for As- trophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Instituto de Astrofísica de Canarias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The Johns Hopkins University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Kavli Institute for the Physics and Mathematics of the Uni- verse (IPMU) / University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' the Korean Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Lawrence Berkeley National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Leibniz Institut für Astro- physik Potsdam (AIP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Max-Planck-Institut für Astronomie (MPIA Heidelberg),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Max-Planck-Institut für Astrophysik (MPA Garching),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Max-Planck-Institut für Extraterrestrische Physik (MPE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' National Astronomical Observatories of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' New Mexico State Univer- sity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' New York University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' University of Notre Dame,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Observatário Nacional / MCTI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Shanghai Astronomical Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' United Kingdom Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Universidad Nacional Autónoma de México,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' University of Arizona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' University of Colorado Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' University of Portsmouth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' University of Utah,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Univer- sity of Virginia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' University of Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' University of Wisconsin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Vanderbilt University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' and Yale University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' For the purpose of open access, the authors will apply a Cre- ative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' DATA AVAILABILITY The spectroscopic data underlying this article are available from SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3 The X-ray data sets underlying this article are available from the references given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The composite spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 1 are available as online supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' REFERENCES Abdurro’uf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, ApJS, 259, 35 Abramowicz M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Fragile P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2013, Living Reviews in Relativity, 16, 1 Abramowicz M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Czerny B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Lasota J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Szuszkiewicz E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1988, ApJ, 332, 646 Alexander D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='08517 Arnaud K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1996, in Jacoby G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Barnes J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', eds, Astronomical Society of the Pacific Conference Series Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 101, Astronomical Data Analysis Software and Systems V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 17 Assef R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, ApJ, 934, 101 Avni Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Tananbaum H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1982, ApJ, 262, L17 Avni Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Tananbaum H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1986, ApJ, 305, 83 3 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='sdss4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='org/dr17/ Bachev R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Marziani P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Sulentic J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Zamanov R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Calvani M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Dultzin- Hacyan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2004, ApJ, 617, 171 Baldwin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1977, ApJ, 214, 679 Baldwin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Netzer H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1978, ApJ, 226, 1 Baskin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Laor A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2004, MNRAS, 350, L31 Baskin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Laor A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2005, MNRAS, 356, 1029 Baskin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Laor A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2018, MNRAS, 474, 1970 Baskin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Laor A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Hamann F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2013, MNRAS, 432, 1525 Begelman M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', McKee C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Shields G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1983, ApJ, 271, 70 Blandford R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Payne D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1982, MNRAS, 199, 883 Boller T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Freyberg M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Trümper J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Haberl F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Voges W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Nandra K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2016, A&A, 588, A103 Boroson T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Green R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1992, ApJS, 80, 109 Bottorff M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Korista K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Shlosman I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2000, ApJ, 537, 134 Brodzeller A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Dawson K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, AJ, 163, 110 Bustamante S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Springel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2019, MNRAS, 490, 4133 Calistro Rivera G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2021, A&A, 649, A102 Casebeer D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Leighly K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Baron E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2006, ApJ, 637, 157 Castor J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Abbott D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Klein R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1975, ApJ, 195, 157 Chajet L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Hall P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2013, MNRAS, 429, 3214 Chaussidon E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='08511 Coatman L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Hewett P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Banerji M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Richards G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2016, MNRAS, 461, 647 Coatman L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Hewett P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Banerji M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Richards G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Hennawi J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Prochaska J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2017, MNRAS, 465, 2120 Coatman L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Hewett P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Banerji M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Richards G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Hennawi J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Prochaska J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2019, MNRAS, 486, 5335 Coffey D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2019, A&A, 625, A123 Collin S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Kawaguchi T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Peterson B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Vestergaard M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2006, A&A, 456, 75 Comparat J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2020, A&A, 636, A97 Croom S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2002, MNRAS, 337, 275 Czerny B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Hryniewicz K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2011, A&A, 525, L8 Dalla Bontà E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2020, ApJ, 903, 112 Davidson K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Netzer H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1979, Reviews of Modern Physics, 51, 715 Dibai E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1980, Soviet Ast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 24, 389 Done C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Davis S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Jin C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Blaes O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ward M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2012, MNRAS, 420, 1848 Du P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Wang J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2019, ApJ, 886, 42 Dwelly T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2017, MNRAS, 469, 1065 Eltvedt A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='07324 Elvis M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2000, ApJ, 545, 63 Elvis M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2017, ApJ, 847, 56 Elvis M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1994, ApJS, 95, 1 Emmering R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Blandford R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Shlosman I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1992, ApJ, 385, 460 Fabian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2012, ARA&A, 50, 455 Fabian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Vasudevan R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Gandhi P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2008, MNRAS, 385, L43 Fawcett V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Alexander D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Rosario D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Klindt L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Lusso E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Morabito L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Calistro Rivera G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, MNRAS, 513, 1254 Ferland G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2013, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Mex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Astrofis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 49, 137 Ferland G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2017, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Mex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Astrofis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 53, 385 Ferland G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Done C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Jin C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Landt H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ward M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2020, MNRAS, 494, 5917 Fonseca Alvarez G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2020, ApJ, 899, 73 Freedman W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2001, ApJ, 553, 47 Fu S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Brandt W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Zou F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Laor A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Garmire G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ni Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Timlin John D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Xue Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, ApJ, 934, 97 Fukumura K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Kazanas D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Contopoulos I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Behar E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2010, ApJ, 715, 636 Gaskell C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1982, ApJ, 263, 79 Gaskell C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Goosmann R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2013, ApJ, 769, 30 Gaskell C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Goosmann R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2016, Ap&SS, 361, 67 Giustini M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Proga D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2019, A&A, 630, A94 Gordon C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Arnaud K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2021, PyXspec: Python interface to XSPEC spectral- fitting program, Astrophysics Source Code Library, record ascl:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='014 (ascl:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='014) Haardt F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Maraschi L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1991, ApJ, 380, L51 Hewett P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Wild V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2010, MNRAS, 405, 2302 Higginbottom N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Proga D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Knigge C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Long K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Matthews J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Sim S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2014, ApJ, 789, 19 MNRAS 000, 1–20 (2023) C iv and He ii emission in quasars 15 Hoare M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Drew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1991, MNRAS, 249, 452 Ishibashi W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Fabian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ricci C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Celotti A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2018, MNRAS, 479, 3335 Jensen T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2016, ApJ, 833, 199 Jin C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ward M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Done C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Gelbord J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2012, MNRAS, 420, 1825 Jun H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Assef R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Carroll C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Hickox R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Kim Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Lee J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ricci C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Stern D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2021, ApJ, 906, 21 Just D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Brandt W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Shemmer O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Steffen A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Schneider D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Chartas G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Garmire G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2007, ApJ, 665, 1004 Kaspi S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Maoz D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Netzer H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Peterson B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Vestergaard M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Jannuzi B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2005, ApJ, 629, 61 Khadka N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ratra B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, MNRAS, 510, 2753 Khadka N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Martínez-Aldama M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Zajaček M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Czerny B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ratra B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022a, MNRAS, 513, 1985 Khadka N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Zajaček M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Panda S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Martínez-Aldama M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ratra B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022b, MNRAS, 515, 3729 King A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Pringle J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Hofmann J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2008, MNRAS, 385, 1621 Kollmeier J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2017, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='03234 Konigl A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Kartje J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1994, ApJ, 434, 446 Krawczyk C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Richards G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Mehta S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Vogeley M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Gallagher S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Leighly K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ross N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Schneider D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2013, ApJS, 206, 4 Krolik J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Kallman T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1988, ApJ, 324, 714 Krolik J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Voit G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1998, ApJ, 497, L5 Kruczek N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2011, AJ, 142, 130 Kubota A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Done C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2018, MNRAS, 480, 1247 Kwan J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Krolik J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1981, ApJ, 250, 478 Laha S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Reynolds C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Reeves J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Kriss G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Guainazzi M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Smith R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Veilleux S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Proga D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2021, Nature Astronomy, 5, 13 Lai S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, MNRAS, 513, 1801 Lansbury G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Banerji M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Fabian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Temple M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2020, MNRAS, 495, 2652 Laor A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Davis S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2014, MNRAS, 438, 3024 Leighly K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2004, ApJ, 611, 125 Leighly K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Moore J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2004, ApJ, 611, 107 Leighly K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Halpern J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Jenkins E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Casebeer D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2007, ApJS, 173, 1 Liu T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2020, ApJS, 250, 32 Liu T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, A&A, 661, A5 Lusso E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Risaliti G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2016, ApJ, 819, 154 Lusso E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Risaliti G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2017, A&A, 602, A79 Lusso E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2020, A&A, 642, A150 Lusso E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2021, A&A, 653, A158 Lyke B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2020, ApJS, 250, 8 Lynden-Bell D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1969, Nature, 223, 690 Maddox N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2017, MNRAS, 470, 2314 Marlar A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Shemmer O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Brotherton M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Richards G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Dix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, ApJ, 931, 41 Martínez-Aldama M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Zajaček M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Czerny B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Panda S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2020, ApJ, 903, 86 Mathews W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ferland G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1987, ApJ, 323, 456 Matthews J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Knigge C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Long K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2017, MNRAS, 467, 2571 Mazzucchelli C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2017, ApJ, 849, 91 Mejía-Restrepo J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Trakhtenbrot B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Lira P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Netzer H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2018, MNRAS, 478, 1929 Merloni A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2019, The Messenger, 175, 42 Meyer R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Bosman S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ellis R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2019, MNRAS, 487, 3305 Mitchell J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Done C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ward M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Kynoch D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Hagen S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Lusso E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Landt H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='11977 Mizumoto M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Done C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Tomaru R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Edwards I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2019, MNRAS, 489, 1152 Murray N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Chiang J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1995, ApJ, 454, L105 Murray N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Chiang J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Grossman S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Voit G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1995, ApJ, 451, 498 Nagao T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Marconi A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Maiolino R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2006, A&A, 447, 157 Ni Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2018, MNRAS, 480, 5184 Ni Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, MNRAS, 511, 5251 Nomura M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ohsuga K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2017, MNRAS, 465, 2873 Nomura M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ohsuga K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Done C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2020, MNRAS, 494, 3616 Novikov I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Thorne K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1973, in Black Holes (Les Astres Occlus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' pp 343–450 Osterbrock D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ferland G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2006, Astrophysics of gaseous nebulae and active galactic nuclei Petrosian V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Singal J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Mutchnick S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, ApJ, 935, L19 Petrucci P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ursini F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', De Rosa A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Bianchi S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Cappi M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Matt G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Dadina M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Malzac J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2018, A&A, 611, A59 Proga D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2007, ApJ, 661, 693 Proga D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Kallman T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2004, ApJ, 616, 688 Proga D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Stone J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Kallman T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2000, ApJ, 543, 686 Rankine A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Hewett P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Banerji M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Richards G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2020, MNRAS, 492, 4553 Rankine A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Matthews J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Hewett P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Banerji M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Morabito L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Richards G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2021, MNRAS, 502, 4154 Reynolds C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2019, Nature Astronomy, 3, 41 Ricci C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, ApJ, 938, 67 Richards G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Vanden Berk D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Reichard T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Hall P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Schneider D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', SubbaRao M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Thakar A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', York D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2002, AJ, 124, 1 Richards G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2006, ApJS, 166, 470 Richards G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2011, AJ, 141, 167 Riess A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Casertano S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Yuan W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Bowers J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Macri L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Zinn J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Scolnic D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2021, ApJ, 908, L6 Risaliti G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Elvis M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2010, A&A, 516, A89 Risaliti G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Lusso E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2015, ApJ, 815, 33 Risaliti G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Lusso E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2017, Astronomische Nachrichten, 338, 329 Risaliti G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Lusso E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2019, Nature Astronomy, 3, 272 Rivera A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Richards G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Hewett P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Rankine A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2020, ApJ, 899, 96 Rivera A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Richards G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Gallagher S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', McCaffrey T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Rankine A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Hewett P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Shemmer O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, ApJ, 931, 154 Rodríguez Hidalgo P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Rankine A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, ApJ, 939, L24 Sacchi A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, A&A, 663, L7 Salvato M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, A&A, 661, A3 Salvestrini F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Risaliti G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Bisogni S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Lusso E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Vignali C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2019, A&A, 631, A120 Schindler J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2020, ApJ, 905, 51 Schneider D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2010, AJ, 139, 2360 Sesana A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Barausse E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Dotti M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Rossi E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2014, ApJ, 794, 104 Shang Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Wills B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Robinson E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Wills D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Laor A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Xie B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Yuan J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2003, ApJ, 586, 52 Shang Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2005, ApJ, 619, 41 Shen Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2013, Bulletin of the Astronomical Society of India, 41, 61 Shen Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ho L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2014, Nature, 513, 210 Shen Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Liu X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2012, ApJ, 753, 125 Shen Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Greene J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Strauss M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Richards G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Schneider D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2008, ApJ, 680, 169 Shen Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2011, ApJS, 194, 45 Shen Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2019, ApJ, 873, 35 Sisk-Reynés J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Reynolds C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Matthews J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Smith R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, MNRAS, 514, 2568 Śniegowska M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Marziani P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Czerny B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Panda S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Martínez-Aldama M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', del Olmo A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', D’Onofrio M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2021, ApJ, 910, 115 Steffen A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Strateva I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Brandt W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Alexander D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Koekemoer A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Lehmer B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Schneider D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Vignali C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2006, AJ, 131, 2826 Stevans M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Shull J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Danforth C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Tilton E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2014, ApJ, 794, 75 Sulentic J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Marziani P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2015, Frontiers in Astronomy and Space Sciences, 2, 6 Sulentic J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Zwitter T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Marziani P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Dultzin-Hacyan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2000, ApJ, 536, L5 Sulentic J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Bachev R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Marziani P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Negrete C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Dultzin D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2007, ApJ, 666, 757 Sun J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Shen Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2015, ApJ, 804, L15 Temple M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Banerji M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Hewett P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Coatman L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Maddox N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Peroux C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2019, MNRAS, 487, 2594 Temple M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ferland G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Rankine A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Hewett P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Badnell N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ballance C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Del Zanna G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Dufresne R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2020, MNRAS, 496, 2565 Temple M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Banerji M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Hewett P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Rankine A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Richards G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2021a, MNRAS, 501, 3061 Temple M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ferland G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Rankine A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Chatzikos M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Hewett P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2021b, MNRAS, 505, 3247 Temple M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Hewett P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Banerji M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2021c, MNRAS, 508, 737 Timlin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Brandt W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ni Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Luo B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Pu X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Schneider D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Vivek M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Yi W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2020, MNRAS, 492, 719 MNRAS 000, 1–20 (2023) 16 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Temple et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Timlin John D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Brandt W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Laor A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2021, MNRAS, 504, 5556 Titarchuk L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1994, ApJ, 434, 570 Vanden Berk D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Wesolowski S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Yeckley M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Marcinik J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Quash- nock J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Machia L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Wu J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2020, MNRAS, 493, 2745 Vasudevan R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Fabian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2007, MNRAS, 381, 1235 Vasudevan R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Fabian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2009, MNRAS, 392, 1124 Verner D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ferland G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Korista K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Yakovlev D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1996, ApJ, 465, 487 Vestergaard M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Osmer P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2009, ApJ, 699, 800 Vestergaard M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Wilkes B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2001, ApJS, 134, 1 Vietri G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2018, A&A, 617, A81 Vietri G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2020, A&A, 644, A175 Wang S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, ApJ, 925, 121 Wild V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Hewett P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2005, MNRAS, 358, 1083 Wilkes B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1984, MNRAS, 207, 73 Wills B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Laor A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Brotherton M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Wills D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Wilkes B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ferland G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Shang Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1999, ApJ, 515, L53 Wolf J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2020, MNRAS, 492, 3580 Wolfire M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Vallini L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Chevance M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, ARA&A, 60, 247 Woods D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Klein R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Castor J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', McKee C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Bell J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1996, ApJ, 461, 767 Xu F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Bian F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Shen Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Zuo W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Fan X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Zhu Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2018, MNRAS, 480, 345 Yang X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Ablimit K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Li Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2021a, ApJ, 914, 31 Yang J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2021b, ApJ, 923, 262 York D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2000, AJ, 120, 1579 Yu Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022a, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='05491 Yu W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Richards G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Vogeley M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Moreno J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Graham M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022b, ApJ, 936, 132 Zakamska N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Greene J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2014, MNRAS, 442, 784 Zanstra H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 1929, Publs Dom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Victoria, 4, 209 Zappacosta L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2020, A&A, 635, L5 Zhang X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Lu Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2019, ApJ, 873, 101 Zhu Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Bu D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Yang X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Yuan F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', Lin W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2022, MNRAS, 513, 1141 Zubovas K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', King A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=', 2012, ApJ, 745, L34 APPENDIX A: THE ROLE OF X-RAYS IN PHOTOIONIZATION This appendix shows that the X-ray portion of the SED is an insignif- icant source of ionization for typical AGN emission line regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This is surprising since the SEDs shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2 have a signifi- cant fraction of their power at high energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The ultraviolet–X-ray hardness ratio (𝛼ox) does correlate with emission line properties, as summarized above, but this is not likely a direct relation with the observed X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' To illustrate this point, we use the hardest and softest SEDs from our model grid (Section 3), corresponding to �𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='027 and �𝑚 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='000 at 𝑀BH = 1010𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' These SEDs are shown in the top panel of Figure A1, and have 𝛼ox = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='05 in the hardest and 𝛼ox = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='05 in the softest cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The photoionization rate for a given shell 𝑛 is Γ𝑛 = ∫ ∞ 𝜈0 𝜎𝜈𝜙𝜈 𝑑𝜈[s−1] (A1) where 𝜈0, 𝜎𝜈, and 𝜙𝜈 are the photoionization threshold of shell 𝑛, its energy-dependent photoionization cross section [cm2] , and the flux of ionizing photons [cm−2 s−1 Hz−1] (Osterbrock & Ferland 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The total photoionization rate is the sum over all shells, Γtotal = ∑︁ 𝑛 Γ𝑛[s−1] (A2) The flux of ionizing photons 𝜙𝜈 enters in the photoionization rate (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This is the ratio 𝜙𝜈 = 4𝜋𝜈𝐽𝜈/(ℎ𝜈2) and is shown in the lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The photon flux near 2 keV is typically C K He, C L H 2 keV φν Softest SED (αox=-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='05) φν Hardest SED (αox=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='05) φν φν [cm-2 Hz-1 s-1] 10−6 10−3 1 1000 106 Hz 1014 1015 1016 1017 1018 1019 4πν Jν C K He, C L H 2 keV ν Fν Softest SED (αox=-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='05) ν Fν Hardest SED (αox=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='05) 4π ν Jν [erg cm-2 s-1] 105 106 107 108 109 1014 1015 1016 1017 1018 1019 Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The upper panel shows the hardest and softest SEDs presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The lower panel shows the flux of photons 𝜙𝜈 for the same models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Some important energies are indicated by the hashed lines near the bottom of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' These show the ionization potentials of ground-state H0 and He2+, the L and K shells of C2+, and 2 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The flux of ionizing photons is orders of magnitude smaller at 2 keV compared to the flux at the He ii and C iv photoionization edges even for the hardest SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' ∼6 dex fainter than the value near the peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Photoionization is photon- counting and there are relatively few high-energy photons despite their significant energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The photon flux is multiplied by the photoionization cross section to derive the photoionization rate (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We concentrate on C2+ since photoionization of that ion produces C3+ and C iv 𝜆1549 emis- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The shell-dependent cross sections for photoionization of C2+, taken from Verner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (1996), are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Both the 1s2 K shell in the X-ray and the lower energy 2s2 L shell are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Both shells have two electrons and, as expected, the peak photoionization cross sections are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' MNRAS 000, 1–20 (2023) C iv and He ii emission in quasars 17 C2+ cross section K L σphoto [cm2] 10−21 10−20 10−19 10−18 Hz 1014 1015 1016 1017 1018 1019 Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The K and L shell photoionization cross sections for C2+ are shown as a function of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The cross section are from the calculations by Verner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' These, together with the flux of photons (lower panel of Figure A1), determine the photoionization rate (Equation A1) and the effects of light upon matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Photoionization rates for H0 and the K and L shells of C2+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Shell ΓSoftest [s−1] ΓHardest [s−1] H0 K 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='24e+00 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='52e+00 C2+ L 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='66e-01 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='66e-01 C2+ K 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='49e-03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='71e-02 The photoionization rates for H0 and the two shells of C2+ are listed in Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The C2+ K-shell rate is 17 to 60 times smaller then the L-shell rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Both are ∼ 10 − 30 times smaller than the H0 photoionization rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' From this comparison we expect that the effects of the EUV and XUV will be more important than the X-ray, mainly due to the larger number of softer photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Figure A3 shows the rate at which photons interact with matter for our two reference SEDs and a solar composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Calculations are done with Cloudy version 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='01, as last described by Ferland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Cloud parameters are typical of the C iv emitting region of an AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The vertical axis is the total light-matter interaction rate at a particular frequency and is the product of the photon flux and the total gas opacity, evaluated for the appropriate chemical composition and degree of ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The 50-912 Å ultraviolet region is ∼7 dex more interactive than 2 keV X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The result that the X-rays hardly matter at all is surprising given the shape of the SEDs in the upper panel of Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This is because photoionization is photon counting, and the paucity of X-ray photons cannot make up for their great energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' High-energy photons would dominate the physics if softer parts of the SED were extinguished so only X-rays strike the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Indeed, this is the ‘XDR’ model of atomic and molecular regions of clouds exposed to ionizing radiation (Wolfire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' It would be difficult to detect this XDR emission C K He, C L H 2 keV Light-matter interaction rate 2 keV φν σν Softest SED (αox=-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='05) φν σν Hardest SED (αox=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='05) φν σν [s-1] 10−3 1 106 109 Hz 1014 1015 1016 1017 1018 1019 Figure A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This shows the rate that photons interact with matter, the product of the flux of photons and the total gas opacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' X-rays interact with matter with a rate about ∼ 7 dex slower than the rate near the peak rate around 50 − 912Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' since emission from the gas which absorbed the softer radiation would be far stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This is discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1 of Ferland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The ultraviolet luminosity of a realistic SED has more power than the relatively hard X-ray portion that drives an XDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The full SED striking a cloud produces successive H+/H0/H2 layers, which are brighter than the deep X-ray heated regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Emission from regions powered by lower-energy light would dominate over the XDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Correlations between the X-ray portion of the SED and emission- line properties are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' These are likely due to other correlations with the softer portions of the SED, as evident in Figure 2, and not the X-rays themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' APPENDIX B: ADDITIONAL OBSERVATIONS In this Appendix we present additional observational results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' First, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' B1 we present the He ii EW, the C iv blueshift and 𝛼ox as a function of the FWHM of Mg ii 𝜆2800 and the 3000 Å continuum luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' These two parameters are measured directly from the SDSS spectroscopy and photometry respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' By contrast, the plots in the main text show observed properties as a function of 𝑀BH ∝ 𝐿0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 3000FWHM2 Mg ii (B1) and 𝐿/𝐿Edd ∝ 𝐿3000/𝑀BH ∝ 𝐿0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 3000FWHM−2 Mg ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (B2) Given that both 𝑀BH and 𝐿/𝐿Edd depend on the observed parameters FWHMMg ii and 𝐿3000, this might lead to induced correlations in the 𝑀BH–𝐿/𝐿Edd space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' However, in practice our inferred 𝑀BH–𝐿/𝐿Edd space is simply a rotation and reflection of the FWHMMg ii–𝐿3000 space, where we (reassuringly) still see clear trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Furthermore, we see the same He ii behaviour as a function of FWHMMg ii and 𝐿3000 in the X-ray detected sub-sample as in our full sample, meaning that MNRAS 000, 1–20 (2023) 18 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Temple et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 1045 1046 1047 νLν|3000˚ A [erg s−1] All sources 10000 2500 5000 7500 FWHM(MgII) [km s−1] 1045 1046 1047 νLν|3000˚ A [erg s−1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 1 2 4 EW(HeII λ1640) [˚A] 0 250 500 750 1000 1250 1500 1750 2000 CIV λ1550 blueshift [km s−1] 1045 1046 1047 νLν|3000˚ A [erg s−1] X-ray detected sources 10000 2500 5000 7500 FWHM(MgII) [km s−1] 1045 1046 1047 νLν|3000˚ A [erg s−1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 1 2 4 EW(HeII λ1640) [˚A] −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='9 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='7 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='4 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 αox Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Observed quasar properties as a function of the FWHM of Mg ii 𝜆2800 and the 3000 Å continuum luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Left panel: The median He ii EW (top) and C iv blueshift (bottom) in our full sample of 191 391 objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Right panel: The median He ii EW (top) and 𝛼ox (bottom) in our sub-sample of 5325 X-ray detected sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The He ii behaviour is identical in both panels (modulo the sample size), suggesting that our X-ray detected sub-sample is not biased in terms of its ultraviolet emission properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Moreover, clear differences are seen in the behaviour of He ii and 𝛼ox within the X-ray subsample: the strongest He ii emission is seen only at low Mg ii FWHM while the strongest 2 keV X-ray emission is seen only at the lowest 3000 Å luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' our X-ray detected objects are not obviously biased compared to our full sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Second, we show the 2 keV X-ray continuum and He ii 𝜆1640 line luminosities in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Assuming no changes in the BLR covering factor, and that the He ii continuum is optically thick, 𝐿He ii can be taken as a proxy for the continuum luminosity at 54 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' These two observables show qualitatively different behaviour: contours of constant 𝐿2 keV are largely aligned with lines of constant 𝑀BH, which is consistent with the assumption in qsosed that the hard X-ray power law component emits a constant fraction of the Eddington luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' He ii behaves in a much more complex manner, with the gradient vector of increasing 𝐿He ii changing depending on the location in the 𝑀BH–𝐿/𝐿Edd space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' APPENDIX C: BOLOMETRIC CORRECTIONS In the data, we apply a fixed bolometric correction of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='15 to estimate 𝐿bol from 𝜈𝐿𝜈 at 𝜈 = 3000Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In reality, the bolometric correction will vary as a function of 𝑀BH and 𝐿/𝐿Edd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' We discussed the vari- ation of the bolometric correction from the qsosed model grid in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1, showing a range in 𝑓bol by a factor of ≈ 2 − 3, where 𝑓bol ≡ 𝜈𝐿𝜈|3000Å/𝐿bol is calculated from each individual qsosed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Although a true ‘Apples versus Apples’ comparison is only really possible with full knowledge of the intrinsic SED, in com- paring our qsosed models with data we tried to match scalings and biases in the data introduced by the fixed bolometric correction by applying appropriate transformations to the qsosed outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' For the SMBH mass, the estimate in the data from the Mg ii line width but also requires an estimate of the line formation radius, for which we follow the usual method and assume that the BLR radius scales as 𝑅BLR ∝ 𝐿1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The 𝐿 in this expression should really be some appro- priate ionizing luminosity, but 𝐿bol is normally used and we follow this convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' As a result, the bolometric correction enters into the SMBH mass estimate and implies a bias in the SMBH mass estimates with respect to the true SMBH mass by factor of ( 𝑓bol/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='15)−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' As a result, when plotting 𝑀BH along the 𝑥-axis of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 5 and 4, we apply the scaling 𝑀BH = ( 𝑓bol/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='15)−1/2𝑀BH, q, (C1) where 𝑀BH, q denotes the input qsosed grid value (the ‘true’ SMBH mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' For 𝐿/𝐿Edd, correction factors appear in both the numer- ator and denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 𝐿Edd ∝ 𝑀BH, introducing a bias factor ( 𝑓bol/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='15)−1/2 into the Eddington ratio estimate, while the numera- tor is 𝐿bol and so contains a straightforward bias factor of 𝑓bol/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' As a result, the relationship between the 𝐿/𝐿Edd plotted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 5 and 4, and the dimensionless, Eddington-scaled accretion rate used as input to qsosed is given by 𝐿/𝐿Edd = �𝑚 × ( 𝑓bol/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='15) × ( 𝑓bol/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='15)−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' (C2) MNRAS 000, 1–20 (2023) C iv and He ii emission in quasars 19 1045 1046 1047 νLν|3000˚ A [erg s−1] 108 109 1010 MBH [M⊙] 10−2 10−1 100 L/LEdd 1042 1043 1044 LHeII λ1640 [erg s−1] 1045 1046 1047 νLν|3000˚ A [erg s−1] 108 109 1010 MBH [M⊙] 10−2 10−1 100 L/LEdd 1044 1045 νLν|2 keV [erg s−1] Figure B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' He ii line luminosity (left) and 2 keV X-ray continuum luminosity (right) as a function of SMBH mass, luminosity and Eddington ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Contours of constant 𝐿2 keV are largely aligned with lines of constant 𝑀BH, consistent with the assumption in qsosed that the hard X-ray emission is equal to 2 per cent of the Eddington limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 108 109 1010 MBH,q [M⊙] 10−2 10−1 100 ˙m ≡ ˙M/ ˙MEdd 108 109 1010 MBH,q [M⊙] 10−2 10−1 100 (ν Lν|3000˚ A,q/LEdd) × 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='15 108 109 1010 MBH ≡ MBH,q × (fbol/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='15)−1/2 [M⊙] 10−2 10−1 100 L/LEdd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1 L228/L1640 Figure C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' An illustration of how the qsosed predictions change with differing treatments of the bolometric correction, focusing on the bottom-right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Left panel: the model outputs plotted as a function of the intrinsic, input values of �𝑚 and 𝑀BH, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Centre panel: as in the left panel, but with the 𝑦-axis is replaced with 𝐿bol/𝐿Edd where 𝐿bol is calculated from 𝐿3000 using a constant bolometric correction of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Right panel: as in the centre panel, but with the 𝑥-axis scaled by ( 𝑓bol/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='15)−1/2 and the 𝑦-axis scaled by ( 𝑓bol/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='15)1/2 to capture the impact of the bolometric correction on SMBH mass estimates in the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' See the main Appendix text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The effect of introducing these scaling factors as transformations from the initial qsosed grid is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' C1, to show how the right-hand panels of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 4 and 5 would change if we had made a different presentation choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The scale factors twist and distort the simulation grid slightly from the original uniform parameter space, but, overall, the effects are fairly modest because only square-root terms distinguish the rightmost panel from the original input grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' APPENDIX D: BLACK HOLE SPIN In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' D1 we show how the predictions of 𝛼ox and our He ii 𝜆1640 EW proxy (𝐿228/𝐿1640) from qsosed change if we instead con- sider a maximally spinning SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' While the qualitative trends in the 𝐿228/𝐿1640 are broadly in line with the low spin case, the 𝑎∗ = 1 models fail to reproduce the observed low values of 𝛼ox at high Eddington fractions and SMBH masses (see discussion in Sec- MNRAS 000, 1–20 (2023) 20 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Temple et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 1044 1045 1046 1047 νLν (erg s−1) MBH = 109 M⊙ 10−1 100 ˙m 1014 1015 1016 1017 1018 1019 ν (Hz) 1044 1045 1046 1047 νLν (erg s−1) He II αox ˙m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='15 108 109 1010 MBH [M⊙] 1045 1046 1047 ν Lν|3000˚ A [erg s−1] 108 109 1010 MBH [M⊙] 10−2 10−1 100 L/LEdd −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='9 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='7 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='4 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2 αox 1045 1046 1047 ν Lν|3000˚ A [erg s−1] 108 109 1010 MBH [M⊙] 10−2 10−1 100 L/LEdd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='1 L228/L1640 Figure D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Left panel: As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2, but for the maximally spinning case, 𝑎∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The grey shaded area shows the range of the SEDs shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2 for the non-spinning case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Centre and right panels: qsosed predictions for 𝛼ox and 𝐿228/𝐿1640 for a maximally spinning SMBH with 𝑎∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='998 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' The reason for this can be understood from the left-hand panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' D1, where we show the qsosed broadband spectrum (the analogue to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2) for the maximally spinning case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' Inspection of the high �𝑚 models in the top-panel reveals that the 𝛼ox behaviour is driven by a combination of stronger X-rays and the movement of the peak of the thermal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' At high spin, the thermal peak moves blueward to higher energies, such that the lower frequency pivot point falls further from the peak and has lower flux compared to the low spin model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' In qsosed, this behaviour comes about in a slightly convoluted way, but is driven by the decrease of the ra- dius 𝑅warm (and corresponding temperature increase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This decrease happens because 𝑅warm = 2𝑅hot, and 𝑅hot must move inwards as spin increases, because 𝑅ISCO moves closer to the SMBH so 𝑅hot must also decrease from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' 2 of Kubota & Done (2018) to maintain the model assumption that the dissipated power is 2 per cent of the Eddington luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' One could clearly construct other models in which the critical radii change in different ways when the spin is changed, which is partly why we caution against over-interpreting the fact that maximally spins appear difficult to reconcile with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} +page_content=' MNRAS 000, 1–20 (2023)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfzwIE/content/2301.02675v1.pdf'} diff --git a/K9E3T4oBgHgl3EQfAglp/content/tmp_files/2301.04258v1.pdf.txt b/K9E3T4oBgHgl3EQfAglp/content/tmp_files/2301.04258v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..78f42e83dd48bb7e982918c1f4efc55dd4ad52c8 --- /dev/null +++ b/K9E3T4oBgHgl3EQfAglp/content/tmp_files/2301.04258v1.pdf.txt @@ -0,0 +1,2360 @@ +CARD: Semantic Segmentation with Efficient Class-Aware Regularized Decoder (Tech report) +CARD: Semantic Segmentation with Efficient Class-Aware +Regularized Decoder +Ye Huang1, Di Kang2, Liang Chen3, Wenjing Jia4, Xiangjian He5, Lixin +Duan1, Xuefei Zhe2 and Linchao Bao2* +1Shenzhen Institute for Advanced Study, UESTC, China. +2Tencent AI Lab, China. +3Fujian Normal University, China. +4University of Technology Sydney, Australia. +5University of Nottingham Ningbo China, China. +*Corresponding author(s). E-mail(s): linchaobao@gmail.com; +Contributing authors: edward.ye.huang@qq.com; di.kang@outlook.com; +liang.chen@outlook.com; Wenjing.Jia@uts.edu.au; xiangjian.he@gmail.com; +lxduan@gmail.com; zhexuefei@outlook.com; +Abstract +Semantic segmentation has recently achieved notable advances by exploiting “class-level” contex- +tual information during learning, e.g., the Object Contextual Representation (OCR) and Context +Prior (CPNet) approaches. However, these approaches simply concatenate class-level information +to pixel features to boost the pixel representation learning, which cannot fully utilize intra-class +and inter-class contextual information. Moreover, these approaches learn soft class centers based +on coarse mask prediction, which is prone to error accumulation. To better exploit class level +information, we propose a universal Class-Aware Regularization (CAR) approach to optimize the +intra-class variance and inter-class distance during feature learning, motivated by the fact that +humans can recognize an object by itself no matter which other objects it appears with. More- +over, we design a dedicated decoder for CAR (named CARD), which consists of a novel spatial +token mixer and an upsampling module, to maximize its gain for existing baselines while being +highly efficient in terms of computational cost. Specifically, CAR consists of three novel loss func- +tions. The first loss function encourages more compact class representations within each class, the +second directly maximizes the distance between different class centers, and the third further pushes +the distance between inter-class centers and pixels. Furthermore, the class center in our approach +is directly generated from ground truth instead of from the error-prone coarse prediction. CAR +can be directly applied to most existing segmentation models during training, including OCR and +CPNet, and can largely improve their accuracy at no additional inference overhead. Extensive experi- +ments and ablation studies conducted on multiple benchmark datasets demonstrate that the proposed +CAR can boost the accuracy of all baseline models by up to 2.23% mIOU with superior gen- +eralization ability. CARD outperforms state-of-the-art approaches on multiple benchmarks with a +highly efficient architecture. The code will be available at https://github.com/edwardyehuang/CAR. +Keywords: Class-aware regularizations, semantic segmentation +1 +arXiv:2301.04258v1 [cs.CV] 11 Jan 2023 + +2 +Article Title +1 Introduction +The task of semantic segmentation is to predict +a class label for each pixel in an image. It is a +fundamental computer vision task that serves as +a critical building block for various downstream +tasks, such as scene understanding, image editing, +self-driving vehicles, etc. After the seminal work +FCN [1] that used fully convolutional networks to +make the dense per-pixel segmentation task more +efficient, many FCN-based approaches [2, 3] have +been proposed and greatly advanced the segmen- +tation accuracy on various benchmarks. Among +these methods, many of them have focused on +better fusing spatial domain context information +to obtain more powerful feature representations +(termed pixel features in this work) for the final +per-pixel classification. For example, DeepLab [3] +and PSPNet [2] utilized multi-scale features via +constructing feature pyramids. +Recently, methods based on dot-product self- +attention (SA) have become very popular since +they can easily capture the long-range relationship +between pixels +[4–12]. SA aggregates informa- +tion dynamically (by different attention maps for +different inputs) and selectively (using weighted +averaging spatial features according to their sim- +ilarity scores). Significant progresses have been +made by using multi-scale and self-attention tech- +niques during spatial information aggregation. +As complements to the above methods, many +recent works have proposed various modules to +utilize class-level contextual information. The +class-level information is often represented by the +class center/context prior which are the mean fea- +tures of each class in the images. OCR [13] and +ACFNet [14] extract “soft” class centers accord- +ing to the predicted coarse segmentation mask by +using the weighted sum. CPNet [15] proposed a +context prior map/affinity map, which indicates +if two spatial locations belong to the same class, +and used this predicted context prior map for fea- +ture aggregation. However, they [13–15] simply +concatenated these class-level features with the +original pixel features for the final classification. +In this paper, we also focus on utilizing class +level information. Instead of focusing on how to +better extract class-level features like the existing +methods [13–15], we use the simple, but accurate, +average feature according to the GT mask, and +focus on maximizing the inter-class distance dur- +ing feature learning. This is because it mirrors how +humans can robustly recognize an object by itself +no matter what other objects it appears with. +Learning more separable features makes the +features of a class less dependent upon other +classes, resulting in improved generalization abil- +ity, especially when the training set contains only +limited and biased class combinations (e.g., cows +and grass, boats and beach). Fig. 1 illustrates an +example of such a problem, where the classifica- +tion of dog and sheep depends on the classification +of grass class, and has been mis-classified as cow. +In comparison, networks trained with our pro- +posed CAR successfully generalize to these unseen +class combinations. +To better achieve this goal, we propose CAR, +a class-aware regularizations module, that opti- +mizes the class center (intra-class) and inter-class +dependencies during feature learning. Three loss +functions are devised: the first encourages more +compact class representations within each class, +and the other two directly maximize the distance +between different classes. Specifically, an intra- +class center-to-pixel loss (termed as “intra-c2p”, +Eq. (3)) is first devised to produce more compact +representation within a class by minimizing the +distance between all pixels and their class cen- +ter. In our work, a class center is calculated as +the averaged feature of all pixels belonging to the +same class according to the GT mask. More com- +pact intra-class representations leave a relatively +large margin between classes, thus contributing to +more separable representations. Then, an inter- +class center-to-center loss (“inter-c2c”, Eq. (6)) +is devised to maximize the distance between any +two different class centers. This inter-class center- +to-center loss alone does not necessarily produce +separable representations for every individual pix- +els. Therefore, a third inter-class center-to-pixel +loss (“inter-c2p”, Eq. (13)) is proposed to enlarge +the distance between every class center and all +pixels that do not belong to the class. +A preliminary version of this work was pre- +sented in [16], which proposed three class-aware +regularization (CAR) terms and evaluated their +effectiveness and universality by using them as +a direct addon to various state-of-the-art meth- +ods. Although effective, we notice two issues when +using CAR as an addon for some baselines – + +Article Title +3 +Existing +Models +CAR +w/o CAR, easily affected by +other classes (e..g. grass) +Reduce Intra-class center- +to-pixel distance +Reduce Inter-class center- +to-center dependency +Cow +Horse +Grass +Dog +Sheep +Reduce inter-class center- +to-pixel dependency +Fig. 1: The concept of the proposed CAR. Our CAR optimizes existing models with three regularization +targets: 1) reducing pixels’ intra-class distance, 2) reducing inter-class center-to-center dependency, and +3) reducing pixels’ inter-class dependency. As highlighted in this example (indicated with a red dot in the +image), with our CAR, the grass class does not affect the classification of dog/sheep as much as before, +and hence successfully avoids previous (w/o CAR) mis-classification. +inefficiency brought by the baselines (e.g. dila- +tion and self-attention [4]) and decreased gain due +to limited compatibility with the baselines (e.g. +CCNet [17]). In this extension, we design a dedi- +cated class-aware regularized decoder (CARD) to +overcome the aforementioned two issues, resulting +in greatly improved computational cost and better +performance. Specifically, a leading synced axial +attention (SAA) is proposed right before CAR to +make sparse self-attention gain as much accuracy +gain as self-attention, and a lightweight pyra- +mid upsampling module is proposed to replace +the computation-heavy dilated convolution with +minimal accuracy loss (see Fig. 4). +In summary, the contributions of this work are: +1. We propose a universal class-aware regu- +larization module that can be integrated +into various segmentation models to largely +improve the accuracy. +2. We devise three novel regularization terms +to achieve more separable and less class- +dependent feature representations by mini- +mizing the intra-class variance and maximiz- +ing the inter-class distance. +3. We calculate the class centers directly from +ground truth during training, thus avoiding +the error accumulation issue of the existing +methods and introducing no computational +overhead during inference. +4. We +visualize +pixel-level +feature-similarity +heatmaps for the inter-class features learned +with and without our CAR to demonstrate +they are indeed less related to each other. +5. We propose a class-aware regularized decoder +aiming for better efficiency and effectiveness +for various backbones, achieving new state- +of-the-art accuracies on multiple benchmarks +while being highly efficient. +2 Related Work +2.1 Class Center. +In 2019 [13, 14], the concept of class center was +introduced to describe the overall representation +of each class from the categorical context perspec- +tive. In these approaches, the center representa- +tion of each class was determined by calculating +the dot product of the feature map and the +coarse prediction (i.e., weighted average) from an +auxiliary task branch, supervised by the ground +truth [2]. After that, those intra-class centers are +assigned to the corresponding pixels on feature +map. Furthermore, in 2020 [15], a learnable kernel +and one-hot ground truth were used to sepa- +rate the intra-class center from inter-class center, +and then concatenated with the original feature +representation. + +4 +Article Title +All of these works [13–15] have focused on +extracting the intra (inter) class centers, but they +then simply concatenated the resultant class cen- +ters with the original pixel representations to +perform the final logits. We argue that the cat- +egorical context information can be utilized in a +more effective way so as to reduce the inter-class +dependency. +To this end, we propose a CAR approach, +where the extracted class center is used to directly +regularize the feature extraction process so as +to boost the differentiability of the learned fea- +ture representations (see Fig. 1) and reduce their +dependency on other classes. Fig. 2 contrasts the +two different designs. More details of the proposed +CAR are provided in Sect. 3. +2.2 Inter-Class Reasoning. +Recently, [18, 19] studied the class dependency as +a dataset prior and demonstrated that inter-class +reasoning could improve the classification perfor- +mance. For example, a car usually does not appear +in the sky, and therefore the classification of sky +can help reduce the chance of mis-classifying an +object in the sky as a car. However, due to the +limited training data, such class-dependency prior +may also contain bias, especially when the desired +class relation rarely appears in the training set. +Fig. 1 shows such an example. In the training +set, cow and grass are dependent on each other. +However, as shown in this example, when there +is a dog or sheep standing on the grass, the class +dependency learned from the limited training data +may result in errors and predict the target into a +class that appears more often in the training data, +i.e., cow in this case. In our CAR, we design inter- +class and intra-class loss functions to reduce such +inter-class dependency and achieve more robust +segmentation results. +2.3 Spatial Context Aggregation +The spatial token mixer [20] provides the context +aggregation between each pixel’s encoding. One of +the well-used token mixers is Self-Attention. +Self-attention. Self-attention proposed in [4, 21] +has been widely used in semantic segmentation [5– +7, 9]. Specifically, self-attention determines the +similarity between a pixel with every other pixel +in the feature map by calculating their dot prod- +uct, followed by softmax normalization. With this +attention map, the feature representation of a +given pixel is enhanced by aggregating features +from the whole feature map weighted by the +aforementioned attention values, thus easily tak- +ing long-range relationship into consideration and +yielding boosted performance. In self-attention, in +order to achieve correct pixel classification, the +representation of pixels belonging to the same +class should be similar to gain greater weights in +the final representation augmentation. +Sparse self-attention. Although regular self- +attention [4] performs very well for semantic seg- +mentation, its computational cost is too high (i.e. +O(H2W 2), especially for high-resolution input. +Thus, many sparse alternatives of the full self- +attention have been proposed, including axial +attention [22], CCNet [17], and CAA [23], achiev- +ing similar accuracy as self-attention but with +greatly reduced computational cost. +2.4 Maintain the feature map +resolution +In +semantic +segmentation, +most +backbones +including CNN-based [24–29] and Transformer +based [10, 30], are initially designed for image- +level classification, where the resolution of the +intermediate feature maps does not matter. So, +they usually progressively downsample the feature +map to a resolution of 1/32 of the original size +(i.e. output stride = 32), resulting in large enough +receptive field size and greatly saved computation. +Unlike image classification, semantic segmen- +tation is essentially a per-pixel classification task, +where the final output size is identical to the input +image. Thus, upsampling is required at the final +stage if the resolution of the intermediate results +is smaller. However, output stride = 32 feature +map usually miss necessary segmentation details +(e.g. boundaries, thin objects, etc) that cannot +be recovered via bilinear upsampling. Thus, main- +taining higher-resolution feature maps is crucial, +among which dilation convolution that does not +reduce the feature map’s resolution too much or +multi-scale pyramid style feature upsampling (e.g. +UNet/FPN) are wildly adopted. +Dilation. +Early +approaches +apply +dilation +(instead of stride) on the later stages of a CNN to +stop further downsampling of the feature maps, +resulting in output stirde = 8 feature maps. How- +ever, the dilation modification introduces too + +Article Title +5 +Class +Center +Pixel +Features +1-Hot +Coarse +GT +Class +Center +Map +Concat +Features +Logits +(a) Design of OCR, ACFNet and CPNet +Pixel +Features +1-Hot +Real +GT +Class +Center +Map +Logits +Class +Center +Supervision +C2C inter loss +C2P intra loss +C2P inter loss +(b) Our CAR +Fig. 2: The difference between the proposed CAR and previous methods that use class-level information. +Previous models focus on extracting class center while using simple concatenation of the original pixel +feature and the class/context feature for later classification. In contrast, our CAR uses direct supervision +related to class center as regularization during training, resulting in small intra-class variance and low +inter-class dependency. See Fig. 1 and Sec. 3 for details. +much computation and it is not applicable to +Transformer-based backbone. +Pyramid-based +upsampling. +Many +other +approaches [31–35]utilize pyramid-based feature +upsampling by fusing multi-scale features from dif- +ferent levels, achieving similar accuracy to dilation +methods but with much less computation. Repre- +sentative methods including UNet [31], FPN [32], +and JPU [36]. UNet and FPN based methods +usually add low-level fine-grained feature maps +(with optional convolution layers) and high-level +coarse feature maps together. This direct addi- +tion of low-/high-level features sometimes makes +training harder [34]. Instead, JPU concatenates +low-/high-level feature maps that is followed by +multiple parallel dilated convolutions, achieving +better accuracy. We also use JPU like pyramid +upsampling for efficiency but with some modi- +fications to improve convergence and make this +upsampling module compatible with backbones +producing various number of feature maps. +3 Methodology +3.1 Extracting Class Centers from +Ground Truth +Denote a feature map and its corresponding +resized one-hot encoded ground-truth mask as +X ∈ RH×W ×C1 and Y ∈ RH×W ×Nclass, respec- +tively. We first get the spatially flattened class +mask Yflat ∈ RHW ×Nclass and flattened feature +1H, W and C denote images’ height and width, and number +of channels, respectively. +map Xflat ∈ RHW ×C. Then, the class center2, +which is the average features of all pixel features +of a class, can be calculated by: +µimage = YT +flat · Xflat +Nnon-zero +∈ RNclass×C, +(1) +where Nnon-zero denotes the number of non-zero +values in the corresponding map of the ground- +truth mask Y. In our experiments, to alleviate +the negative impact of noisy images, we calcu- +late the class centers using all the training images +in a batch, and denote them as µbatch3 (see the +appendix for details). +3.2 Reducing Intra-class Feature +Variance +3.2.1 Motivation. +More compact intra-class representation can lead +to a relatively larger margin between classes, and +therefore result in more separable features. In +order to reduce the intra-class feature variance, +existing works [4–6, 8, 9, 15] usually use self- +attention to calculate the dot-product similarity +in spatial space to encourage similar pixels to +have a compact distance implicitly. For example, +the self-attention in [4] implicitly pushed the fea- +ture representation of pixels belonging to the same +class to be more similar to each other than those +of pixels belonging to other classes. In our work, +2It is termed as class center in [14] and object region +representations in [13]. +3We use µ and omit the subscript batch for clarity. + +6 +Article Title +Self- +attention +Self- +attention +CAR +Module +3×3×512 +Conv +BN +ReLU +Logits +Logits +A1) ResNet-50 + Self-Attention +A2) ResNet-50 + Self-Attention + CAR +Uper +Uper +CAR +Module +3×3×512 +Conv +BN +ReLU +Logits +Logits +A3) Swin-Tiny + Uper +A4) Swin-Tiny + Uper + CAR +B) CAR Module +1×1×512 +Conv +Bn +ReLU +Inter-class +c2c +Loss +Intra-class +c2p +loss +One-hot +GT +Inter-class +c2p +Loss +Intra-class c2p loss +CAR (Training Only) +Feature +Map +One-hot +GT +Center +Map +MSE with +ignored +mask +D) +Inter-class c2p loss +Non-diag +Mask +MSE with +threshold +One-hot +GT +Feature +Map + +softmax +Replace +diag +values +softmax +MSE with +threshold +Dot Product +Element-Wise +Multiplication +Element-Wise +subtraction +Stop gradients +Transposed +Class +center +ResNet- +50 +ResNet- +50 +Swin-Tiny +Swin-Tiny +Class +center +Class +center +Class +center +Operations +One-hot +GT +Inter-class c2c loss +C) +E) +Fig. 3: The proposed CAR approach. CAR can be inserted into various segmentation models, right +before the logit prediction module (A1-A4). CAR contains three regularization terms, including (C) intra- +class center-to-center loss Lintra-c2p (Sec. 3.2.2), (D) inter-class center-to-center loss Linter-c2c (Sec. 3.3.2), +and (E) inter-class center-to-pixel loss Linter-c2p (Sec. 3.3.3). + +Article Title +7 +we devise a simple intra-class center-to-pixel loss +to guide the training, which can achieve this goal +very effectively and produce improved accuracy. +3.2.2 Intra-class Center-to-pixel Loss. +We define a simple but effective intra-class center- +to-pixel loss to suppress the intra-class feature +variance by penalizing large distance between a +pixel feature and its class center. The Intra-class +Center-to-pixel Loss Lintra-c2p is defined by: +Lintra-c2p = fmse(Dintra-c2p), +(2) +where +Dintra-c2p = (1 − σ)|Yflat · µ − Xflat|. +(3) +In Eq. (3), σ is a spatial mask indicating pix- +els being ignored (i.e., ignore label), Yflat · µ +distributes the class centers µ to the correspond- +ing positions in each image. Thus, our intra-class +loss Lintra-c2p will push the pixel representations +to their corresponding class center, using mean +squared error (MSE) in Eq. (3). +3.3 Maximizing Inter-class +Separation +3.3.1 Motivation. +Humans can robustly recognize an object by itself +regardless which other objects it appears with. +Conversely, if a classifier heavily relies on the +information from other classes to determine the +classification result, it will easily produce wrong +classification results when a rather rare class com- +bination appears during inference. Maximizing +inter-class separation, or in another words, reduc- +ing the inter-class dependency, can therefore help +the network generalize better, especially when the +training set is small or is biased. As shown in +Fig. 1, the dog and sheep are mis-classified as +the cow because cow and grass appear together +more often in the training set. To improve the +robustness of the model, we propose to reduce this +inter-class dependency. To this end, the following +two loss functions are defined. +3.3.2 Inter-class Center-to-center Loss. +The first loss function is to maximize the distance +between any two different class centers. Inspired +by the center loss used in face recognition [37], +we propose to reduce the similarity between class +centers µ, which are the averaged features of each +class calculated according to the GT mask. The +inter-class relation is defined by the dot-product +similarity [21] between any two classes as: +Ac2c = softmax( µT · µ +√ +C +), +Ac2c ∈ RNclass×Nclass. +(4) +Moreover, since we only need to constrain the +inter-class distance, only the non-diagonal ele- +ments are retained for the later loss calculation +as: +Dinter-c2c = +� +1 − eye(Nclass) +� +Ac2c. +(5) +We +only +penalize +larger +similarity +values +between any two different classes than a pre- +defined threshold +ϵ0 +Nclass−1, i.e., +Dinter-c2c = fsum +� +max(Dinter-c2c− +ϵ0 +Nclass − 1, 0) +� +. +(6) +Thus, the inter-class center-to-center loss Linter-c2c +is defined by: +Linter-c2c = fmse(Dinter-c2c). +(7) +Here, a small margin is used in consideration of +the feature space size and the mislabeled ground +truth. +3.3.3 Inter-class Center-to-pixel Loss. +Maximizing only the distances between class cen- +ters does not necessarily result in separable rep- +resentation for every individual pixels. We further +maximize the distance between a class center and +any pixel that does not belong to this class. More +concretely, we first compute the center-to-pixel +dot product as: +Λc2p = µT · Xflat, +Λc2p ∈ RHW ×Nclass. +(8) +Ideally, with the previous loss Linter-c2c, the +features of all pixels belonging to the same class +should be equal to that of the class center. There- +fore, we replace the intra-class dot product with +its ideal value, namely using the class center µ for +calculating the intra-class dot product as: +Λc = diag(µT · µ), +(9) + +8 +Article Title +and the replacement effect is achieved by using +masks as: +Λ′ = Λc2p(1 − Yflat) + Λc. +(10) +This updated dot product Λ′ is then used +to calculate similarity across class axis with a +softmax as: +Ac2p = softmax(Λ′), +Ac2p ∈ RHW ×Nclass. (11) +Similar to the calculation of Linter-c2c in the +previous subsection, we have +Dinter-c2p = (1 − Yflat)Ac2p, +(12) +Dinter-c2p = fsum +� +max(Dinter-c2p− +ϵ1 +Nclass − 1, 0) +� +. +(13) +Thus, +the +Inter-class +Center-to-pixel +Loss +Linter-c2p is defined by: +Linter-c2p = fmse(Dinter-c2p). +(14) +3.4 Differences with OCR, ACFNet, +CPNet and CIPC +Methods that are closely related to ours are +OCR [13], ACFNet [14] and CPNet [15], which +all focus on better utilizing class-level features +and differ on how to extract the class centers and +context features. However, they all use a simple +concatenation to fuse the original pixel fea- +ture and the complementary context feature. For +example, OCR and ACFNet first produce a coarse +segmentation, which is supervised by the GT mask +with a categorical cross-entropy loss, and then use +this predicted coarse mask to generate the (soft) +class centers by weighted summing all the pixel +features. OCR then aggregates these class centers +according to their similarity to the original pixel +feature termed as “pixel-region relation”, resulting +in a “contextual feature”. Slightly differently from +OCR, ACFNet directly uses the probability (from +the predicted coarse mask) to aggregate class cen- +ter, obtaining a similar context feature termed +as “attentional class feature”. CPNet defines an +affinity map, which is a binary map indicating +if two spatial locations belong to the same class. +Then, they use a sub-network to predict their ideal +affinity map and use the soft version affinity map +termed as “Context Prior Map” for feature aggre- +gation, obtaining a class feature (center) and a +context feature. Note that CPNet concatenates +class feature, which is the updated pixel feature, +and the context feature. +We also propose to utilize class-level contex- +tual features. Instead of extracting and fusing +pixel features with sub-networks, we propose three +loss functions to directly regularize training and +encourage the learned features to maintain cer- +tain desired properties. The approach is simple +but more effective thanks to the direct supervision +(validated in Tab. 2). Moreover, our class center +estimate is more accurate because we use the GT +mask. This strategy largely reduces the complex- +ity of the network and introduces no computa- +tional overhead during inference. Furthermore, it +is compatible with all existing methods, including +OCR, ACFNet and CPNet, demonstrating great +generalization capability. +We also notice that Cross-Image Pixel Con- +trast (CIPC) [38] shares a similar high-level goal +as our CAR, i.e., learning more similar repre- +sentations for pixels belonging to the same class +than to a different class. However, the approaches +of achieving this goal are very different. CIPC +is motivated by contrastive learning while our +CAR is motivated by the compositionality of the +scene, for better generalization in the cases of +rare class combinations. Therefore, CIPC adopts +the one-vs-rest style InfoNCE loss, including the +typical pixel-to-pixel loss and a special pixel- +to-center loss. In contrast, (1) we propose an +additional center-to-center loss to regularize the +inter-class dependency explicitly and effectively +(see Table 1); (2) we use one-vs-one style inter- +class losses while CIPC uses one-vs-rest style NCE +loss. Compared to our one-vs-one loss, using one- +vs-rest loss for training does not necessarily result +in small inter-class similarity between the cur- +rent class and every individual “other” classes +and may increase the inter-class similarity among +those “other” classes. (3) we leave margins to +prevent CAR regularizations, which is not the pri- +mary task of pixel classification, from dominating +the learning process. + +Article Title +9 +Class-Aware Regularized Decoder +EJPU +CAR +Module +Backbone +SAA +Logits +OS=32 +OS=8 +Stop gradients +Fig. 4: Overview of the proposed CARD. Class-aware regularized decoder (CARD) is tailored for +the proposed class-aware regularizations with greatly reduced computation cost and minor accuracy loss. +CARD contains an EJPU, which fuses features from different layers (at the same spatial location) to +obtain high-resolution multi-scale and multi-level feature maps, an synced axial attention (SAA) token +mixer, which fuses features from different locations as context aggregation, and CAR to produce less +class-dependent and thus more generalizable pixel features. The output stride (OS) = 8 logit maps are +bilinearly upsampled to the original resolution to make final prediction. +3.5 The Devil Is in the +Architecture’s Detail +The proposed CAR is compatible with many mod- +els as shown in table 2. However, some layers or +operations in existing models may be harmful to +the ability of CAR, where the last 3×3 conv is one +commonly found case in many models [4, 17, 39] +(see A1 and A3 in Fig. 3). A possible reason is +that the network is trained to maximize the sep- +aration between different classes. However, if the +two pixels lie on different sides of the segmenta- +tion boundary, a 3 × 3 conv will merge the pixel +representations from different classes, making the +proposed CAR harder to optimize. In this work, +we provide a simple and optional general modifi- +cation for those models to enhance CAR’s ability, +where we use 1 × 1 conv to replace the original +3 × 3 conv. Existing models like DeepLab [3] are +not required to modify because they are using the +1 × 1 conv as the original settings. Note that, +this is only modification we made in some existing +models, because it is simple and generalized. +We also found some architecture-specified +modifications, yet not generalized, can further +largely improve the performance when employ- +ing CAR on those existing models. For example, +decreasing the filter number to 256 for the last +conv layer of ResNet-50 + Self-Attention + CAR, +or replacing the conv layer after PPM (inside Uper +block, Fig. 3A3) from 3 × 3 to 1 × 1 in Swin-Tiny ++ UperNet. We did not try to exhaustively search +these variants since they did not generalize. +3.6 Class-aware Regularized +Decoder +3.6.1 Motivation +As mentioned in the Sec. 3.5, simply applying +CAR to existing methods without architecture- +specified modification may result in sub-optimal +result. To better utilize CAR for semantic segmen- +tation, we design a novel decoder module tailored +for CAR by taking efficiency and effectiveness into +consideration. +Concretely, the decoder design focuses on three +aspects: 1) compatibility with the proposed CAR, +2) efficient spatial context aggregation, and 3) less +computational overhead (e.g. avoiding dilation +convolution). The resultant class-aware regular- +ized decoder (CARD) is a lightweight, simple +yet effective decoder for semantic segmentation, +achieving good performance via small compu- +tational overhead and reasonable GPU memory +usage together. +3.6.2 Overview of CARD +Fig. 4 presents the overall architecture of the pro- +posed CARD4. CARD first uses our enhanced +joint pyramid upsampling (EJPU) to obtain +higher resolution multiscale feature maps with +output stride (OS) = 8 (Sec. 3.6.4). Then it uses +4In this work, we refer a complete segmentation network as +“model/method/baseline”, which usually consists of a “back- +bone” feature extractor (e.g. ResNet-50, usually pretrained on +a large-scale classification dataset) and a “decoder” that typi- +cally increases the resolution of the feature maps (e.g. EJPU) +and/or conducts multi-scale/global context aggregation as +further enhancement (e.g. SAA). + +10 +Article Title +Q +K +V +Row +Att. Maps +Q +K +V +Column +Att. Maps +Row +Att. Maps +Q +K +V +Column +Att. Maps +a) Vanilla Axial Attention +b) Synced Axial Attention +Fig. 5: Comparison between vanilla axial atten- +tion and our proposed synced axial attention +(SAA). The difference is highlighed in orange in +the figure. In SAA, both column and row attention +maps are obtained from the same set of queries +and keys The column and row attention of SAA +shared the same query/key. +our proposed synced axial attention (SAA), which +is lightweight and more compatible with the fol- +lowing CAR regularizations, to perform global +spatial context aggregation (Sec. 3.6.3). Finally, +the output of the token mixer is optimized by our +proposed CAR to obtain less class-dependent and +more generalizable pixel representation. +This novel design, which is optimized for effi- +ciency and effectiveness, outperforms other state- +of-the-art methods that use up to 3 times compu- +tation of ours, striking to a good balance between +accuracy and computational cost. +3.6.3 Synced Axial Attention +For efficiency and effectiveness, we design a new +synced axial attention (SAA) for CAR since we +notice existing sparse attention method obtains +limited accuracy gain from CAR (e.g. CCNet [17], +only +0.56 in Tab. 2). +Although CAA + CAR achieves considerably +big gain and the best results, we do not consider +CAA for spatial context aggregation because it is +an uncommon operation that has small FLOPs +but has an actual slow speed in some hardware due +to the lack of hardware and software (e.g, GPU +driver/library) support. +Token +mixer. +In +CARD, +we +proposed +an +improved version of multi-head axial attention as +the token mixer, named synced axial attention +(SAA) in Fig. 5. In vanilla axial attention, column +attention (vertical) and row attention (vertical) +are performed separately, i.e. using different input +feature (X and Xcol) and different transforma- +tions. Differently, SAA only computes the query +Q, key K, and value V once, and uses the same +set of query and key to generate both the col- +umn attention map and row attention map. After +the column-wise context aggregation, the update +feature is directly used for row-wise context aggre- +gation according to the row attention maps. Thus, +SAA takes as input consistent feature space when +computing the column and row attention maps, +since they are generated by the same query and +key. Empirically, we find this consistent/synced +attention calculation not only reduces compu- +tation but also improves the performance (see +Tab. 3). Possible reasons may be that using con- +sistent input and shared transformation avoids +potential error accumulation during the attention- +based feature aggregation and directly conducts +optimization in global context (rather than via +two stages in AA or CCNet). +Positional encoding. In CARD, we apply con- +ditional positional encoding (CPE) [40, 41], a +resolutions insensitive positional encoding before +the attention operations. Note that we did not +apply normalization in MaxViT [41] since we +found it harmful to the accuracy. +3.6.4 Enhanced Joint Pyramid +Upsampling (EJPU) +We choose JPU since it integrates better with +multi-scale/global context aggregation modules +(e.g. ASPP [42], self-attention) than other UNet- +like encoder-decoder or FPN [32] (more discussion +in Sec. 2.4). Based on JPU, we make some cru- +cial modifications to improve its convergence and +make it more compatible with the proposed CAR, +resulting in largely improved accuracy (50.76 vs +49.76 in Tab. 4) +Concretely, we notice the initial convergence +speed on the test set (evaluated every 1k train- +ing steps) is slower than the dilation model during +our experiments. The possible reason is that JPU +does not fully utilize the original backbone feature +maps (i.e. highest abstraction level) since they are + +Article Title +11 +Original +JPU +(Channel = J) +Backbone +OS = 8 Feature +OS = 16 Feature +OS = 32 Feature +(C = K) +OS = 8 +(C = J) +OS = 8 +(C = J) +1D vector +(C = K) +MLP +(LeakReLU) +1×1 Conv +BN, ReLU +Global Average +Pooling +1×1 Conv +BN, ReLU +Operations +Feature Map +CPM +(If required) +Pad Map +(C = J - K) +1D vector +(C = J - K) +To +decoder +OS = 32 +(C = J) +Element-Wise +Addition +OS = 8 +(C = J) +C +Up +Up +Identity (K == J) +Stop gradients +C Concat +Up Upsample +Fig. 6: Detailed design of the proposed EJPU. +Similar to ResNet (see Sec. 3.6.4), we add upsam- +pled original backbone features with “residual” +information extracted from modified JPU mod- +ule, which is then fed into the following decoder. +We applied an optional CPM when the channel +numbers differed between the original backbone +and JPU features. Feature alignment is neces- +sary because the channel of the original feature +from the backbone is not grouped and arranged in +multi-scale order. +treated equally with low-level feature maps from +previous stages. In contrast, FCN [1] initialized +the weights to zero for the convolution follow- +ing low-level features before adding them with +the original backbone features. The dilation model +[2, 3] directly use the original backbone feature +and the filter weights in essence. Both FCN and +dilation models have faster convergence than JPU. +Motivated by the above observation, we equip +JPU with a ResBlock-style residual branch that +directly sends the original backbone feature (via +minimal learned transformation if required) to the +later network layers. We detail the modifications +as follows. +Residual branch. To better utilize the well- +trained original feature from the backbone, we +include a residual branch to directly feed the bilin- +early upsampled backbone feature maps to the fol- +lowing network module (bottom in Fig. 6) similar +to FCN and ResBlock. For some backbones, the +output feature channel is not the same as JPU’s +output (i.e. 2048). So a channel padding mod- +ule (CPM) is introduced with as less as possible +learnable transformations only when necessary. +Multi-scale multi-level feature branch. We +adopt JPU [36] style multi-scale multi-level fea- +ture fusion for upsampling to provide complemen- +tary information lost in the original backbone +feature. Specifically, feature maps extracted by +JPU are processed by a 1 × 1 Conv (followed by +BN and ReLU), and then added to the backbone +feature maps element-wisely. This extra convolu- +tion after JPU is introduced to calibrate the JPU +features to the backbone features since JPU has +reordered the channels and the meaning of the +JPU and backbone features in the same dimen- +sion/channel does not correspond any more. Note +that we do not back-propagate gradient to the +highest level backbone feature through JPU and +only keep gradient from the residual branch. +Channel Padding Module. We include an +optional channel padding module (CPM) since +different backbones output feature maps with +different dimensions (i.e. channel numbers). In +order not to interfere the original feature maps +too much, we use as less as possible learnable +transformations to project the feature to required +dimensions (i.e. 2048). Specifically, the original +backbone feature maps go through only a padding +operation and a convolution layer. The padded +feature maps is generated with global average +pooling, dimension projection and unpooling as +shown in Fig. 6 bottom. +In Tab. 5, other simple and intuitive alter- +natives have also been tested, including direct +projection (optionally with BN + ReLU), chan- +nel axis interpolation, or align JPU dimensions +to backbone dimensions, but they are not as + +12 +Article Title +effective as this configuration. The possible rea- +son is that redundant channel information (i.e., +direct projection/channel axis interpolation does +not fully utilized the original well-trained features +and insufficient channel information (i.e., match +dimensions) reduces the network capacity. +4 Experiments +In the following, we first disclosure the implemen- +tation details and the detailed experiment settings +in Sec. 4.1. Then we present various experimental +results on Pascal Context (Sec. 4.2), COCOStuff- +10K +(Sec. +4.3), +COCOStuff-164K +(Sec. +4.4), +and Cityscapes (Sec. 4.5). On Pascal Context +(Sec. 4.2), we conduct thorough ablation studies +(including the effectiveness of individual regular- +ization terms inside CAR (Tab. 1), the applica- +bility of CAR for various baselines (Tab. 2),, the +effectiveness individual components inside CARD +(Tab. 3-5), etc.) and present various visualizations +for in-depth analysis (Fig. 7-8)). +4.1 Implementation Details +Training Settings. For both baselines and CAR +experiments, we apply the settings common to +most works [7–9, 17, 43], including SyncBatch- +Norm, batch size = 16, weight decay (0.001), 0.01 +initial LR, and poly learning decay with SGD dur- +ing training. In addition, for the CNN backbones +(e.g., ResNet), we set output stride = 8 (see [42]). +Training iteration is set to 30k iterations unless +otherwise specified. For the thresholds in Eq. 6 +and Eq. 13, we set ϵ0 = 0.5 and ϵ1 = 0.25. +CARD experiments use the same settings +as +“Baselines ++ +CAR” +unless +stated +other- +wise. For example, CARD experiments compared +with the state-of-the-art methods use AdamW +(instead of SGD) for fair comparison since it is +widely adopted by recent state-of-the-art meth- +ods. Details are described in the corresponding +subsections with the dataset. +Determinism & Reproducibility. Our imple- +mentations are based on the latest NVIDIA deter- +ministic framework (2022), which means exactly +the same results can be always reproduced with +the same hardware and same training settings +(including random seed). To fairly demonstrate +the effectiveness of our CAR, we reimplement +and reproduce all the baselines in our ablative +experiments. +4.2 Experiments on Pascal Context +The Pascal Context [44] 5 dataset is split into +4,998/5,105 for training/test set. We use its 59 +semantic classes following the common practice [7, +13]. Unless otherwise specified, all experiments are +trained on the training set with 30k iterations. +Ablation studies related to “baselines + CAR” +are presented in Sec. 4.2.1, and ablation studies +related to CARD are presented in Sec. 4.2.2. +4.2.1 Ablation Studies of CAR +In +the +following +experiments, +we +first +test +the effectiveness of the individual regulariza- +tion terms in CAR when plugged into different +basic baselines, including the CNN-based and +the Transformer-based baselines as representa- +tives. Then, we test the effectiveness of CAR as +a whole on many other well-known baselines to +demonstrate its universality +CAR on “ResNet-50 + Self-Attention”. +We firstly test the CAR with ResNet-50 + Self- +Attention (w/o image-level block in [7]) to verify +the effectiveness of the proposed loss functions, +i.e., Lintra-c2p, Linter-c2c, and Linter-c2p. As shown +in Tab. 1, using Lintra-c2p directly improves 1.30 +mIOU (48.32 vs 49.62); Introducing Linter-c2c and +Linter-c2p further improves 0.38 mIOU and 0.50 +mIOU; Finally, with all three loss functions, the +proposed CAR improves 2.18 mIOU from the reg- +ular ResNet-50 + Self-attention (48.32 vs 50.50). +CAR on “Swin-Tiny + Uper”. +“Swin-Tiny + Uper” is a totally different archi- +tecture from “ResNet-50 + Self-Attention [4]”. +Swin [30] is a recent Transformer-based backbone +network. Uper [39] is based on the pyramid pool- +ing modules (PPM) [2] and FPN [32], focusing on +extracting multi-scale context information. Simi- +larly, as shown in Tab. 1, after adding CAR, the +performance of Swin-Tiny + Uper also increases +by 1.16, which shows our CAR can generalize to +different architectures well. +5https://www.cs.stanford.edu/∼roozbeh/pascal-context/ + +Article Title +13 +Table 1: Ablation studies of adding CAR to different methods on Pascal Context dataset. All results are +obtained with single scale test without flip. “A” means replacing the 3 × 3 conv with 1 × 1 conv (detailed +in Sec. 4.2.1). CAR improves the performance of different types of backbones (CNN & Transformer) and +head blocks (SA & Uper), showing the generalizability of the proposed CAR. +Methods +Lintra-c2p +Linter-c2c +Linter-c2p +A +mIOU (%) +R1 +ResNet-50 + Self-Attention +- +- +48.32 +R2 +✓ +48.56 +R3 ++ CAR +✓ +49.17 +R4 +✓ +✓ +49.79 +R5 +✓ +✓ +✓ +50.01 +R6 +✓ +✓ +49.62 +R7 +✓ +✓ +✓ +50.00 +R8 +✓ +✓ +✓ +✓ +50.50 +S1 +Swin-Tiny + UperNet +- +- +49.62 +S2 +✓ +49.82 +S3 ++ CAR +✓ +49.01 +S4 +✓ +✓ +50.63 +S5 +✓ +✓ +✓ +50.26 +S6 +✓ +✓ +49.62 +S7 +✓ +✓ +✓ +50.58 +S8 +✓ +✓ +✓ +✓ +50.78 +The devil is in the architecture’s detail. +As mentioned in Sec. 3.5, we find it important to +replace the leading 3 × 3 convolution (in the orig- +inal baseline) with a 1 × 1 convolution (Fig. 3B). +For example, Linter-c2p did not improve the perfor- +mance in Swin-Tiny + Uper (S5 vs S4 in Tab. 1) +until the last 3 × 3 convolution is replaced by a +1 × 1 (S8 vs S7 in Tab. 1). +To keep the simplicity and demonstrate its +generalizability, we use the same network con- +figurations for all the baseline methods. No +architecture-specific modification is made when +conducting ablation studies on existing models for +experiments in Tab. 1- 2. +CAR on various baselines. +After we have verified the effectiveness of each +part of the proposed CAR, we then tested CAR on +multiple well-known baselines. All of the baselines +were reproduced under similar conditions (see +Sect. 4.1). Experimental results shown in Tab. 2 +demonstrate the generalizability of our CAR on +different backbones and methods. +4.2.2 Ablation Studies of CARD +In the following experiments, we test the effec- +tiveness of the proposed CARD. Ablation studies +include the effectiveness of individual components +in CARD (i.e. the spatial token mixer in Tab. 3, +EJPU in Tab. 4 & 5), and a computational cost +analysis in Tab. 6. +Effectiveness of the token mixer +In Tab. 3, we conduct ablation studies of different +token mixer designs in CARD. They are evalu- +ated using a Dilated ResNet-50 with output stride += 8 (“ResNet-50 (D8)”) on the Pascal Context +dataset. All settings are the same as the the ones +tested in Tab. 1 and Tab. 2. +We empirically find using head count = 4 +for the multi-head axial attention achieves best +accuracy (50.91% mIOU), which is slightly better +than self-attention (50.50% mIOU) and cost much +less computation. This computational cost partic- +ularly matters for high-resolution inputs, which is +evaluated in Tab. 6. Compared to another similar +sparse attention based method CCNet, our SAA +brings much more accuracy gain (+1.45 vs +0.41), + +14 +Article Title +Table 2: Ablation studies of adding CAR to different baselines on Pascal Context [44] and COCOStuff- +10K [45]. We deterministically reproduced all the baselines with the same settings. All results are single- +scale without flipping. CAR works very well in most existing methods. ⊠ means reducing the class-level +threshold to 0.25 from 0.5. We found it is sensitive for some model variants to handle a large number +of class. Affinity loss [15] and Auxiliary loss [2] are applied on CPNet and OCR, respectively, since they +highly rely on those losses. +Methods +Backbone +mIOU(%) +Pascal Context +COCO-Stuff10K +FCN [1] +ResNet-50 [25] +47.72 +34.10 +FCN + CAR +ResNet-50 [25] +48.40(+0.68) +34.91(+0.81)⊠ +FCN [1] +ResNet-101 [25] +50.93 +35.93 +FCN + CAR +ResNet-101 [25] +51.39(+0.49) +36.88(+0.95)⊠ +DeepLabV3 [42] +ResNet-50 [25] +48.59 +34.96 +DeepLabV3 + CAR +ResNet-50 [25] +49.53(+0.94) +35.13(+0.17) +DeepLabV3 [42] +ResNet-101 [25] +51.69 +36.92 +DeepLabV3 + CAR +ResNet-101 [25] +52.58(+0.89) +37.39(+0.47) +Self-Attention [4] +ResNet-50 [25] +48.32 +34.35 +Self-Attention + CAR +ResNet-50 [25] +50.50(+2.18) +36.58(+2.23)⊠ +Self-Attention [4] +ResNet-101 [25] +51.59 +36.53 +Self-Attention + CAR +ResNet-101 [25] +52.49(+0.9) +38.15(+1.62) +CCNet [17] +ResNet-50 [25] +49.15 +35.10 +CCNet + CAR +ResNet-50 [25] +49.56(+0.41) +36.39(+1.29) +CCNet [17] +ResNet-101 [25] +51.41 +36.88 +CCNet + CAR +ResNet-101 [25] +51.97(+0.56) +37.56(+0.68) +DANet [5] +ResNet-101 [25] +51.45 +35.80 +DANet + CAR +ResNet-101 [25] +52.57(+1.12) +37.47(+1.67) +CPNet [15] +ResNet-101 [25] +51.29 +36.92 +CPNet + CAR +ResNet-101 [25] +51.98(+0.69) +37.12(+0.20)⊠ +OCR [13] +HRNet-W48 [46] +54.37 +38.22 +OCR + CAR +HRNet-W48 [46] +54.99(+0.62) +39.53(+1.31) +UperNet [39] +Swin-Tiny [30] +49.62 +36.07 +UperNet + CAR +Swin-Tiny [30] +50.78(+1.16) +36.63(+0.56) ⊠ +UperNet [39] +Swin-Large [30] +57.48 +44.25 +UperNet + CAR +Swin-Large [30] +58.97(+1.49) +44.88(+0.63) +CAA [23] +EfficientNet-B5 [28] +57.79 +43.40 +CAA + CAR +EfficientNet-B5 [28] +58.96(+1.17) +43.93(+0.53) +CAA [23] +ConvNext-Large [29] +60.48 +46.49 +CAA + CAR +ConvNext-Large [29] +61.40(+0.92) +46.70(+0.21) +demonstrating SAA is indeed more compatible +with CAR. +As a result (Tab. 3), SAA brings more accu- +racy gain (+1.45) compared to vanilla AA (+1.24) +and CCNet (+0.41), and achieves similar accuracy +to self-attention (50.91 vs 50.50 ) but with much +smaller computational cost. +Effectiveness of EJPU +In Tab. 4, we evaluate the proposed EJPU +of CARD on the Pascal Context dataset with +ResNet-50. All settings are same as the the ones +in Tab. 1 and Tab. 2. +Compared to other approaches, such as origi- +nal JPU and Semantic FPN, EJPU has the closest +performance to the dilation model and even beats +the Semantic FPN with twice filter. Also note + +Article Title +15 +Table 3: Ablation studies of different token mix- +ers’ compatibility with CAR on Pascal Context +dataset using ResNet-50 (D8), where “D8” means +modifying the last convolution layers of the back- +bone to their dilated version to obtain output +stride (OS) = 8 feature maps [42]. “PE” repre- +sents conditional positional encoding in [40, 41]. +“HC” represents number of attention heads. See +Sec. 3.6.3 for details. +Methods +CAR +CPE +HC +mIOU(%) +T1 +Self-Attention [4] +1 +48.32 +T2 +✓ +1 +50.50(+2.18) +T3 +CCNet [17] +1 +49.15 +T4 +✓ +1 +49.56(+0.41) +T5 +Vanilla AA +✓ +4 +49.39 +T6 +✓ +✓ +4 +50.63(+1.24) +T7 +SAA +✓ +4 +49.46 +T8 +✓ +✓ +2 +50.82(+1.36) +T9 +✓ +✓ +1 +50.55(+1.09) +T10 +✓ +✓ +4 +50.91(+1.45) +Table 4: Ablation studies of EJPU in CARD on +Pascal Context dataset using ResNet-50. Com- +pared to Semantic FPN and JPU, the proposed +EJPU achieved the closest performance to the +dilation model. +Mode +CAR +mIOU(%) +Semantic FPN [33] +48.96 +✓ +49.67 (+0.71) +Semantic FPN [33] 2× filters +48.83 +✓ +50.04 (+1.21) +JPU [36] +49.05 +✓ +49.76 (+0.71) +EJPU (Ours) +49.63 +✓ +50.76 (+1.13) +OS = 8 (Dilation) [2, 42] +49.46 +✓ +50.91 (+1.45) +that EJPU is more compatible with CAR since it +brings more accuracy gain (+1.13 vs +0.71). +Effectiveness of CPM in EJPU +In Tab. 5, we compare different options for chan- +nel padding if the dimensions of the backbone and +the JPU are different. We use ConvNeXt-L [29] +(1536 channels) to conduct the experiments. The +remaining settings are the same as in the previ- +ous sections. “Project JPU output” and “Project +Table 5: Ablation studies of CPM inside EJPU +on Pascal Context dataset using ConvNeXt-L, +which outputs 1536 channel feature maps and thus +requires the Channel Padding Module (CPM). +Padding Strategies +mIOU(%) +Project JPU output +61.51 +Project Backbone output +61.68 +Reduce JPU’s conv filters +60.94 +Interpolation +59.43 +CPM +61.99 +Table 6: Computation analysis of CARD. We +provide computational cost in GFLOPs on two +commonly used resolutions 513 × 513 and 1025 × +2049. “SA” is short for “Self-Attention”. “D8” +means modifying the last convolution layers of +the backbone to their dilated version to obtain +output stride (OS) = 8 feature maps [42]. Rows +in CARD have not marked by “w/ EJPU” use +dilated backbone with output stride = 8 (D8). +Method +Backbone +GFLOPs +513 × 513 +1025 × 2049 +SA (CAR) +ResNet-50 (D8) +158.96 +1723.03 +CARD +ResNet-50 (D8) +151.70 +1157.59 +w/ EJPU +ResNet-50 +112.69 (-25%) +887.180 (-23%) +CARD +ConvNeXt-L (D8) +818.14 +6418.79 +w/ EJPU +ConvNeXt-L +262.82 (-67%) +2043.24 (-68%) +CARD +EfficientNet-L2 (D8) +1635.22 +12834.12 +w/ EJPU +EfficientNet-L2 +283.62 (-82%) +2184.76 (-82%) +backbone output” use a 1 × 1 convolution layer +(followed by BN and ReLU) to adjust the channel +number to match the other one. “Reduce JPU’s +conv filters” means reduce the filter numbers of all +the convolution layers in JPU by a same factor to +the backbone feature dimension. Among all these +configurations, CPM achieves the best accuracy. +Computational cost of CARD +Tab. 6 presents computational cost of CARD for +two commonly seen image resolutions. Compared +to the Self-Attention with dilated ResNet-50, +our CARD significantly reduces the computa- +tional cost from 158.96 to 112.69 GFLOPs. EJPU +reduces more computation for larger backbones or +higher-resolution inputs. + +16 +Article Title +Table 7: Comparisons to state-of-the-art meth- +ods on Pascal Context dataset of CARD. Note +that methods marked with ‘∗’ report mIOU from +their papers while the others are obtained with +our implementation. SS means single scale per- +formance w/o flipping. MF means multi-scale +performance w/ flipping. +Methods +Avenue +mIOU(%) +SS +MF +SETR (ViT-L)* [12] +CVPR’21 +- +55.8 +DPT (ViT-Hybrid)* [11] +ICCV’21 +- +60.5 +Segmenter (ViT-L)* [48] +ICCV’21 +- +59.0 +OCNet (HRNet-W48)* [6] +IJCV’21 +- +56.2 +CAA (EfficientNet-B7)* [23] +AAAI’22 +- +60.5 +SegNeXt (MSCAN-L)* [49] +NIPS’22 +59.2 +60.9 +CAA + CAR (ConvNeXt-L) +ECCV’22 +62.7 +63.9 +CARD (ConvNeXt-L) +Ours +63.2 +64.4 +CARD (ConvNeXtV2-L) +Ours +64.0 +64.6 +CARD (EfficientNet-L2) +Ours +66.0 +67.5 +4.2.3 CARD Compared to the +State-of-the-art +In Tab. 7, we equip CARD to stronger back- +bones to compare with state-of-the-art methods +on Pascal Context dataset. The reported mIOU +of compared methods with “*” comes from their +respective paper instead of our reproduction. +We +train +CARD +with +ConvNeXt-L +[29], +ConvNeXtV2-L [47] and EfficientNet-L2 [28] as +backbone, using AdamW optimizer, an initial +learning rate of 4e-5, while the other settings +remain the same as the experiments in our abla- +tion studies. The AdamW optimizer improved +the performance of CARD (ConvNeXt-L) from +61.99% (shown in Tab. 5, trained by SGD) +to 63.20%. As shown in Tab. 7, CARD out- +performs other state-of-the-art approaches when +using ConvNeXt-L and ConvNeXtV2-L. +With +even +stronger +backbone +such +as +EfficientNet-L2, CARD achieves 66.0% mIOU +under single-scale setting and 67.5% mIOU under +multi-scale with flipping setting. +4.2.4 Visualization of Class +Dependency Maps +In Fig. 7, we present the class dependency maps +calculated on the complete Pascal Context test +set, where every pixel stores the dot-product sim- +ilarities between every two class centers. The +maps indicate the inter-class dependency obtained +with the standard ResNet-50 + Self-Attention and +Swin-Tiny + UperNet, and the effect of applying +our CAR. A hotter color means that the class has +higher dependency on the corresponding class, and +vice versa. According to Fig. 7 a1-a2, we can easily +observe that the inter-class dependency has been +significantly reduced with CAR on ResNet50 + +Self-Attention. Fig. 7 b1-b2 show a similar trend +when tested with different backbones and head +blocks. This partially explains the reason why +baselines with CAR generalize better on rarely +seen class combinations (Figs. 1 and 8). Interest- +ingly, we find that the class-dependency issue is +more serious in Swin-Tiny + Uper, but our CAR +can still reduce its dependency level significantly. +4.2.5 Visualization of Pixel-relation +Maps +In Fig. 8, we visualize the pixel-to-pixel relation +energy map, based on the dot-product similarity +between a red-dot marked pixel and other pixels, +as well as the predicted results for different meth- +ods, for comparison. Examples are from Pascal +Context test set. As we can see, with CAR super- +vision, the existing models focus better on objects +themselves rather than other objects. Therefore, +this reduces the possibility of the classification +errors because of the class-dependency bias. +4.3 Experiments on COCOStuff-10K +COCOStuff-10K dataset [45] 6 is widely used for +evaluating the robustness of semantic segmenta- +tion models [8, 13]. The COCOStuff-10k dataset is +a very challenging dataset containing 171 labeled +classes and 9000/1000 images for training/test. +4.3.1 CAR on Different Baselines +In Tab. 2, all of the tested baselines gain perfor- +mance boost ranging from 0.17% to 2.23% with +our proposed CAR on COCOStuff-10K dataset. +This demonstrates the generalization ability of our +CAR when handling a large number of classes. +4.3.2 CARD Compared to the +State-of-the-art +In Tab. 8, we equip CARD to ConvNeXt-L +to compare with state-of-the-art methods on +6https://github.com/nightrome/cocostuff10k + +Article Title +17 +a1. ResNet50 + Self-Attention +a2. ResNet50 + Self-Attention + CAR +b1. Swin-Tiny + UperNet +b2. Swin-Tiny + UperNet + CAR +a1. ResNet50 + Self-Attention +a2. ResNet50 + Self-Attention + CAR +b1. Swin-Tiny + UperNet +b2. Swin-Tiny + UperNet + CAR +Fig. 7: Class dependency maps generated on Pascal Context test set. One may zoom in to see class names. +A hotter color means that the class has higher dependency to the corresponding class, and vice versa. It +is obvious that our CAR reduces the inter-class dependency, thus providing better generalizability. +Table 8: Comparisons to state-of-the-art meth- +ods on COCOStuff10k dataset of CARD. Note +that methods marked with ‘∗’ report mIOU from +their papers while the others are obtained with +our implementation. SS means single scale per- +formance w/o flipping. MF means multi-scale +performance w/ flipping. +Methods +Avenue +mIOU(%) +SS +MF +OCR (HRNet-W48)* [13] +ECCV’20 +- +45.2 +OCNet (HRNet-W48)* [6] +IJCV’21 +- +40.0 +CAA (EfficientNet-B7)* [23] +AAAI’22 +- +45.4 +RankSeg (ViT-L)* [50] +ECCV’22 +- +47.9 +CAA + CAR (ConvNeXt-L) +ECCV’22 +48.2 +48.8 +CARD (ConvNeXt-L) +Ours +48.9 +50.0 +COCOStuff-10K dataset. The reported mIOU of +compared methods with “*” comes from their +respective paper instead of our reproduction. We +trained CARD with ConvNeXt-L using AdamW +optimizer, an initial learning rate of 4e-5, while the +other settings remain the same as the experiments +in our ablation studies. As shown in Tab. 8, CARD +(ConvNeXt-L) surpasses the other methods with +a large margin. +Table 9: Comparisons to state-of-the-art meth- +ods on COCOStuff164K dataset of CARD. Note +that methods marked with ‘∗’ report mIOU from +their papers while the others are obtained with +our implementation. SS means single scale per- +formance w/o flipping. MF means multi-scale +performance w/ flipping. +Methods +Avenue +mIOU(%) +SS +MF +SegFormer (MiT-B5)* [51] +NIPS’21 +- +46.7 +CAA (EfficientNet-B5)* [23] +AAAI’22 +- +47.3 +SegNeXt (MSCAN-L)* [49] +NIPS’22 +46.5 +47.2 +CARD (ConvNeXt-L) +Ours +48.9 +49.6 +CARD (EfficientNet-L2) +Ours +50.2 +50.9 +4.4 Experiments on +COCOStuff-164K +4.4.1 CARD Compared to the +State-of-the-art +COCOStuff-164k 7 is the full set of COCOStuff- +10K, which becomes a new popular benchmark +starting from 2021. Training settings are the same +as COCOStuff-10k (Sec. 4.3.2), except the total +training iteration is set to 80k. As shown in +Tab. 9, the proposed CARD outperforms previous +approaches by a large margin. +7https://github.com/nightrome/cocostuff + +0.0 +1.0 +o'S +0'3 +0'4 +0'2 +o'e +8.0 +e.018 +Article Title +a1 +W/O CAR +W/ CAR +a1 +W/O CAR +W/ CAR +a3 +W/O CAR +W/ CAR +a3 +W/O CAR +W/ CAR +a4 +W/O CAR +W/ CAR +a4 +W/O CAR +W/ CAR +a5 +W/O CAR +W/ CAR +a5 +W/O CAR +W/ CAR +a6 +W/O CAR +W/ CAR +a6 +W/O CAR +W/ CAR +Image/GT +Heatmap +Prediction +Image/GT +Heatmap +Prediction +a2 +W/O CAR +W/ CAR +a2 +W/O CAR +W/ CAR +(a) ResNet50 + Self-Attention +b1 +b2 +b3 +b4 +b5 +b6 +Image/GT +Heatmap +Prediction +W/O CAR +W/ CAR +W/O CAR +W/ CAR +W/O CAR +W/ CAR +W/O CAR +W/ CAR +W/O CAR +W/ CAR +W/O CAR +W/ CAR +b1 +b2 +b3 +b4 +b5 +b6 +Image/GT +Heatmap +Prediction +W/O CAR +W/ CAR +W/O CAR +W/ CAR +W/O CAR +W/ CAR +W/O CAR +W/ CAR +W/O CAR +W/ CAR +W/O CAR +W/ CAR +(b) Swin-Tiny + UperNet +Fig. 8: Visualization of the feature similarity between a given pixel (marked with a red dot in the image) +and all pixels, as well as the segmentation results on Pascal Context test set. A hotter color denotes +larger similarity value. Apparently, our CAR reduces the inter-class dependency and exhibits better +generalization ability, where energies are better restrained in the intra-class pixels. + +ATA9 YAWHDIA +2unh3TAVIR9Article Title +19 +4.4.2 Visualization of CARD +COCOStuff-164k results in Fig. 9 compare Seg- +NeXt [49], and our CARD. As it can be seen, our +CARD can segment the uncommon objects and +complex scenes very well. +Image +GT +SegNeXt-L +CARD +(ConvNeXt-L) +Fig. +9: Eamples of the results obtained on +the +COCOStuff-164K +dataset +with +our +pro- +posed CARD (ConvNeXt-L) in comparison with +SegNeXt-L [49] and the ground truth. +4.5 Experiments on Cityscapes +4.5.1 CARD Compared to the +State-of-the-art +Cityscapes [53]8 contains 2975/500/1525 images +for training/validation/test. We adopt AdamW, +batch size = 8, 80K training iterations in total and +8https://www.cityscapes-dataset.com/ +Table 10: Comparisons to state-of-the-art meth- +ods on Cityscapes validation set of CARD. Note +that methods marked with ‘∗’ report mIOU from +their papers while the others are obtained with +our implementation. SS means single scale per- +formance w/o flipping. MF means multi-scale +performance w/ flipping. +Methods +Ref +mIOU(%) +SS +MF +Axial-DeepLab-L* [52] +ECCV-2020 +- +81.5 +SETR (ViT-L)* [51] +CVPR-2021 +- +82.2 +Segmenter (ViT-L)* [48] +ICCV-2021 +- +81.3 +HRFormer-B * [49] +NIPS-2021 +- +82.6 +CARD (ResNet-50) +Ours +79.8 +81.6 +CARD (ConvNeXt-L) +Ours +82.8 +83.6 +1000 steps linear warmup when training CARD +with ConvNeXt-Large. +4.5.2 Visualization of CARD +Cityscapes +results +in +Fig. +10 +compare +Segmenter(ViT-L) [48], and our CARD. As it can +be seen, our CARD can segment the hard class +(e.g, rider vs person) very well, which is very +useful for the autopilot. +5 Conclusion +In this paper, we have aimed to make a bet- +ter use of class level context information. We +first proposed a universal class-aware regular- +izations (CAR) approach, which minimizes the +intra-class feature variance and maximize the +inter-class separation simultaneously, to regularize +the training process and boost the differentiabil- +ity of the learned pixel representations without +extra computation during inference. Then we pro- +posed a class-aware regularized decoder (CARD), +which is designed for better effectiveness and effi- +ciency tailored for the proposed CAR. Extensive +experiments conducted on various benchmarks +and thorough ablation studies have validated the +effectiveness of the proposed CAR, which has +boosted the existing models’ performance by up +to 2.18% mIOU on Pascal Context and 2.23% on +COCOStuff-10k with no extra inference overhead. +And the proposed CARD achieved state-of-the-art +performance on multiple benchmarks while using +much less computation. + +q62b6cial +beuobmoo.lismp@b6121ss.21obno20 +Article Title +Image +GT +Segmenter +(ViT-L) +CARD (ConvNeXt-L) +terrain +void +road +sky +sidewalk +building +wall +fance +pole +traffic light +traffic sign +vegetation +person +rider +car +truck +bus +train +motorcycle +bicycle +84.2% +69.5% +75.5% +67.9% +77.7% +65.4% +92.2% +82.4% +86.4% +75.4% +Fig. 10: Some visual examples on the Cityscapes. We compare our proposed CARD with Segmenter +(ViT-L) [48]. We show mIOU for each predicted mask on the bottom left corner. +Acknowledgement This research depends on the NVIDIA determinism framework. We appreciate the +support from @duncanriach and @reedwm at NVIDIA and TensorFlow team. +We also thank OpenI (https://openi.org.cn) for providing GPUs to conduct experiments. + +Article Title +21 +References +[1] Long, J., Shelhamer, E., Darrell, T.: Fully +convolutional networks for semantic segmen- +tation. In: CVPR (2015) +[2] Zhao, H., Shi, J., Qi, X., Wang, X., Jia, J.: +Pyramid scene parsing network. In: CVPR +(2017) +[3] Chen, L.-C., Papandreou, G., Kokkinos, I., +Murphy, K., Yuille, A.L.: Deeplab: Semantic +image segmentation with deep convolutional +nets, atrous convolution, and fully connected +crfs. IEEE TPAMI (2017) +[4] Wang, X., Girshick, R., Gupta, A., He, K.: +Non-local neural networks. In: CVPR (2018) +[5] Fu, J., Liu, J., Tian, H., Li, Y., Bao, Y., Fang, +Z., Lu, H.: Dual attention network for scene +segmentation. In: CVPR (2019) +[6] Yuan, Y., Huang, L., Guo, J., Zhang, C., +Chen, X., Wang, J.: Ocnet: Object context +network for scene parsing. IJCV (2021) +[7] Zhang, H., Zhan, H., Wang, C., Xie, J.: +Semantic correlation promoted shape-variant +context for segmentation. In: CVPR (2019) +[8] Li, X., Zhong, Z., Wu, J., Yang, Y., Lin, +Z., Liu, H.: Expectation-maximization atten- +tion networks for semantic segmentation. In: +ICCV (2019) +[9] Zhu, Z., Xu, M., Bai, S., Huang, T., Bai, +X.: Asymmetric non-local neural networks for +semantic segmentation. In: ICCV (2019) +[10] Dosovitskiy, A., Beyer, L., Kolesnikov, A., +Weissenborn, D., Zhai, X., Unterthiner, T., +Dehghani, M., Minderer, M., Heigold, G., +Gelly, S., Uszkoreit, J., Houlsby, N.: An image +is worth 16x16 words: Transformers for image +recognition at scale. In: ICLR (2021) +[11] Ranftl, R., Bochkovskiy, A., Koltun, V.: +Vision transformers for dense prediction. In: +ICCV (2021) +[12] Sixiao, Z., Jiachen, L., Hengshuang, Z., Xia- +tian, Z., Zekun, L., Yabiao, W., Yanwei, +F., Jianfeng, F., Tao, X., H.S., T.P., Li, +Z.: Rethinking semantic segmentation from a +sequence-to-sequence perspective with trans- +formers. In: CVPR (2021) +[13] Yuan, Y., Chen, X., Wang, J.: Object- +contextual representations for semantic seg- +mentation. In: ECCV (2020) +[14] Zhang, F., Chen, Y., Li, Z., Hong, Z., Liu, J., +Ma, F., Han, J., Ding, E.: Acfnet: Attentional +class feature network for semantic segmenta- +tion. In: ICCV (2019) +[15] Yu, C., Wang, J., Gao, C., Yu, G., Shen, C., +Sang, N.: Context prior for scene segmenta- +tion. In: CVPR (2020) +[16] Huang, Y., Kang, D., Chen, L., Zhe, X., Jia, +W., Bao, L., He, X.: Car: Class-aware regular- +izations for semantic segmentation. In: ECCV +(2022) +[17] Huang, Z., Wang, X., Wei, Y., Huang, L., Shi, +H., Liu, W., Huang, T.S.: Ccnet: Criss-cross +attention for semantic segmentation. IEEE +TPAMI (2020) +[18] Choi, S., Kim, J.T., Choo, J.: Cars can’t fly +up in the sky: Improving urban-scene segmen- +tation via height-driven attention networks. +In: CVPR (2020) +[19] Liu, M., Schonfeld, D., Tang, W.: Exploit +visual dependency relations for semantic seg- +mentation. In: CVPR (2021) +[20] Yu, W., Luo, M., Zhou, P., Si, C., Zhou, Y., +Wang, X., Feng, J., Yan, S.: Metaformer is +actually what you need for vision. In: CVPR +(2022) +[21] Vaswani, A., Shazeer, N., Parmar, N., Uszko- +reit, J., Jones, L., Gomez, A.N., �Lukasz +Kaiser, Polosukhin, I.: Attention is all you +need. In: NeurIPS (2017) +[22] Ho, J., Kalchbrenner, N., Weissenborn, D., +Salimans, T.: Axial Attention in Multidimen- +sional Transformers (2019) + +22 +Article Title +[23] Huang, Y., Kang, D., Jia, W., He, X., liu, +L.: Channelized axial attention - considering +channel relation within spatial attention for +semantic segmentation. In: Proceedings of the +AAAI Conference on Artificial Intelligence +(2022) +[24] Simonyan, K., Zisserman, A.: Very deep +convolutional networks for large-scale image +recognition. In: ICLR (2015) +[25] He, K., Zhang, X., Ren, S., Sun, J.: Deep +residual learning for image recognition. In: +CVPR (2016) +[26] Chollet, F.: Xception: Deep learning with +depthwise separable convolutions. In: CVPR +(2017) +[27] Zhang, H., Wu, C., Zhang, Z., Zhu, Y., +Zhang, +Z., +Lin, +H., +Sun, +Y., +He, +T., +Muller, J., Manmatha, R., Li, M., Smola, +A.: Resnest: Split-attention networks. arXiv +preprint arXiv:2004.08955 (2020) +[28] Mingxing, +T., +Quoc, +L.: +Efficientnet: +Rethinking model scaling for convolutional +neural networks. In: International Conference +on Machine Learning (2019) +[29] Liu, Z., Mao, H., Wu, C.-Y., Feichtenhofer, +C., Darrell, T., Xie, S.: A convnet for the +2020s. In: CVPR (2022) +[30] Liu, Z., Lin, Y., Cao, Y., Hu, H., Wei, Y., +Zhang, Z., Lin, S., Guo, B.: Swin transformer: +Hierarchical vision transformer using shifted +windows. In: ICCV (2021) +[31] Ronneberger, O., Fischer, P., Brox, T.: U-net: +Convolutional networks for biomedical image +segmentation. MICCAI (2015) +[32] Lin, T.-Y., Doll´a, P., Girshick, R., He, K., +Hariharan, B., Belongie, S.: Feature pyra- +mid networks for object detection. In: CVPR +(2017) +[33] Kirillov, A., Girshick, R., He, K., Doll´ar, +P.: Panoptic feature pyramid networks. In: +CVPR (2019) +[34] Chen, +L.-C., +Zhu, +Y., +Papandreou, +G., +Schroff, F., Adam, H.: Encoder-decoder with +atrous separable convolution for semantic +image segmentation. In: ECCV (2018) +[35] Cheng, B., Misra, I., Schwing, A.G., Kir- +illov, A., Girdhar, R.: Masked-attention mask +transformer for universal image segmenta- +tion. In: CVPR (2022) +[36] Wu, H., Zhang, J., Huang, K., Liang, K., +Yizhou, Y.: FastFCN: Rethinking Dilated +Convolution in the Backbone for Semantic +Segmentation (2019) +[37] Wen1, Y., Zhang, K., Li, Z., Qiao, Y.: Dis- +criminative feature learning approach for +deep face recognition. In: ECCV (2016) +[38] Wang, +W., +Zhou, +T., +Yu, +F., +Dai, +J., +Konukoglu, E., Van Gool, L.: Exploring cross- +image pixel contrast for semantic segmenta- +tion. In: ICCV, pp. 7303–7313 (2021) +[39] Xiao, T., Liu, Y., Zhou, B., Jiang, Y., Sun, J.: +Unified perceptual parsing for scene under- +standing. In: ECCV (2018) +[40] Chu, X., Tian, Z., Zhang, B., Wang, X., +Wei, X., Xia, H., Shen, C.: Conditional +positional encodings for vision transformers. +arXiv preprint arXiv:2102.10882 (2021) +[41] Tu, Z., Talebi, H., Zhang, H., Yang, F., Pey- +man Milanfar, A.B., Li, Y.: Maxvit: Multi- +axis vision transformer. In: ECCV (2022) +[42] Chen, L.-C., Papandreou, G., Schroff, F., +Adam, H.: Rethinking Atrous Convolution +for Semantic Image Segmentation (2017) +[43] Zhang, H., Dana, K., Shi, J., Zhang, Z., +Wang, X., Tyagi, A., Agrawal, A.: Con- +text encoding for semantic segmentation. In: +CVPR (2018) +[44] Mottaghi, R., Chen, X., Liu, X., Cho, N.-G., +Lee, S.-W., Fidler, S., Urtasun, R., Yuille, A.: +The role of context for object detection and +semantic segmentation in the wild. In: CVPR +(2014) + +Article Title +23 +[45] Caesar, H., Uijlings, J., Ferrari, V.: COCO- +Stuff: Thing and Stuff Classes in Context. In: +CVPR (2018) +[46] Wang, J., Sun, K., Cheng, T., Jiang, B., +Deng, C., Zhao, Y., Liu, D., Mu, Y., Tan, +M., Wang, X., Liu, W., Xiao, B.: Deep high- +resolution representation learning for visual +recognition. IEEE TPAMI (2020) +[47] Woo, +S., +Debnath, +S., +Hu, +R., +Chen, +X., Liu, Z., Kweon, I.S., Xie, S.: Con- +vnext v2: Co-designing and scaling convnets +with masked autoencoders. arXiv preprint +arXiv:2301.00808 (2023) +[48] Strudel, R., Garcia, R., Laptev, I., Schmid, +C.: Segmenter: Transformer for semantic seg- +mentation. In: ICCV (2021) +[49] Guo, M.-H., Lu, C.-Z., Hou, Q., Liu, Z.-N., +Cheng, M.-M., Hu, S.-M.: Segnext: Rethink- +ing convolutional attention design for seman- +tic segmentation. In: NeurIPS (2022) +[50] He, H., Yuan, Y., Yue, X., Hu, H.: Rankseg: +Adaptive pixel classification with image cat- +egory ranking for segmentation. In: ECCV +(2022) +[51] Xie, E., Wang, W., Yu, Z., Anandkumar, A., +Alvarez, J.M., Luo, P.: Segformer: Simple and +efficient design for semantic segmentation +with transformers. In: NeurIPS (2021) +[52] Wang, H., Zhu, Y., Green, B., Adam, H., +Yuille, A., Chen, L.-C.: Axial-deeplab: Stand- +alone axial-attention for panoptic segmenta- +tion. In: ECCV (2020) +[53] Marius, C., Mohamed, O., Sebastian, R., +Timo, R., Markus, E., Rodrigo, B., Uwe, F., +Roth, S., Bernt, S.: The cityscapes dataset +for semantic urban scene understanding. In: +CVPR (2016) + diff --git a/K9E3T4oBgHgl3EQfAglp/content/tmp_files/load_file.txt b/K9E3T4oBgHgl3EQfAglp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7e66a2938425baad4044a4f007f4aff1dee7d60e --- /dev/null +++ b/K9E3T4oBgHgl3EQfAglp/content/tmp_files/load_file.txt @@ -0,0 +1,1509 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf,len=1508 +page_content='CARD: Semantic Segmentation with Efficient Class-Aware Regularized Decoder (Tech report) CARD: Semantic Segmentation with Efficient Class-Aware Regularized Decoder Ye Huang1, Di Kang2, Liang Chen3, Wenjing Jia4, Xiangjian He5, Lixin Duan1, Xuefei Zhe2 and Linchao Bao2* 1Shenzhen Institute for Advanced Study, UESTC, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 2Tencent AI Lab, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3Fujian Normal University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4University of Technology Sydney, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 5University of Nottingham Ningbo China, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' E-mail(s): linchaobao@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Contributing authors: edward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='ye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='huang@qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='kang@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='chen@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Wenjing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Jia@uts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='au;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' xiangjian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='he@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' lxduan@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' zhexuefei@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Abstract Semantic segmentation has recently achieved notable advances by exploiting “class-level” contex- tual information during learning, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', the Object Contextual Representation (OCR) and Context Prior (CPNet) approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' However, these approaches simply concatenate class-level information to pixel features to boost the pixel representation learning, which cannot fully utilize intra-class and inter-class contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Moreover, these approaches learn soft class centers based on coarse mask prediction, which is prone to error accumulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' To better exploit class level information, we propose a universal Class-Aware Regularization (CAR) approach to optimize the intra-class variance and inter-class distance during feature learning, motivated by the fact that humans can recognize an object by itself no matter which other objects it appears with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' More- over, we design a dedicated decoder for CAR (named CARD), which consists of a novel spatial token mixer and an upsampling module, to maximize its gain for existing baselines while being highly efficient in terms of computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Specifically, CAR consists of three novel loss func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' The first loss function encourages more compact class representations within each class, the second directly maximizes the distance between different class centers, and the third further pushes the distance between inter-class centers and pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Furthermore, the class center in our approach is directly generated from ground truth instead of from the error-prone coarse prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' CAR can be directly applied to most existing segmentation models during training, including OCR and CPNet, and can largely improve their accuracy at no additional inference overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Extensive experi- ments and ablation studies conducted on multiple benchmark datasets demonstrate that the proposed CAR can boost the accuracy of all baseline models by up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='23% mIOU with superior gen- eralization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' CARD outperforms state-of-the-art approaches on multiple benchmarks with a highly efficient architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' The code will be available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='com/edwardyehuang/CAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Keywords: Class-aware regularizations, semantic segmentation 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='04258v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='CV] 11 Jan 2023 2 Article Title 1 Introduction The task of semantic segmentation is to predict a class label for each pixel in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' It is a fundamental computer vision task that serves as a critical building block for various downstream tasks, such as scene understanding, image editing, self-driving vehicles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' After the seminal work FCN [1] that used fully convolutional networks to make the dense per-pixel segmentation task more efficient, many FCN-based approaches [2, 3] have been proposed and greatly advanced the segmen- tation accuracy on various benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Among these methods, many of them have focused on better fusing spatial domain context information to obtain more powerful feature representations (termed pixel features in this work) for the final per-pixel classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' For example, DeepLab [3] and PSPNet [2] utilized multi-scale features via constructing feature pyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Recently, methods based on dot-product self- attention (SA) have become very popular since they can easily capture the long-range relationship between pixels [4–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' SA aggregates informa- tion dynamically (by different attention maps for different inputs) and selectively (using weighted averaging spatial features according to their sim- ilarity scores).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Significant progresses have been made by using multi-scale and self-attention tech- niques during spatial information aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' As complements to the above methods, many recent works have proposed various modules to utilize class-level contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' The class-level information is often represented by the class center/context prior which are the mean fea- tures of each class in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' OCR [13] and ACFNet [14] extract “soft” class centers accord- ing to the predicted coarse segmentation mask by using the weighted sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' CPNet [15] proposed a context prior map/affinity map, which indicates if two spatial locations belong to the same class, and used this predicted context prior map for fea- ture aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' However, they [13–15] simply concatenated these class-level features with the original pixel features for the final classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In this paper, we also focus on utilizing class level information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Instead of focusing on how to better extract class-level features like the existing methods [13–15], we use the simple, but accurate, average feature according to the GT mask, and focus on maximizing the inter-class distance dur- ing feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' This is because it mirrors how humans can robustly recognize an object by itself no matter what other objects it appears with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Learning more separable features makes the features of a class less dependent upon other classes, resulting in improved generalization abil- ity, especially when the training set contains only limited and biased class combinations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', cows and grass, boats and beach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 1 illustrates an example of such a problem, where the classifica- tion of dog and sheep depends on the classification of grass class, and has been mis-classified as cow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In comparison, networks trained with our pro- posed CAR successfully generalize to these unseen class combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' To better achieve this goal, we propose CAR, a class-aware regularizations module, that opti- mizes the class center (intra-class) and inter-class dependencies during feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Three loss functions are devised: the first encourages more compact class representations within each class, and the other two directly maximize the distance between different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Specifically, an intra- class center-to-pixel loss (termed as “intra-c2p”, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' (3)) is first devised to produce more compact representation within a class by minimizing the distance between all pixels and their class cen- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In our work, a class center is calculated as the averaged feature of all pixels belonging to the same class according to the GT mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' More com- pact intra-class representations leave a relatively large margin between classes, thus contributing to more separable representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Then, an inter- class center-to-center loss (“inter-c2c”, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' (6)) is devised to maximize the distance between any two different class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' This inter-class center- to-center loss alone does not necessarily produce separable representations for every individual pix- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Therefore, a third inter-class center-to-pixel loss (“inter-c2p”, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' (13)) is proposed to enlarge the distance between every class center and all pixels that do not belong to the class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' A preliminary version of this work was pre- sented in [16], which proposed three class-aware regularization (CAR) terms and evaluated their effectiveness and universality by using them as a direct addon to various state-of-the-art meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Although effective, we notice two issues when using CAR as an addon for some baselines – Article Title 3 Existing Models CAR w/o CAR, easily affected by other classes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='.g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' grass) Reduce Intra-class center- to-pixel distance Reduce Inter-class center- to-center dependency Cow Horse Grass Dog Sheep Reduce inter-class center- to-pixel dependency Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 1: The concept of the proposed CAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Our CAR optimizes existing models with three regularization targets: 1) reducing pixels’ intra-class distance, 2) reducing inter-class center-to-center dependency, and 3) reducing pixels’ inter-class dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' As highlighted in this example (indicated with a red dot in the image), with our CAR, the grass class does not affect the classification of dog/sheep as much as before, and hence successfully avoids previous (w/o CAR) mis-classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' inefficiency brought by the baselines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' dila- tion and self-attention [4]) and decreased gain due to limited compatibility with the baselines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' CCNet [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In this extension, we design a dedi- cated class-aware regularized decoder (CARD) to overcome the aforementioned two issues, resulting in greatly improved computational cost and better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Specifically, a leading synced axial attention (SAA) is proposed right before CAR to make sparse self-attention gain as much accuracy gain as self-attention, and a lightweight pyra- mid upsampling module is proposed to replace the computation-heavy dilated convolution with minimal accuracy loss (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In summary, the contributions of this work are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We propose a universal class-aware regu- larization module that can be integrated into various segmentation models to largely improve the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We devise three novel regularization terms to achieve more separable and less class- dependent feature representations by mini- mizing the intra-class variance and maximiz- ing the inter-class distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We calculate the class centers directly from ground truth during training, thus avoiding the error accumulation issue of the existing methods and introducing no computational overhead during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We visualize pixel-level feature-similarity heatmaps for the inter-class features learned with and without our CAR to demonstrate they are indeed less related to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We propose a class-aware regularized decoder aiming for better efficiency and effectiveness for various backbones, achieving new state- of-the-art accuracies on multiple benchmarks while being highly efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 2 Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='1 Class Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In 2019 [13, 14], the concept of class center was introduced to describe the overall representation of each class from the categorical context perspec- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In these approaches, the center representa- tion of each class was determined by calculating the dot product of the feature map and the coarse prediction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', weighted average) from an auxiliary task branch, supervised by the ground truth [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' After that, those intra-class centers are assigned to the corresponding pixels on feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Furthermore, in 2020 [15], a learnable kernel and one-hot ground truth were used to sepa- rate the intra-class center from inter-class center, and then concatenated with the original feature representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4 Article Title All of these works [13–15] have focused on extracting the intra (inter) class centers, but they then simply concatenated the resultant class cen- ters with the original pixel representations to perform the final logits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We argue that the cat- egorical context information can be utilized in a more effective way so as to reduce the inter-class dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' To this end, we propose a CAR approach, where the extracted class center is used to directly regularize the feature extraction process so as to boost the differentiability of the learned fea- ture representations (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 1) and reduce their dependency on other classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 2 contrasts the two different designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' More details of the proposed CAR are provided in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2 Inter-Class Reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Recently, [18, 19] studied the class dependency as a dataset prior and demonstrated that inter-class reasoning could improve the classification perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' For example, a car usually does not appear in the sky, and therefore the classification of sky can help reduce the chance of mis-classifying an object in the sky as a car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' However, due to the limited training data, such class-dependency prior may also contain bias, especially when the desired class relation rarely appears in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 1 shows such an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In the training set, cow and grass are dependent on each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' However, as shown in this example, when there is a dog or sheep standing on the grass, the class dependency learned from the limited training data may result in errors and predict the target into a class that appears more often in the training data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', cow in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In our CAR, we design inter- class and intra-class loss functions to reduce such inter-class dependency and achieve more robust segmentation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='3 Spatial Context Aggregation The spatial token mixer [20] provides the context aggregation between each pixel’s encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' One of the well-used token mixers is Self-Attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Self-attention proposed in [4, 21] has been widely used in semantic segmentation [5– 7, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Specifically, self-attention determines the similarity between a pixel with every other pixel in the feature map by calculating their dot prod- uct, followed by softmax normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' With this attention map, the feature representation of a given pixel is enhanced by aggregating features from the whole feature map weighted by the aforementioned attention values, thus easily tak- ing long-range relationship into consideration and yielding boosted performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In self-attention, in order to achieve correct pixel classification, the representation of pixels belonging to the same class should be similar to gain greater weights in the final representation augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Sparse self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Although regular self- attention [4] performs very well for semantic seg- mentation, its computational cost is too high (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' O(H2W 2), especially for high-resolution input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Thus, many sparse alternatives of the full self- attention have been proposed, including axial attention [22], CCNet [17], and CAA [23], achiev- ing similar accuracy as self-attention but with greatly reduced computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='4 Maintain the feature map resolution In semantic segmentation, most backbones including CNN-based [24–29] and Transformer based [10, 30], are initially designed for image- level classification, where the resolution of the intermediate feature maps does not matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' So, they usually progressively downsample the feature map to a resolution of 1/32 of the original size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' output stride = 32), resulting in large enough receptive field size and greatly saved computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Unlike image classification, semantic segmen- tation is essentially a per-pixel classification task, where the final output size is identical to the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Thus, upsampling is required at the final stage if the resolution of the intermediate results is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' However, output stride = 32 feature map usually miss necessary segmentation details (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' boundaries, thin objects, etc) that cannot be recovered via bilinear upsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Thus, main- taining higher-resolution feature maps is crucial, among which dilation convolution that does not reduce the feature map’s resolution too much or multi-scale pyramid style feature upsampling (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' UNet/FPN) are wildly adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Dilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Early approaches apply dilation (instead of stride) on the later stages of a CNN to stop further downsampling of the feature maps, resulting in output stirde = 8 feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' How- ever, the dilation modification introduces too Article Title 5 Class Center Pixel Features 1-Hot Coarse GT Class Center Map Concat Features Logits (a) Design of OCR, ACFNet and CPNet Pixel Features 1-Hot Real GT Class Center Map Logits Class Center Supervision C2C inter loss C2P intra loss C2P inter loss (b) Our CAR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 2: The difference between the proposed CAR and previous methods that use class-level information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Previous models focus on extracting class center while using simple concatenation of the original pixel feature and the class/context feature for later classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In contrast, our CAR uses direct supervision related to class center as regularization during training, resulting in small intra-class variance and low inter-class dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 1 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' much computation and it is not applicable to Transformer-based backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Pyramid-based upsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Many other approaches [31–35]utilize pyramid-based feature upsampling by fusing multi-scale features from dif- ferent levels, achieving similar accuracy to dilation methods but with much less computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Repre- sentative methods including UNet [31], FPN [32], and JPU [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' UNet and FPN based methods usually add low-level fine-grained feature maps (with optional convolution layers) and high-level coarse feature maps together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' This direct addi- tion of low-/high-level features sometimes makes training harder [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Instead, JPU concatenates low-/high-level feature maps that is followed by multiple parallel dilated convolutions, achieving better accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We also use JPU like pyramid upsampling for efficiency but with some modi- fications to improve convergence and make this upsampling module compatible with backbones producing various number of feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3 Methodology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='1 Extracting Class Centers from Ground Truth Denote a feature map and its corresponding resized one-hot encoded ground-truth mask as X ∈ RH×W ×C1 and Y ∈ RH×W ×Nclass, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We first get the spatially flattened class mask Yflat ∈ RHW ×Nclass and flattened feature 1H, W and C denote images’ height and width, and number of channels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' map Xflat ∈ RHW ×C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Then, the class center2, which is the average features of all pixel features of a class, can be calculated by: µimage = YT flat · Xflat Nnon-zero ∈ RNclass×C, (1) where Nnon-zero denotes the number of non-zero values in the corresponding map of the ground- truth mask Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In our experiments, to alleviate the negative impact of noisy images, we calcu- late the class centers using all the training images in a batch, and denote them as µbatch3 (see the appendix for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2 Reducing Intra-class Feature Variance 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='1 Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' More compact intra-class representation can lead to a relatively larger margin between classes, and therefore result in more separable features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In order to reduce the intra-class feature variance, existing works [4–6, 8, 9, 15] usually use self- attention to calculate the dot-product similarity in spatial space to encourage similar pixels to have a compact distance implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' For example, the self-attention in [4] implicitly pushed the fea- ture representation of pixels belonging to the same class to be more similar to each other than those of pixels belonging to other classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In our work, 2It is termed as class center in [14] and object region representations in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3We use µ and omit the subscript batch for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Article Title ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Self- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Self- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='CAR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='3×3×512 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='BN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Logits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Logits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='A1) ResNet-50 + Self-Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='A2) ResNet-50 + Self-Attention + CAR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Uper ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Uper ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='CAR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='3×3×512 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='BN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Logits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Logits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='A3) Swin-Tiny + Uper ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='A4) Swin-Tiny + Uper + CAR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='B) CAR Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='1×1×512 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Bn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Inter-class ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='c2c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Intra-class ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='c2p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='One-hot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='GT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Inter-class ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='c2p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Intra-class c2p loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='CAR (Training Only) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Map ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='One-hot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='GT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Center ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Map ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='MSE with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='ignored ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='mask ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='D) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Inter-class c2p loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Non-diag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Mask ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='MSE with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='threshold ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='One-hot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='GT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Feature ' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='threshold ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Dot Product ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Element-Wise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Multiplication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Element-Wise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='subtraction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Stop gradients ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Transposed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Class ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='center ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='ResNet- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='ResNet- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Swin-Tiny ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Swin-Tiny ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Class ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='center ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Class ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='center ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Class ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='center ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Operations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='One-hot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='GT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Inter-class c2c loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='C) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3: The proposed CAR approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' CAR can be inserted into various segmentation models, right before the logit prediction module (A1-A4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' CAR contains three regularization terms, including (C) intra- class center-to-center loss Lintra-c2p (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2), (D) inter-class center-to-center loss Linter-c2c (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2), and (E) inter-class center-to-pixel loss Linter-c2p (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Article Title 7 we devise a simple intra-class center-to-pixel loss to guide the training, which can achieve this goal very effectively and produce improved accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2 Intra-class Center-to-pixel Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We define a simple but effective intra-class center- to-pixel loss to suppress the intra-class feature variance by penalizing large distance between a pixel feature and its class center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' The Intra-class Center-to-pixel Loss Lintra-c2p is defined by: Lintra-c2p = fmse(Dintra-c2p), (2) where Dintra-c2p = (1 − σ)|Yflat · µ − Xflat|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' (3) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' (3), σ is a spatial mask indicating pix- els being ignored (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', ignore label), Yflat · µ distributes the class centers µ to the correspond- ing positions in each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Thus, our intra-class loss Lintra-c2p will push the pixel representations to their corresponding class center, using mean squared error (MSE) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='3 Maximizing Inter-class Separation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='1 Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Humans can robustly recognize an object by itself regardless which other objects it appears with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Conversely, if a classifier heavily relies on the information from other classes to determine the classification result, it will easily produce wrong classification results when a rather rare class com- bination appears during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Maximizing inter-class separation, or in another words, reduc- ing the inter-class dependency, can therefore help the network generalize better, especially when the training set is small or is biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 1, the dog and sheep are mis-classified as the cow because cow and grass appear together more often in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' To improve the robustness of the model, we propose to reduce this inter-class dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' To this end, the following two loss functions are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2 Inter-class Center-to-center Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' The first loss function is to maximize the distance between any two different class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Inspired by the center loss used in face recognition [37], we propose to reduce the similarity between class centers µ, which are the averaged features of each class calculated according to the GT mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' The inter-class relation is defined by the dot-product similarity [21] between any two classes as: Ac2c = softmax( µT · µ √ C ), Ac2c ∈ RNclass×Nclass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' (4) Moreover, since we only need to constrain the inter-class distance, only the non-diagonal ele- ments are retained for the later loss calculation as: Dinter-c2c = � 1 − eye(Nclass) � Ac2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' (5) We only penalize larger similarity values between any two different classes than a pre- defined threshold ϵ0 Nclass−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Dinter-c2c = fsum � max(Dinter-c2c− ϵ0 Nclass − 1, 0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' (6) Thus, the inter-class center-to-center loss Linter-c2c is defined by: Linter-c2c = fmse(Dinter-c2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' (7) Here, a small margin is used in consideration of the feature space size and the mislabeled ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='3 Inter-class Center-to-pixel Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Maximizing only the distances between class cen- ters does not necessarily result in separable rep- resentation for every individual pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We further maximize the distance between a class center and any pixel that does not belong to this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' More concretely, we first compute the center-to-pixel dot product as: Λc2p = µT · Xflat, Λc2p ∈ RHW ×Nclass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' (8) Ideally, with the previous loss Linter-c2c, the features of all pixels belonging to the same class should be equal to that of the class center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' There- fore, we replace the intra-class dot product with its ideal value, namely using the class center µ for calculating the intra-class dot product as: Λc = diag(µT · µ), (9) 8 Article Title and the replacement effect is achieved by using masks as: Λ′ = Λc2p(1 − Yflat) + Λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' (10) This updated dot product Λ′ is then used to calculate similarity across class axis with a softmax as: Ac2p = softmax(Λ′), Ac2p ∈ RHW ×Nclass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' (11) Similar to the calculation of Linter-c2c in the previous subsection, we have Dinter-c2p = (1 − Yflat)Ac2p, (12) Dinter-c2p = fsum � max(Dinter-c2p− ϵ1 Nclass − 1, 0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' (13) Thus, the Inter-class Center-to-pixel Loss Linter-c2p is defined by: Linter-c2p = fmse(Dinter-c2p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' (14) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='4 Differences with OCR, ACFNet, CPNet and CIPC Methods that are closely related to ours are OCR [13], ACFNet [14] and CPNet [15], which all focus on better utilizing class-level features and differ on how to extract the class centers and context features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' However, they all use a simple concatenation to fuse the original pixel fea- ture and the complementary context feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' For example, OCR and ACFNet first produce a coarse segmentation, which is supervised by the GT mask with a categorical cross-entropy loss, and then use this predicted coarse mask to generate the (soft) class centers by weighted summing all the pixel features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' OCR then aggregates these class centers according to their similarity to the original pixel feature termed as “pixel-region relation”, resulting in a “contextual feature”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Slightly differently from OCR, ACFNet directly uses the probability (from the predicted coarse mask) to aggregate class cen- ter, obtaining a similar context feature termed as “attentional class feature”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' CPNet defines an affinity map, which is a binary map indicating if two spatial locations belong to the same class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Then, they use a sub-network to predict their ideal affinity map and use the soft version affinity map termed as “Context Prior Map” for feature aggre- gation, obtaining a class feature (center) and a context feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Note that CPNet concatenates class feature, which is the updated pixel feature, and the context feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We also propose to utilize class-level contex- tual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Instead of extracting and fusing pixel features with sub-networks, we propose three loss functions to directly regularize training and encourage the learned features to maintain cer- tain desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' The approach is simple but more effective thanks to the direct supervision (validated in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Moreover, our class center estimate is more accurate because we use the GT mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' This strategy largely reduces the complex- ity of the network and introduces no computa- tional overhead during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Furthermore, it is compatible with all existing methods, including OCR, ACFNet and CPNet, demonstrating great generalization capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We also notice that Cross-Image Pixel Con- trast (CIPC) [38] shares a similar high-level goal as our CAR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', learning more similar repre- sentations for pixels belonging to the same class than to a different class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' However, the approaches of achieving this goal are very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' CIPC is motivated by contrastive learning while our CAR is motivated by the compositionality of the scene, for better generalization in the cases of rare class combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Therefore, CIPC adopts the one-vs-rest style InfoNCE loss, including the typical pixel-to-pixel loss and a special pixel- to-center loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In contrast, (1) we propose an additional center-to-center loss to regularize the inter-class dependency explicitly and effectively (see Table 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' (2) we use one-vs-one style inter- class losses while CIPC uses one-vs-rest style NCE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Compared to our one-vs-one loss, using one- vs-rest loss for training does not necessarily result in small inter-class similarity between the cur- rent class and every individual “other” classes and may increase the inter-class similarity among those “other” classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' (3) we leave margins to prevent CAR regularizations, which is not the pri- mary task of pixel classification, from dominating the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Article Title 9 Class-Aware Regularized Decoder EJPU CAR Module Backbone SAA Logits OS=32 OS=8 Stop gradients Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4: Overview of the proposed CARD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Class-aware regularized decoder (CARD) is tailored for the proposed class-aware regularizations with greatly reduced computation cost and minor accuracy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' CARD contains an EJPU, which fuses features from different layers (at the same spatial location) to obtain high-resolution multi-scale and multi-level feature maps, an synced axial attention (SAA) token mixer, which fuses features from different locations as context aggregation, and CAR to produce less class-dependent and thus more generalizable pixel features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' The output stride (OS) = 8 logit maps are bilinearly upsampled to the original resolution to make final prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='5 The Devil Is in the Architecture’s Detail The proposed CAR is compatible with many mod- els as shown in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' However, some layers or operations in existing models may be harmful to the ability of CAR, where the last 3×3 conv is one commonly found case in many models [4, 17, 39] (see A1 and A3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' A possible reason is that the network is trained to maximize the sep- aration between different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' However, if the two pixels lie on different sides of the segmenta- tion boundary, a 3 × 3 conv will merge the pixel representations from different classes, making the proposed CAR harder to optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In this work, we provide a simple and optional general modifi- cation for those models to enhance CAR’s ability, where we use 1 × 1 conv to replace the original 3 × 3 conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Existing models like DeepLab [3] are not required to modify because they are using the 1 × 1 conv as the original settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Note that, this is only modification we made in some existing models, because it is simple and generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We also found some architecture-specified modifications, yet not generalized, can further largely improve the performance when employ- ing CAR on those existing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' For example, decreasing the filter number to 256 for the last conv layer of ResNet-50 + Self-Attention + CAR, or replacing the conv layer after PPM (inside Uper block, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3A3) from 3 × 3 to 1 × 1 in Swin-Tiny + UperNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We did not try to exhaustively search these variants since they did not generalize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='6 Class-aware Regularized Decoder 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='1 Motivation As mentioned in the Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='5, simply applying CAR to existing methods without architecture- specified modification may result in sub-optimal result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' To better utilize CAR for semantic segmen- tation, we design a novel decoder module tailored for CAR by taking efficiency and effectiveness into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Concretely, the decoder design focuses on three aspects: 1) compatibility with the proposed CAR, 2) efficient spatial context aggregation, and 3) less computational overhead (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' avoiding dilation convolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' The resultant class-aware regular- ized decoder (CARD) is a lightweight, simple yet effective decoder for semantic segmentation, achieving good performance via small compu- tational overhead and reasonable GPU memory usage together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2 Overview of CARD Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4 presents the overall architecture of the pro- posed CARD4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' CARD first uses our enhanced joint pyramid upsampling (EJPU) to obtain higher resolution multiscale feature maps with output stride (OS) = 8 (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Then it uses 4In this work, we refer a complete segmentation network as “model/method/baseline”, which usually consists of a “back- bone” feature extractor (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' ResNet-50, usually pretrained on a large-scale classification dataset) and a “decoder” that typi- cally increases the resolution of the feature maps (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' EJPU) and/or conducts multi-scale/global context aggregation as further enhancement (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' SAA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 10 Article Title Q K V Row Att.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Maps Q K V Column Att.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Maps Row Att.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Maps Q K V Column Att.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Maps a) Vanilla Axial Attention b) Synced Axial Attention Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 5: Comparison between vanilla axial atten- tion and our proposed synced axial attention (SAA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' The difference is highlighed in orange in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In SAA, both column and row attention maps are obtained from the same set of queries and keys The column and row attention of SAA shared the same query/key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' our proposed synced axial attention (SAA), which is lightweight and more compatible with the fol- lowing CAR regularizations, to perform global spatial context aggregation (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Finally, the output of the token mixer is optimized by our proposed CAR to obtain less class-dependent and more generalizable pixel representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' This novel design, which is optimized for effi- ciency and effectiveness, outperforms other state- of-the-art methods that use up to 3 times compu- tation of ours, striking to a good balance between accuracy and computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='3 Synced Axial Attention For efficiency and effectiveness, we design a new synced axial attention (SAA) for CAR since we notice existing sparse attention method obtains limited accuracy gain from CAR (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' CCNet [17], only +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='56 in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Although CAA + CAR achieves considerably big gain and the best results, we do not consider CAA for spatial context aggregation because it is an uncommon operation that has small FLOPs but has an actual slow speed in some hardware due to the lack of hardware and software (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='g, GPU driver/library) support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Token mixer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In CARD, we proposed an improved version of multi-head axial attention as the token mixer, named synced axial attention (SAA) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In vanilla axial attention, column attention (vertical) and row attention (vertical) are performed separately, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' using different input feature (X and Xcol) and different transforma- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Differently, SAA only computes the query Q, key K, and value V once, and uses the same set of query and key to generate both the col- umn attention map and row attention map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' After the column-wise context aggregation, the update feature is directly used for row-wise context aggre- gation according to the row attention maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Thus, SAA takes as input consistent feature space when computing the column and row attention maps, since they are generated by the same query and key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Empirically, we find this consistent/synced attention calculation not only reduces compu- tation but also improves the performance (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Possible reasons may be that using con- sistent input and shared transformation avoids potential error accumulation during the attention- based feature aggregation and directly conducts optimization in global context (rather than via two stages in AA or CCNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Positional encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In CARD, we apply con- ditional positional encoding (CPE) [40, 41], a resolutions insensitive positional encoding before the attention operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Note that we did not apply normalization in MaxViT [41] since we found it harmful to the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='4 Enhanced Joint Pyramid Upsampling (EJPU) We choose JPU since it integrates better with multi-scale/global context aggregation modules (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' ASPP [42], self-attention) than other UNet- like encoder-decoder or FPN [32] (more discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Based on JPU, we make some cru- cial modifications to improve its convergence and make it more compatible with the proposed CAR, resulting in largely improved accuracy (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='76 vs 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='76 in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4) Concretely, we notice the initial convergence speed on the test set (evaluated every 1k train- ing steps) is slower than the dilation model during our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' The possible reason is that JPU does not fully utilize the original backbone feature maps (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' highest abstraction level) since they are Article Title 11 Original JPU (Channel = J) Backbone OS = 8 Feature OS = 16 Feature OS = 32 Feature (C = K) OS = 8 (C = J) OS = 8 (C = J) 1D vector (C = K) MLP (LeakReLU) 1×1 Conv BN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' ReLU Global Average Pooling 1×1 Conv BN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' ReLU Operations Feature Map CPM (If required) Pad Map (C = J - K) 1D vector (C = J - K) To decoder OS = 32 (C = J) Element-Wise Addition OS = 8 (C = J) C Up Up Identity (K == J) Stop gradients C Concat Up Upsample Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 6: Detailed design of the proposed EJPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Similar to ResNet (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='4), we add upsam- pled original backbone features with “residual” information extracted from modified JPU mod- ule, which is then fed into the following decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We applied an optional CPM when the channel numbers differed between the original backbone and JPU features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Feature alignment is neces- sary because the channel of the original feature from the backbone is not grouped and arranged in multi-scale order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' treated equally with low-level feature maps from previous stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In contrast, FCN [1] initialized the weights to zero for the convolution follow- ing low-level features before adding them with the original backbone features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' The dilation model [2, 3] directly use the original backbone feature and the filter weights in essence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Both FCN and dilation models have faster convergence than JPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Motivated by the above observation, we equip JPU with a ResBlock-style residual branch that directly sends the original backbone feature (via minimal learned transformation if required) to the later network layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We detail the modifications as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Residual branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' To better utilize the well- trained original feature from the backbone, we include a residual branch to directly feed the bilin- early upsampled backbone feature maps to the fol- lowing network module (bottom in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 6) similar to FCN and ResBlock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' For some backbones, the output feature channel is not the same as JPU’s output (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 2048).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' So a channel padding mod- ule (CPM) is introduced with as less as possible learnable transformations only when necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Multi-scale multi-level feature branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We adopt JPU [36] style multi-scale multi-level fea- ture fusion for upsampling to provide complemen- tary information lost in the original backbone feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Specifically, feature maps extracted by JPU are processed by a 1 × 1 Conv (followed by BN and ReLU), and then added to the backbone feature maps element-wisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' This extra convolu- tion after JPU is introduced to calibrate the JPU features to the backbone features since JPU has reordered the channels and the meaning of the JPU and backbone features in the same dimen- sion/channel does not correspond any more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Note that we do not back-propagate gradient to the highest level backbone feature through JPU and only keep gradient from the residual branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Channel Padding Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We include an optional channel padding module (CPM) since different backbones output feature maps with different dimensions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' channel numbers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In order not to interfere the original feature maps too much, we use as less as possible learnable transformations to project the feature to required dimensions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 2048).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Specifically, the original backbone feature maps go through only a padding operation and a convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' The padded feature maps is generated with global average pooling, dimension projection and unpooling as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 6 bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 5, other simple and intuitive alter- natives have also been tested, including direct projection (optionally with BN + ReLU), chan- nel axis interpolation, or align JPU dimensions to backbone dimensions, but they are not as 12 Article Title effective as this configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' The possible rea- son is that redundant channel information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', direct projection/channel axis interpolation does not fully utilized the original well-trained features and insufficient channel information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', match dimensions) reduces the network capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4 Experiments In the following, we first disclosure the implemen- tation details and the detailed experiment settings in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Then we present various experimental results on Pascal Context (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2), COCOStuff- 10K (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='3), COCOStuff-164K (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='4), and Cityscapes (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' On Pascal Context (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2), we conduct thorough ablation studies (including the effectiveness of individual regular- ization terms inside CAR (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 1), the applica- bility of CAR for various baselines (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 2),, the effectiveness individual components inside CARD (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3-5), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=') and present various visualizations for in-depth analysis (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 7-8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='1 Implementation Details Training Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' For both baselines and CAR experiments, we apply the settings common to most works [7–9, 17, 43], including SyncBatch- Norm, batch size = 16, weight decay (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='001), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='01 initial LR, and poly learning decay with SGD dur- ing training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In addition, for the CNN backbones (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', ResNet), we set output stride = 8 (see [42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Training iteration is set to 30k iterations unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' For the thresholds in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 6 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 13, we set ϵ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='5 and ϵ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' CARD experiments use the same settings as “Baselines + CAR” unless stated other- wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' For example, CARD experiments compared with the state-of-the-art methods use AdamW (instead of SGD) for fair comparison since it is widely adopted by recent state-of-the-art meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Details are described in the corresponding subsections with the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Determinism & Reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Our imple- mentations are based on the latest NVIDIA deter- ministic framework (2022), which means exactly the same results can be always reproduced with the same hardware and same training settings (including random seed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' To fairly demonstrate the effectiveness of our CAR, we reimplement and reproduce all the baselines in our ablative experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2 Experiments on Pascal Context The Pascal Context [44] 5 dataset is split into 4,998/5,105 for training/test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We use its 59 semantic classes following the common practice [7, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Unless otherwise specified, all experiments are trained on the training set with 30k iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Ablation studies related to “baselines + CAR” are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='1, and ablation studies related to CARD are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='1 Ablation Studies of CAR In the following experiments, we first test the effectiveness of the individual regulariza- tion terms in CAR when plugged into different basic baselines, including the CNN-based and the Transformer-based baselines as representa- tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Then, we test the effectiveness of CAR as a whole on many other well-known baselines to demonstrate its universality CAR on “ResNet-50 + Self-Attention”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We firstly test the CAR with ResNet-50 + Self- Attention (w/o image-level block in [7]) to verify the effectiveness of the proposed loss functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Lintra-c2p, Linter-c2c, and Linter-c2p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 1, using Lintra-c2p directly improves 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='30 mIOU (48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='32 vs 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='62);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Introducing Linter-c2c and Linter-c2p further improves 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='38 mIOU and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='50 mIOU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Finally, with all three loss functions, the proposed CAR improves 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='18 mIOU from the reg- ular ResNet-50 + Self-attention (48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='32 vs 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' CAR on “Swin-Tiny + Uper”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' “Swin-Tiny + Uper” is a totally different archi- tecture from “ResNet-50 + Self-Attention [4]”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Swin [30] is a recent Transformer-based backbone network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Uper [39] is based on the pyramid pool- ing modules (PPM) [2] and FPN [32], focusing on extracting multi-scale context information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Simi- larly, as shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 1, after adding CAR, the performance of Swin-Tiny + Uper also increases by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='16, which shows our CAR can generalize to different architectures well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 5https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='edu/∼roozbeh/pascal-context/ Article Title 13 Table 1: Ablation studies of adding CAR to different methods on Pascal Context dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' All results are obtained with single scale test without flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' “A” means replacing the 3 × 3 conv with 1 × 1 conv (detailed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' CAR improves the performance of different types of backbones (CNN & Transformer) and head blocks (SA & Uper), showing the generalizability of the proposed CAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Methods Lintra-c2p Linter-c2c Linter-c2p A mIOU (%) R1 ResNet-50 + Self-Attention 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='32 R2 ✓ 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='56 R3 + CAR ✓ 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='17 R4 ✓ ✓ 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='79 R5 ✓ ✓ ✓ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='01 R6 ✓ ✓ 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='62 R7 ✓ ✓ ✓ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='00 R8 ✓ ✓ ✓ ✓ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='50 S1 Swin-Tiny + UperNet 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='62 S2 ✓ 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='82 S3 + CAR ✓ 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='01 S4 ✓ ✓ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='63 S5 ✓ ✓ ✓ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='26 S6 ✓ ✓ 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='62 S7 ✓ ✓ ✓ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='58 S8 ✓ ✓ ✓ ✓ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='78 The devil is in the architecture’s detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='5, we find it important to replace the leading 3 × 3 convolution (in the orig- inal baseline) with a 1 × 1 convolution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' For example, Linter-c2p did not improve the perfor- mance in Swin-Tiny + Uper (S5 vs S4 in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 1) until the last 3 × 3 convolution is replaced by a 1 × 1 (S8 vs S7 in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' To keep the simplicity and demonstrate its generalizability, we use the same network con- figurations for all the baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' No architecture-specific modification is made when conducting ablation studies on existing models for experiments in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 1- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' CAR on various baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' After we have verified the effectiveness of each part of the proposed CAR, we then tested CAR on multiple well-known baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' All of the baselines were reproduced under similar conditions (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Experimental results shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 2 demonstrate the generalizability of our CAR on different backbones and methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2 Ablation Studies of CARD In the following experiments, we test the effec- tiveness of the proposed CARD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Ablation studies include the effectiveness of individual components in CARD (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' the spatial token mixer in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3, EJPU in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4 & 5), and a computational cost analysis in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Effectiveness of the token mixer In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3, we conduct ablation studies of different token mixer designs in CARD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' They are evalu- ated using a Dilated ResNet-50 with output stride = 8 (“ResNet-50 (D8)”) on the Pascal Context dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' All settings are the same as the the ones tested in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 1 and Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We empirically find using head count = 4 for the multi-head axial attention achieves best accuracy (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='91% mIOU), which is slightly better than self-attention (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='50% mIOU) and cost much less computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' This computational cost partic- ularly matters for high-resolution inputs, which is evaluated in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Compared to another similar sparse attention based method CCNet, our SAA brings much more accuracy gain (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='45 vs +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='41), 14 Article Title Table 2: Ablation studies of adding CAR to different baselines on Pascal Context [44] and COCOStuff- 10K [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We deterministically reproduced all the baselines with the same settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' All results are single- scale without flipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' CAR works very well in most existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' ⊠ means reducing the class-level threshold to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='25 from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We found it is sensitive for some model variants to handle a large number of class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Affinity loss [15] and Auxiliary loss [2] are applied on CPNet and OCR, respectively, since they highly rely on those losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Methods Backbone mIOU(%) Pascal Context COCO-Stuff10K FCN [1] ResNet-50 [25] 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='72 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='10 FCN + CAR ResNet-50 [25] 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='40(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='68) 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='91(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='81)⊠ FCN [1] ResNet-101 [25] 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='93 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='93 FCN + CAR ResNet-101 [25] 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='39(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='49) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='88(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='95)⊠ DeepLabV3 [42] ResNet-50 [25] 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='59 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='96 DeepLabV3 + CAR ResNet-50 [25] 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='53(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='94) 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='13(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='17) DeepLabV3 [42] ResNet-101 [25] 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='69 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='92 DeepLabV3 + CAR ResNet-101 [25] 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='58(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='89) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='39(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='47) Self-Attention [4] ResNet-50 [25] 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='32 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='35 Self-Attention + CAR ResNet-50 [25] 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='50(+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='18) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='58(+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='23)⊠ Self-Attention [4] ResNet-101 [25] 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='59 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='53 Self-Attention + CAR ResNet-101 [25] 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='49(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='9) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='15(+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='62) CCNet [17] ResNet-50 [25] 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='15 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='10 CCNet + CAR ResNet-50 [25] 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='56(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='41) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='39(+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='29) CCNet [17] ResNet-101 [25] 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='41 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='88 CCNet + CAR ResNet-101 [25] 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='97(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='56) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='56(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='68) DANet [5] ResNet-101 [25] 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='45 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='80 DANet + CAR ResNet-101 [25] 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='57(+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='12) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='47(+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='67) CPNet [15] ResNet-101 [25] 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='29 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='92 CPNet + CAR ResNet-101 [25] 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='98(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='69) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='12(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='20)⊠ OCR [13] HRNet-W48 [46] 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='37 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='22 OCR + CAR HRNet-W48 [46] 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='99(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='62) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='53(+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='31) UperNet [39] Swin-Tiny [30] 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='62 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='07 UperNet + CAR Swin-Tiny [30] 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='78(+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='16) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='63(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='56) ⊠ UperNet [39] Swin-Large [30] 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='48 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='25 UperNet + CAR Swin-Large [30] 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='97(+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='49) 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='88(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='63) CAA [23] EfficientNet-B5 [28] 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='79 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='40 CAA + CAR EfficientNet-B5 [28] 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='96(+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='17) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='93(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='53) CAA [23] ConvNext-Large [29] 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='48 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='49 CAA + CAR ConvNext-Large [29] 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='40(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='92) 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='70(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='21) demonstrating SAA is indeed more compatible with CAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' As a result (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3), SAA brings more accu- racy gain (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='45) compared to vanilla AA (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='24) and CCNet (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='41), and achieves similar accuracy to self-attention (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='91 vs 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='50 ) but with much smaller computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Effectiveness of EJPU In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4, we evaluate the proposed EJPU of CARD on the Pascal Context dataset with ResNet-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' All settings are same as the the ones in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 1 and Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Compared to other approaches, such as origi- nal JPU and Semantic FPN, EJPU has the closest performance to the dilation model and even beats the Semantic FPN with twice filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Also note Article Title 15 Table 3: Ablation studies of different token mix- ers’ compatibility with CAR on Pascal Context dataset using ResNet-50 (D8), where “D8” means modifying the last convolution layers of the back- bone to their dilated version to obtain output stride (OS) = 8 feature maps [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' “PE” repre- sents conditional positional encoding in [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' “HC” represents number of attention heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='3 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Methods CAR CPE HC mIOU(%) T1 Self-Attention [4] 1 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='32 T2 ✓ 1 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='50(+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='18) T3 CCNet [17] 1 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='15 T4 ✓ 1 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='56(+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='41) T5 Vanilla AA ✓ 4 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='39 T6 ✓ ✓ 4 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='63(+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='24) T7 SAA ✓ 4 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='46 T8 ✓ ✓ 2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='82(+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='36) T9 ✓ ✓ 1 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='55(+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='09) T10 ✓ ✓ 4 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='91(+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='45) Table 4: Ablation studies of EJPU in CARD on Pascal Context dataset using ResNet-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Com- pared to Semantic FPN and JPU, the proposed EJPU achieved the closest performance to the dilation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Mode CAR mIOU(%) Semantic FPN [33] 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='96 ✓ 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='67 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='71) Semantic FPN [33] 2× filters 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='83 ✓ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='04 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='21) JPU [36] 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='05 ✓ 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='76 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='71) EJPU (Ours) 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='63 ✓ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='76 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='13) OS = 8 (Dilation) [2, 42] 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='46 ✓ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='91 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='45) that EJPU is more compatible with CAR since it brings more accuracy gain (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='13 vs +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='71).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Effectiveness of CPM in EJPU In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 5, we compare different options for chan- nel padding if the dimensions of the backbone and the JPU are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We use ConvNeXt-L [29] (1536 channels) to conduct the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' The remaining settings are the same as in the previ- ous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' “Project JPU output” and “Project Table 5: Ablation studies of CPM inside EJPU on Pascal Context dataset using ConvNeXt-L, which outputs 1536 channel feature maps and thus requires the Channel Padding Module (CPM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Padding Strategies mIOU(%) Project JPU output 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='51 Project Backbone output 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='68 Reduce JPU’s conv filters 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='94 Interpolation 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='43 CPM 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='99 Table 6: Computation analysis of CARD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We provide computational cost in GFLOPs on two commonly used resolutions 513 × 513 and 1025 × 2049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' “SA” is short for “Self-Attention”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' “D8” means modifying the last convolution layers of the backbone to their dilated version to obtain output stride (OS) = 8 feature maps [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Rows in CARD have not marked by “w/ EJPU” use dilated backbone with output stride = 8 (D8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Method Backbone GFLOPs 513 × 513 1025 × 2049 SA (CAR) ResNet-50 (D8) 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='96 1723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='03 CARD ResNet-50 (D8) 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='70 1157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='59 w/ EJPU ResNet-50 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='69 (-25%) 887.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='180 (-23%) CARD ConvNeXt-L (D8) 818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='14 6418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='79 w/ EJPU ConvNeXt-L 262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='82 (-67%) 2043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='24 (-68%) CARD EfficientNet-L2 (D8) 1635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='22 12834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='12 w/ EJPU EfficientNet-L2 283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='62 (-82%) 2184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='76 (-82%) backbone output” use a 1 × 1 convolution layer (followed by BN and ReLU) to adjust the channel number to match the other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' “Reduce JPU’s conv filters” means reduce the filter numbers of all the convolution layers in JPU by a same factor to the backbone feature dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Among all these configurations, CPM achieves the best accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Computational cost of CARD Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 6 presents computational cost of CARD for two commonly seen image resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Compared to the Self-Attention with dilated ResNet-50, our CARD significantly reduces the computa- tional cost from 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='96 to 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='69 GFLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' EJPU reduces more computation for larger backbones or higher-resolution inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 16 Article Title Table 7: Comparisons to state-of-the-art meth- ods on Pascal Context dataset of CARD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Note that methods marked with ‘∗’ report mIOU from their papers while the others are obtained with our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' SS means single scale per- formance w/o flipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' MF means multi-scale performance w/ flipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Methods Avenue mIOU(%) SS MF SETR (ViT-L)* [12] CVPR’21 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='8 DPT (ViT-Hybrid)* [11] ICCV’21 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='5 Segmenter (ViT-L)* [48] ICCV’21 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='0 OCNet (HRNet-W48)* [6] IJCV’21 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2 CAA (EfficientNet-B7)* [23] AAAI’22 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='5 SegNeXt (MSCAN-L)* [49] NIPS’22 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='9 CAA + CAR (ConvNeXt-L) ECCV’22 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='7 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='9 CARD (ConvNeXt-L) Ours 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='4 CARD (ConvNeXtV2-L) Ours 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='0 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='6 CARD (EfficientNet-L2) Ours 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='3 CARD Compared to the State-of-the-art In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 7, we equip CARD to stronger back- bones to compare with state-of-the-art methods on Pascal Context dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' The reported mIOU of compared methods with “*” comes from their respective paper instead of our reproduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We train CARD with ConvNeXt-L [29], ConvNeXtV2-L [47] and EfficientNet-L2 [28] as backbone, using AdamW optimizer, an initial learning rate of 4e-5, while the other settings remain the same as the experiments in our abla- tion studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' The AdamW optimizer improved the performance of CARD (ConvNeXt-L) from 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='99% (shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 5, trained by SGD) to 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 7, CARD out- performs other state-of-the-art approaches when using ConvNeXt-L and ConvNeXtV2-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' With even stronger backbone such as EfficientNet-L2, CARD achieves 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='0% mIOU under single-scale setting and 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='5% mIOU under multi-scale with flipping setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='4 Visualization of Class Dependency Maps In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 7, we present the class dependency maps calculated on the complete Pascal Context test set, where every pixel stores the dot-product sim- ilarities between every two class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' The maps indicate the inter-class dependency obtained with the standard ResNet-50 + Self-Attention and Swin-Tiny + UperNet, and the effect of applying our CAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' A hotter color means that the class has higher dependency on the corresponding class, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 7 a1-a2, we can easily observe that the inter-class dependency has been significantly reduced with CAR on ResNet50 + Self-Attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 7 b1-b2 show a similar trend when tested with different backbones and head blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' This partially explains the reason why baselines with CAR generalize better on rarely seen class combinations (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 1 and 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Interest- ingly, we find that the class-dependency issue is more serious in Swin-Tiny + Uper, but our CAR can still reduce its dependency level significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='5 Visualization of Pixel-relation Maps In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 8, we visualize the pixel-to-pixel relation energy map, based on the dot-product similarity between a red-dot marked pixel and other pixels, as well as the predicted results for different meth- ods, for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Examples are from Pascal Context test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' As we can see, with CAR super- vision, the existing models focus better on objects themselves rather than other objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Therefore, this reduces the possibility of the classification errors because of the class-dependency bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='3 Experiments on COCOStuff-10K COCOStuff-10K dataset [45] 6 is widely used for evaluating the robustness of semantic segmenta- tion models [8, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' The COCOStuff-10k dataset is a very challenging dataset containing 171 labeled classes and 9000/1000 images for training/test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='1 CAR on Different Baselines In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 2, all of the tested baselines gain perfor- mance boost ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='17% to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='23% with our proposed CAR on COCOStuff-10K dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' This demonstrates the generalization ability of our CAR when handling a large number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2 CARD Compared to the State-of-the-art In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 8, we equip CARD to ConvNeXt-L to compare with state-of-the-art methods on 6https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='com/nightrome/cocostuff10k Article Title 17 a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' ResNet50 + Self-Attention a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' ResNet50 + Self-Attention + CAR b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Swin-Tiny + UperNet b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Swin-Tiny + UperNet + CAR a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' ResNet50 + Self-Attention a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' ResNet50 + Self-Attention + CAR b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Swin-Tiny + UperNet b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Swin-Tiny + UperNet + CAR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 7: Class dependency maps generated on Pascal Context test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' One may zoom in to see class names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' A hotter color means that the class has higher dependency to the corresponding class, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' It is obvious that our CAR reduces the inter-class dependency, thus providing better generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Table 8: Comparisons to state-of-the-art meth- ods on COCOStuff10k dataset of CARD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Note that methods marked with ‘∗’ report mIOU from their papers while the others are obtained with our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' SS means single scale per- formance w/o flipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' MF means multi-scale performance w/ flipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Methods Avenue mIOU(%) SS MF OCR (HRNet-W48)* [13] ECCV’20 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2 OCNet (HRNet-W48)* [6] IJCV’21 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='0 CAA (EfficientNet-B7)* [23] AAAI’22 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='4 RankSeg (ViT-L)* [50] ECCV’22 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='9 CAA + CAR (ConvNeXt-L) ECCV’22 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='8 CARD (ConvNeXt-L) Ours 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='9 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='0 COCOStuff-10K dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' The reported mIOU of compared methods with “*” comes from their respective paper instead of our reproduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We trained CARD with ConvNeXt-L using AdamW optimizer, an initial learning rate of 4e-5, while the other settings remain the same as the experiments in our ablation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 8, CARD (ConvNeXt-L) surpasses the other methods with a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Table 9: Comparisons to state-of-the-art meth- ods on COCOStuff164K dataset of CARD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Note that methods marked with ‘∗’ report mIOU from their papers while the others are obtained with our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' SS means single scale per- formance w/o flipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' MF means multi-scale performance w/ flipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Methods Avenue mIOU(%) SS MF SegFormer (MiT-B5)* [51] NIPS’21 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='7 CAA (EfficientNet-B5)* [23] AAAI’22 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='3 SegNeXt (MSCAN-L)* [49] NIPS’22 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='5 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2 CARD (ConvNeXt-L) Ours 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='9 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='6 CARD (EfficientNet-L2) Ours 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='4 Experiments on COCOStuff-164K 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='1 CARD Compared to the State-of-the-art COCOStuff-164k 7 is the full set of COCOStuff- 10K, which becomes a new popular benchmark starting from 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Training settings are the same as COCOStuff-10k (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2), except the total training iteration is set to 80k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 9, the proposed CARD outperforms previous approaches by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 7https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='com/nightrome/cocostuff 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content="0 o'S 0'3 0'4 0'2 o'e 8." metadata={'source': 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feature similarity between a given pixel (marked with a red dot in the image) and all pixels, as well as the segmentation results on Pascal Context test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' A hotter color denotes larger similarity value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Apparently, our CAR reduces the inter-class dependency and exhibits better generalization ability, where energies are better restrained in the intra-class pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' ATA9 YAWHDIA 2unh3TAVIR9Article Title 19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2 Visualization of CARD COCOStuff-164k results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 9 compare Seg- NeXt [49], and our CARD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' As it can be seen, our CARD can segment the uncommon objects and complex scenes very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Image GT SegNeXt-L CARD (ConvNeXt-L) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 9: Eamples of the results obtained on the COCOStuff-164K dataset with our pro- posed CARD (ConvNeXt-L) in comparison with SegNeXt-L [49] and the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='5 Experiments on Cityscapes 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='1 CARD Compared to the State-of-the-art Cityscapes [53]8 contains 2975/500/1525 images for training/validation/test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We adopt AdamW, batch size = 8, 80K training iterations in total and 8https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='cityscapes-dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='com/ Table 10: Comparisons to state-of-the-art meth- ods on Cityscapes validation set of CARD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Note that methods marked with ‘∗’ report mIOU from their papers while the others are obtained with our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' SS means single scale per- formance w/o flipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' MF means multi-scale performance w/ flipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Methods Ref mIOU(%) SS MF Axial-DeepLab-L* [52] ECCV-2020 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='5 SETR (ViT-L)* [51] CVPR-2021 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2 Segmenter (ViT-L)* [48] ICCV-2021 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='3 HRFormer-B * [49] NIPS-2021 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='6 CARD (ResNet-50) Ours 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='8 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='6 CARD (ConvNeXt-L) Ours 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='8 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='6 1000 steps linear warmup when training CARD with ConvNeXt-Large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2 Visualization of CARD Cityscapes results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 10 compare Segmenter(ViT-L) [48], and our CARD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' As it can be seen, our CARD can segment the hard class (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='g, rider vs person) very well, which is very useful for the autopilot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 5 Conclusion In this paper, we have aimed to make a bet- ter use of class level context information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We first proposed a universal class-aware regular- izations (CAR) approach, which minimizes the intra-class feature variance and maximize the inter-class separation simultaneously, to regularize the training process and boost the differentiabil- ity of the learned pixel representations without extra computation during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Then we pro- posed a class-aware regularized decoder (CARD), which is designed for better effectiveness and effi- ciency tailored for the proposed CAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Extensive experiments conducted on various benchmarks and thorough ablation studies have validated the effectiveness of the proposed CAR, which has boosted the existing models’ performance by up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='18% mIOU on Pascal Context and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='23% on COCOStuff-10k with no extra inference overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' And the proposed CARD achieved state-of-the-art performance on multiple benchmarks while using much less computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' q62b6cial beuobmoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='lismp@b6121ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='21obno20 Article Title Image GT Segmenter (ViT-L) CARD (ConvNeXt-L) terrain void road sky sidewalk building wall fance pole traffic light traffic sign vegetation person rider car truck bus train motorcycle bicycle 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2% 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='5% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='5% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='9% 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='7% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='4% 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='2% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='4% 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='4% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='4% Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 10: Some visual examples on the Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We compare our proposed CARD with Segmenter (ViT-L) [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We show mIOU for each predicted mask on the bottom left corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Acknowledgement This research depends on the NVIDIA determinism framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We appreciate the support from @duncanriach and @reedwm at NVIDIA and TensorFlow team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' We also thank OpenI (https://openi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='cn) for providing GPUs to conduct experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' Article Title 21 References [1] Long, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Shelhamer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Darrell, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Fully convolutional networks for semantic segmen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: CVPR (2015) [2] Zhao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Qi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Jia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Pyramid scene parsing network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: CVPR (2017) [3] Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Papandreou, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Kokkinos, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Murphy, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Yuille, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' : Deeplab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected crfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' IEEE TPAMI (2017) [4] Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Girshick, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Gupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', He, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Non-local neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: CVPR (2018) [5] Fu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Tian, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Bao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Fang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Lu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Dual attention network for scene segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: CVPR (2019) [6] Yuan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Huang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Guo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Ocnet: Object context network for scene parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' IJCV (2021) [7] Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zhan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Xie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Semantic correlation promoted shape-variant context for segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: CVPR (2019) [8] Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zhong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Lin, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Expectation-maximization atten- tion networks for semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: ICCV (2019) [9] Zhu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Xu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Bai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Huang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Bai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Asymmetric non-local neural networks for semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: ICCV (2019) [10] Dosovitskiy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Beyer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Kolesnikov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Weissenborn, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zhai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Unterthiner, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Dehghani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Minderer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Heigold, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Gelly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Uszkoreit, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Houlsby, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': An image is worth 16x16 words: Transformers for image recognition at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: ICLR (2021) [11] Ranftl, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Bochkovskiy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Koltun, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Vision transformers for dense prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: ICCV (2021) [12] Sixiao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Jiachen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Hengshuang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Xia- tian, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zekun, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Yabiao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Yanwei, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Jianfeng, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Tao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Rethinking semantic segmentation from a sequence-to-sequence perspective with trans- formers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: CVPR (2021) [13] Yuan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Object- contextual representations for semantic seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: ECCV (2020) [14] Zhang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Hong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Ma, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Han, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Ding, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Acfnet: Attentional class feature network for semantic segmenta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: ICCV (2019) [15] Yu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Gao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Yu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Shen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Sang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Context prior for scene segmenta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: CVPR (2020) [16] Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Kang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zhe, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Jia, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Bao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Car: Class-aware regular- izations for semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: ECCV (2022) [17] Huang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Wei, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Huang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Shi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Liu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Huang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' : Ccnet: Criss-cross attention for semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' IEEE TPAMI (2020) [18] Choi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Choo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Cars can’t fly up in the sky: Improving urban-scene segmen- tation via height-driven attention networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: CVPR (2020) [19] Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Schonfeld, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Tang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Exploit visual dependency relations for semantic seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: CVPR (2021) [20] Yu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Luo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zhou, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Si, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Feng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Yan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Metaformer is actually what you need for vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: CVPR (2022) [21] Vaswani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Shazeer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Parmar, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Uszko- reit, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Jones, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Gomez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', �Lukasz Kaiser, Polosukhin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Attention is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: NeurIPS (2017) [22] Ho, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Kalchbrenner, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Weissenborn, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Salimans, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Axial Attention in Multidimen- sional Transformers (2019) 22 Article Title [23] Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Kang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Jia, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Channelized axial attention - considering channel relation within spatial attention for semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: Proceedings of the AAAI Conference on Artificial Intelligence (2022) [24] Simonyan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zisserman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Very deep convolutional networks for large-scale image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: ICLR (2015) [25] He, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Ren, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Sun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Deep residual learning for image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: CVPR (2016) [26] Chollet, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Xception: Deep learning with depthwise separable convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: CVPR (2017) [27] Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zhu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Lin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Sun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', He, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Muller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Manmatha, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Smola, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Resnest: Split-attention networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' arXiv preprint arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='08955 (2020) [28] Mingxing, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Quoc, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Efficientnet: Rethinking model scaling for convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: International Conference on Machine Learning (2019) [29] Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Mao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Feichtenhofer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Darrell, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Xie, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': A convnet for the 2020s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: CVPR (2022) [30] Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Cao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Hu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Wei, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Lin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Guo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Swin transformer: Hierarchical vision transformer using shifted windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: ICCV (2021) [31] Ronneberger, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Fischer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Brox, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': U-net: Convolutional networks for biomedical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' MICCAI (2015) [32] Lin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Doll´a, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Girshick, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', He, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Hariharan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Belongie, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Feature pyra- mid networks for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: CVPR (2017) [33] Kirillov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Girshick, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', He, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Doll´ar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Panoptic feature pyramid networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: CVPR (2019) [34] Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zhu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Papandreou, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Schroff, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Adam, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Encoder-decoder with atrous separable convolution for semantic image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: ECCV (2018) [35] Cheng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Misra, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Schwing, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Kir- illov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Girdhar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Masked-attention mask transformer for universal image segmenta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: CVPR (2022) [36] Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Huang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Liang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Yizhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': FastFCN: Rethinking Dilated Convolution in the Backbone for Semantic Segmentation (2019) [37] Wen1, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Qiao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Dis- criminative feature learning approach for deep face recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: ECCV (2016) [38] Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zhou, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Yu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Dai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Konukoglu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Van Gool, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Exploring cross- image pixel contrast for semantic segmenta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: ICCV, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' 7303–7313 (2021) [39] Xiao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zhou, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Sun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Unified perceptual parsing for scene under- standing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: ECCV (2018) [40] Chu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Tian, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Wei, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Xia, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Shen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Conditional positional encodings for vision transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' arXiv preprint arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='10882 (2021) [41] Tu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Talebi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Yang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Pey- man Milanfar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Maxvit: Multi- axis vision transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: ECCV (2022) [42] Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Papandreou, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Schroff, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Adam, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Rethinking Atrous Convolution for Semantic Image Segmentation (2017) [43] Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Dana, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Tyagi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Agrawal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Con- text encoding for semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: CVPR (2018) [44] Mottaghi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Cho, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Fidler, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Urtasun, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Yuille, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': The role of context for object detection and semantic segmentation in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: CVPR (2014) Article Title 23 [45] Caesar, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Uijlings, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Ferrari, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': COCO- Stuff: Thing and Stuff Classes in Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: CVPR (2018) [46] Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Sun, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Cheng, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Jiang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Deng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Liu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Mu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Tan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Liu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Xiao, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Deep high- resolution representation learning for visual recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' IEEE TPAMI (2020) [47] Woo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Debnath, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Hu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Kweon, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Xie, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Con- vnext v2: Co-designing and scaling convnets with masked autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' arXiv preprint arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='00808 (2023) [48] Strudel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Garcia, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Laptev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Schmid, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Segmenter: Transformer for semantic seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: ICCV (2021) [49] Guo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Lu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Hou, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Cheng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Hu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Segnext: Rethink- ing convolutional attention design for seman- tic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: NeurIPS (2022) [50] He, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Yuan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Yue, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Hu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Rankseg: Adaptive pixel classification with image cat- egory ranking for segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: ECCV (2022) [51] Xie, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Yu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Anandkumar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Alvarez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Luo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Segformer: Simple and efficient design for semantic segmentation with transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: NeurIPS (2021) [52] Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Zhu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Green, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Adam, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Yuille, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': Axial-deeplab: Stand- alone axial-attention for panoptic segmenta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: ECCV (2020) [53] Marius, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Mohamed, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Sebastian, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Timo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Markus, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Rodrigo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Uwe, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Roth, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=', Bernt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=': The cityscapes dataset for semantic urban scene understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} +page_content=' In: CVPR (2016)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E3T4oBgHgl3EQfAglp/content/2301.04258v1.pdf'} diff --git a/KNE2T4oBgHgl3EQfUwfR/content/tmp_files/2301.03817v1.pdf.txt b/KNE2T4oBgHgl3EQfUwfR/content/tmp_files/2301.03817v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..664859eb99a2bc437fd3b6fb684c23c0bc98d6d0 --- /dev/null +++ b/KNE2T4oBgHgl3EQfUwfR/content/tmp_files/2301.03817v1.pdf.txt @@ -0,0 +1,2166 @@ +IEEE TRANSACTIONS ON SIGNAL PROCESSING , VOL. XX, NO. XX, 2022 +1 +RIS-Assisted Joint Uplink Communication and +Imaging: Phase Optimization and Bayesian +Echo Decoupling +Shengyu Zhu, Zehua Yu, Qinghua Guo, Senior Member, IEEE, Jinshan Ding, Member, IEEE, +Qiang Cheng, Senior Member, IEEE, and Tie Jun Cui, Fellow, IEEE +Abstract—Achieving integrated sensing and communication +(ISAC) via uplink transmission is challenging due to the unknown +waveform and the coupling of communication and sensing echoes. +In this paper, a joint uplink communication and imaging system +is proposed for the first time, where a reconfigurable intelligent +surface (RIS) is used to manipulate the electromagnetic signals +for echo decoupling at the base station (BS). Aiming to enhance +the transmission gain in desired directions and generate required +radiation pattern in the region of interest (RoI), a phase optimiza- +tion problem for RIS is formulated, which is high dimensional +and nonconvex with discrete constraints. To tackle this problem, a +back propagation based phase design scheme for both continuous +and discrete phase models is developed. Moreover, the echo +decoupling problem is tackled using the Bayesian method with +the factor graph technique, where the problem is represented by +a graph model which consists of difficult local functions. Based +on the graph model, a message-passing algorithm is derived, +which can efficiently cooperate with the adaptive sparse Bayesian +learning (SBL) to achieve joint communication and imaging. +Numerical results show that the proposed method approaches +the relevant lower bound asymptotically, and the communication +performance can be enhanced with the utilization of imaging +echoes. +Index Terms—integrated sensing and communication (ISAC), +joint communication and imaging, reconfigurable intelligent sur- +face (RIS), factor graph, information metamaterials. +I. INTRODUCTION +I +NTEGRATED sensing and communication (ISAC) has +been recognized as a promising technology for the next- +generation wireless networks [1]. ISAC not only allows com- +munication and radar systems to share spectrum resources, but +also enables communication and radar sensing functionalities +simultaneously, which significantly reduces hardware resource +consumption. However, this new framework introduces great +challenges to hardware design and signal processing, thus +This work was supported by the Basic Scientific Center of Information +Metamaterials of the National Natural Science Foundation of China under +Grant 6228810001 and also supported by the National Natural Science +Foundation of China under Grant 62171358. +Shengyu Zhu and Jinshan Ding are with the National Laboratory of +Radar Signal Processing, Xidian University, Xi’an 710071, China (e-mail: +ding@xidian.edu.cn). +Zehua Yu is with the School of Electronic Engineering, Xidian University, +Xi’an 710071, China (e-mail: yuzehua@xidian.edu.cn). +Qinghua Guo is with the School of Electrical, Computer, and Telecom- +munications Engineering, University of Wollongong, Wollongong, Australia +(e-mail: qguo@uow.edu.au). +Qiang Cheng and Tie Jun Cui are with the State Key Laboratory of +Millimeter Waves, Southeast University, Nanjing 210096, China (e-mail: +tjcui@seu.edu.cn). +bringing a research upsurge in system design 2]–[6, optimal +joint waveform design [7]–[11], exploration of joint processing +algorithms [12]–[14], and resource allocation [15]–[17]. +In the past years, information metamaterials have opened +a new era of real-time digital regulation of electromagnetic +waves [18]–[29]. As one representative of information meta- +materials, reconfigurable intelligent surface (RIS) is expected +to break the dependence of traditional sensing and commu- +nication on the channel environment. RIS bridges the gap +between ISAC and information metamaterials by configuring +the electromagnetic environment more flexibly, quickly and +intelligently, which is highly expected to better serve the next +generation of wireless communication and senisng. Essen- +tially, it is a low-cost passive system that achieves control +of electromagnetic beam energy by tuning the phase of array +elements. It is shown in [18] that the highly controllable +reflection of RIS can be practically achieved by leveraging +the existing digitally reconfigurable or programmable meta- +surface. RIS was first applied as a wireless relay to redirect +electromagnetic signals in the field of wireless communication +[30]–[32], thereby overcoming the adverse effects in natural +environments. In addition, it can also be used as a new +communication architecture transmitter to implement various +modulation [33]–[40], e.g., FSK [33], PSK [35], and MIMO +[38]. Recently, RIS has been employed to achieve environment +awareness and parameters estimation. A new self-sensing RIS +architecture was proposed in [41], where the performance of +different benchmark sensing systems in the cases of with and +without RIS was compared. The authors in [42] and [43] shed +light on the interplay among the system parameters, including +the radar-RIS distance, the RIS size and the location of the +prospective target. The results show that the radar system can +achieve the optimal performance when the RIS is deployed +in the near field of the radar arrays on both the transmitter +and receiver sides. The authors in [44] and [45] proposed to +sense humans, recognize their gestures and physiological state +simultaneously by utilizing the programmable metasurface and +Wi-Fi signals. +Recently, RIS-assisted ISAC system has attracted significant +research interest. A Dual-Functional Radar-Communication +(DFRC) system with RIS deployed near the communication +devices has been proposed in [46], [47]. By optimizing the +Cramer-Rao bound (CRB) of DOA estimation, the constant- +waveform and RIS phase shifts are designed jointly to mitigate +the mitigating multi-user interference (MUI). The RIS-assisted +arXiv:2301.03817v1 [eess.SP] 10 Jan 2023 + +IEEE TRANSACTIONS ON SIGNAL PROCESSING , VOL. XX, NO. XX, 2022 +2 +ISAC system in the cases of congested and obstructed channels +has been investigated in [48] and [49], respectively, where the +PSM of the RIS and the precoding of the base station (BS) are +optimized jointly to improve the SNR at receiver. To explore +the potential of multiple RISs in assisting ISAC, the authors in +[50] propose a double-RIS-assisted ISAC system, where two +RISs are deployed to enhance the communication performance +while suppressing mutual interference. The beam patterns +of RISs and radar are optimized jointly based on penalized +dual decomposition (PDD) to enhane the system performance. +In [51], the authors propose an RIS-aided localization and +communication system, where the theoretical performance +limits of localization and communication is derived for both +near-field and far-field scenarios. Numerical results show that +with the assistance of multiple RISs, both spectral efficiency +and localization accuracy can be improved significantly. +Most of the existing literature in ISAC focus on the uti- +lization of downlink transmission. On the contrary, ISAC in +uplink transmission is underexplored, technically because it is +much more challenging compared with downlink counterparts +due to unknown signal waveform and coupling of communi- +cation and sensing echoes. In this article, we propose a new +RIS-assisted ISAC system, where the uplink transmission is +exploited to achieve joint communication and imaging. In con- +trast to existing works [46]–[50] that achieve communication +and sensing at different receivers, the proposed uplink ISAC +system performs communication and imaging at one receiver, +where the communication performance can be enhanced with +the utilization of sensing echo. We show that the radiation +pattern can be controlled expectedly with the proposed phase +design scheme and the factor graph technique is very suitable +for echo decoupling for uplink ISAC system. The contribution +of this work can be summarized as follows +• A novel RIS-assisted uplink joint communication and +imaging system is presented for the first time, to the +best of our knowledge. It is shown that by properly +designing the phase shift of the RIS, the radiation pattern +can be modulated as desired, thereby allowing joint +communication and imaging at the BS with only one RF +chain. +• A phase optimization problem based on the requirements +of the system is formulated. A back propagation based +phase design scheme with the combination of temperature +parameter is proposed to achieve efficient phase optimiza- +tion for both continuous and discrete phase models. +• A factor graph representation of the joint communication +and imaging is established, and then, an efficient message +passing algorithm is successfully developed to decoupling +echoes at the BS. It is also demonstrated that the com- +munication performance can be enhanced by making full +use of imaging echoes. +The article is organized as follows. Section II briefly intro- +duces the system and the signal model used. In Section III, the +phase design problem is formulated and a back propagation +based optimization method is proposed. In Section IV, a +joint maximum a posteriori estimation problem is established. +Then the problem is represented by a graph model, based on +Fig. 1: Diagram of a RIS-assisted joint uplink imaging and +communication system. +which we propose an efficient message passing algorithm to +achieve joint communication and imaging. Numerical results +are provided in Section V. Section VI concludes this article. +Notations: Throughout this paper, column vectors and matri- +ces are denoted by bold lower-case and bold upper-case letters, +respectively. The notation (·)H denotes the conjugate operator +and (·)−1 denotes inverse operator. | · |, ∥ · ∥ and ∥ · ∥F are the +l2 norm, the l2 norm and the Frobenius norm, respectively. +The transpose operation is denoted by (·)T. We use E{·} +to denote the expectation operator. The notation N(x; m, V) +denotes a Gaussian distribution of x with mean vector m and +covariance matrix V. In many cases, we also use the inverse of +the covariance matrix V, which is denoted by W and called +weight matrix as in [52], [53]. The notation Ia denotes an +identity matrix of a size a×a. The notation ∝ denotes equality +of functions up to a scale factor. We use ⊙ to represent the +inner product, and +� +∼a F(·) denotes integral over all variables +in F(·) except a. The notation diag(·) denotes the diagonal +operation and δ(·) is the Dirac delta function. +II. SYSTEM MODEL +We consider an uplink RIS-assisted ISAC system in 2D +plane as shown in Fig.1, which consists of a user equipment +(UE), a BS, a RIS and a region of interest (RoI). Assume +there is an obstacle between the UE and the BS. The RIS +is deployed between the UE and BS to customize the radio +environment for the signal from the UE side and reflect to the +BS, and its center is considered as the origin of the coordinate +system. The UE sends communication signals to the BS with +the assistant of the RIS. Due to the high path loss, it is also +assumed that the signals with more than two reflections are +negligible. We aim to perform communication at the BS while +realizing reconstruction of the RoI. +The UE and the BS are located at the far field of the RIS, in +directions of θU and θB from the origin, respectively, and both +of which are assumed to adopt the single-antenna structure. +The RIS is a uniform linear array (ULA) with N elements that +are half a wavelength apart. The RoI is divided into an equi- + +RIS +90° +h +.06- +0 +B +H +0 +UE +.0 +Base station +Rol +ObstacleIEEE TRANSACTIONS ON SIGNAL PROCESSING , VOL. XX, NO. XX, 2022 +3 +Fig. 2: Diagram of overlapped receive signal model. +spaced grid with M pixel units. We denote the angle vector +of the RoI from the origin as θS = [θS,1, θS,2, . . . , θS,M]T . +It is noted that at the transmitted signals reach the BS +through two paths, i.e., UE-RIS-BS, namely communication +link and UE-RoI-RIS-BS, namely imagine link. Supposing that +the UE transmits a communication frame of length L, due to +the difference of propagation delay, the signal received at the +BS falls into 3 categories: 1) non-overlapped communication +echo, 2) overlapped communication and imaging echo and 3) +non-overlapped imaging echo. A generic model of received +signal is shown in Fig.2. Specifically, the sampled base band +received signal at time index t can be expressed as +y(t)= +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +αcgTΘ(t)hcx(t) +� +�� +� +UE→RIS→BS ++w(t), +t ≤ k +αcgTΘ(t)hcx(t) +� +�� +� +UE→RIS→BS ++ αIgTΘ(t)HIσx(t − k) +� +�� +� +UE→RoI→RIS→BS ++w(t), +k < t < L +αIgTΘ(t)HIσx(t − k) +� +�� +� +UE→RoI→RIS→BS ++w(t), +L < t ≤ L + k +(1) +where we define the variables and symbols list below +• αc and αI denote the attenuation coefficients of com- +munication and imaging links, respectively, which are +assumed to have been obtained by the pilot signals. +• The steering vectors are g = a(θB) ∈ CN×1, hc = +a(θU) ∈ CN×1 and HI = a(θS) ∈ CN×M, where +a(θ) = +� +1, ejπ sin(θ), . . . , ejπ(N−1) sin(θ)�T . +• Θ(t) = diag +� +ejθt,1, . . . , ejθt,N � +denotes the phase shift +matrix of RIS at the time index t, where θt,n +∈ +{0, +2π +2nbit , ..., 2π(2nbit−1) +2nbit +}, nbit denotes the number of +discrete values [45], [54]. When nbit → +∞, it means +that the phase shift of RIS can be controlled continuous. +• k indicates the time delay between communication and +imaging links, which is assumed to has been estimated +by the pilot signals [55]. +• x(t) +denotes +the +communication +symbol +sent +by +the +UE. +In +this +paper, +we +consider +QPSK +codes +for +communication, +i.e., +x(t) +∈ +{exp(j π +4 ), exp(j 3π +4 ), exp(j 5π +4 ), exp(j 7π +4 )}. +• σ ∈ CM×1 denotes the vector of scattering coefficients +of the RoI, which is assumed to be sparse. +• w(t) is the i.i.d. complex Gaussian noise with zero mean +and variance ξ2. +III. PHASE SHIFT DESIGN SCHEME +In this section, a phase optimization problem is established +based on the requirement of the joint communication and +imaging system. We then present a back propagation based +phase design scheme in the both cases of continuous and +discrete phase models. +A. Phase Shifts Optimization Formulation +The phase shift {Θ(t)} is critical since it determines the +gain of UE-RIS-BE and RoI-RIS-BS links simultaneously, and +both communication and imaging performance should be con- +sidered in optimizing {Θ(t)}. For communication functional- +ity, we hope to reflect more radiated energy from the UE to the +BS after modulation by the RIS, i.e., maximizing gTΘ(t)hc. +For imaging functionality, on the one hand, we hope to gather +more radiated energy reflected by the RoI and reflect it to +the BS, i.e., maximizing gT Θ(t)HI. On the other hand, +according to the compressed sensing theory [56], [57], when +reconstructing σ, we need to modify {Θ(t), k < t ≤ L + k} +to minimize the correlation between columns of the sensing +matrix which given by +G = +� +��� +gT Θ(k)HI +... +gT Θ(L + k)HI +� +��� . +(2) +The orthogonality of G can be evaluated by the mean of +non-diagonal elements of R(G), i.e., ∥R(G) − IM∥F /(M 2− +M), where R(G) denotes the correlation coefficient matrix +of columns of G. Based on the above consideration, we can +formulate the following phase shift optimization problem +max +Θ(t) +L+k +� +t=1 +ρ +��gT Θ(t)hc +��2 + (1 − ρ) +��gT Θ(t)HI +��2 +(3a) +s.t. ∥R(G) − IM∥F +M 2 − M +≤ η0, +(3b) +θt,n ∈ +� +0, 2π +2nbit , ..., 2π(2nbit − 1) +2nbit +� +. +(3c) +The cost function (3a) indicates that we expect to concentrate +energy in the RoI and the direction of communication as much +as possible, where ρ ∈ [0, 1] is used to realize a trade-off +between communication and imaging performance. η0 is a +threshold between 0 and 1 to constrain the orthogonality of G. +The constraint (3c) denotes the feasible set of phase shift. We +noted that (3) is non-convex, highly nonlinear and there are +numerous coupled variables to be optimized, making it highly +challenging to solve the problem directly. +B. Back Propagation Based Phase Shift Optimization +Note that (3b) indicates a strongly coupled constraint on +{Θ(t), 1 < t ≤ L + k}, while the constraint (3c) makes the +problem even more challenging. To solve this problem, we +relax the original problem (3) into two subproblems and +turn to find the sub-optimal solution. Specifically, we pro- +pose to first initialize {Θ(t), 1 ≤ t ≤ L + k} by minimizing +∥R(G) − IM∥ /(M 2−M) subject to the constraint (3c). With + +Communication Signals +Imaging Signals +人 +1 +2 +3 +Time Delay +OverlapIEEE TRANSACTIONS ON SIGNAL PROCESSING , VOL. XX, NO. XX, 2022 +4 +Fig. 3: The flowchart of the proposed phase optimization +method for continuous and discrete phase model. +an appropriate start point of {Θ(t), 1 ≤ t ≤ L + k}, we can +further modify it to maximize (3a) until (3b) and (3c) can +not be satisfied. With this in mind, we first need to solve the +following problem +min +Θ(t) +∥R(G) − IM∥F +M 2 − M +(4) +s.t. (3c). +However, the above problem is still challenging due to the +coupled cost function. Since Gaussian matrix is recognized +to be suit for a sensing matrix [58], we convert (4) into the +following problem +min +Θ(t) ∥RL×N − G∥F +(5) +s.t. (3c). +where RL×N is a Gaussian matrix in the size of L × N. +Inspired by the color section method in [59], we intro- +duce an increasing temperature parameter α and the softmax +function into back propagation framework to solve (5), where +the temperature parameter in the lth iteration is calculated by +α = 1 + (rl)2 and r is a factor to adjust the increasing rate, +and the softmax operation of α|wl +n| is defined as +Fsoft max(α|wl +n|) = +exp(α|wl +n|) +�2nbit +s=1 exp(α +��wln,s +��) +(6) +The corresponding flowchart of the proposed method is +shown in Fig. 3, where {Θ(t)} can be updated by alternating +iterative forward and backward propagation procedures. In +the forward propagation, assume that wn in the (l − 1)th +Algorithm 1 Discrete Phase Optimization Algorithm +1: Initialize the parameters: weights w1 +t,n, learning rate η and +scale factor r +2: while Γ do +3: +α = 1 + (rl)2 +4: +∀n, ˜wl +t,n = soft max(α|wl +t,n|) +5: +∀n, θl +t,n = ˜wl +t,n ⊙ [0, +2π +2nbit , ..., 2π(2nbit−1) +2nbit +] +6: +Θl(t) = diag +� +ejθl +t,1, . . . , ejθl +t,N +� +7: +Cl +t = Cost function(Θl(t)) +8: +Wl+1 +t += SGD +� +Wl +t, η, ∂Cl +t +∂Wl +t +� +9: +l = l + 1 +10: end while +iteration has been obtained, which denoted by wl−1 +n +, then ˜wl +n +is calculated by making softmax operation for α|wl−1 +n +|. It is +noteworthy that α increases with the iteration, and the softmax +operation essentially plays a role of selector. Specifically, +the softmax operation will push the larger element in wn +approaching 1 while the smaller elements approaching 0 with +the increase of α. Then θn can be obtained by making inner +product of ˜wl +n and s. After applying the above procedure for +all {θ}, we can update {Θ(t)} in parallel thereby calculate the +loss in the lth iteration. In the backward propagation, the chain +rule is utilized to update the gradients of all variables except +s and thereby update w by stochastic gradient descent (SGD) +[60]. We can finally obtain {Θ(t)} along with corresponding +w after the forward/backward propagation iterated alternately +until convergence. The steps of the proposed method for +solving (5) is summarized in Algorithm 1. +Based the initialized {Θ(t)} , we need to further modify +{Θ(t)} by solving the following problem +min +Θ(t) +L+k +� +t=1 +1 +ρ∥gT Θ(t)hc∥2 + (1 − ρ)∥gT Θ(t)HI∥2 +(7) +s.t. (3c). +Note that (3a) is rewritten to be a minimized form to facilitate +the calculation of loss. Fortunately, it is also worth noting that +(7) can be solved in the same way as (4). All we need to do is +execute Algorithm 1, where the loss is calculated by the cost +function in (7) and the iteration condition Γ is given by (3b). +In practical, when the phase shift can be controlled contin- +uously, i.e., nbit → +∞, the constraint (3c) can be removed +from (3), which makes (3) greatly simplified. In this case, we +can first initialize Θ by minimizing ∥R(G) − IM∥ /(M 2 − +M). Then {Θ(t)} can be further optimized by minimize +L+k +� +t=1 +1 +ρ∥gT Θ(t)hc∥2+(1−ρ)∥gT Θ(t)HI∥2 , which can be solved ef- +ficient by back propagation procedure in the dashed box of +Fig.3. +IV. MESSAGE PASSING BASED JOINT COMMUNICATION +AND IMAGING +In this part, we investigate joint communication and imaging +at the BS. The problem is first formulated into a maxi- +mum posteriori probability problem, which is nonconvex and + +loss +Forward +ac +a0(k) +Backward +(T+)0 +①(t) +00(k) +Continuous phase +0 +optimization +N +00. +2元 +2元(2nbit -1) +s= +0 +Me +2nbir +2 nbit +公V +W2 +W. +W +2nbit +↑ +F +ow +softmax +Ow +↑ +Discrete phase +W +W2 +w +optimization +2nbitIEEE TRANSACTIONS ON SIGNAL PROCESSING , VOL. XX, NO. XX, 2022 +5 +strongly coupled. To solve it, a factor graph representation is +then established, based on which an efficient iterative message +passing algorithm is derived to estimate σ and x jointly . +A. Joint Communication and Imaging Formulation +According to the Bayes’ theorem, the joint posterior distri- +bution of x and σ conditioned on y is given by +p(x, σ|y) ∝ p(y|x, σ)p(σ)p(x) +(8) +where y = [y(1), ..., y(L + k)]T ∈ C(L+k)×1 and x = +[x(1), ..., x(L)]T ∈ CL×1; p(x) and p(σ) denote the prior +for x and σ, respectively. For QPSK modulation, we have +p(x) = +L +� +t=1 +p(x(t)) = +L +� +t=1 +1 +4 +4 +� +i=1 +δ[x(t) − ej( π +2 i− π +4 )]. +(9) +Following [61], it is assumed that the elements in σ are +independent and following the two-layer sparsity-promoting +prior, i.e., +p(σ) = p(σ|γ)p(γ|ϵ, η) += +M +� +m=1 +N +� +σm|0, γ−1 +m +� +· +M +� +m=1 +Ga(γ−1 +m |ϵ, η), +(10) +where the precision vector γ = [γ1, ..., γM]H. +Based on (1), p(y|x, σ) can be decomposed into the product +of three categories of probability, i.e., +p(y|x, σ)= +k +� +t=1 +p (y(t)|x(t))· +L +� +t=k+1 +p ((y(t)|x(t), x(t − k), σ) +· +L+k +� +t=L+1 +p (y(t)|x(t − k), σ) , +(11) +where +p (y(t)|x(t)) ∝ exp +� +− +� +y(t) − αcgTΘ(t)hcx(t) +�2 +ξ2 +� +, t ≤ k +(12) +p(y(t)|x(t), x(t − k), σ) ∝ +exp +� +− +� +y(t) − αcgTΘ(t)hcx(t) − αIgTΘ(t)HIσx(t − k) +�2 +ξ2 +� +, +k < t ≤ L +(13) +p(y(t)|x(t − k), σ) ∝ exp +� +− +� +y(t) − αIgTΘ(t)HIσx(t − k) +�2 +ξ2 +� +, +L < t ≤ L + k +(14) +Based on the decomposition in (9) - (14), we aim to find x +and σ that maximize the joint posterior distribution p(x, σ|y) +in (8). However, it is a challenging problem as σ and x is +strongly coupled in different time segments. +TABLE I: Definitions of Variables and Functions +at +∆= αcgTΘ(t)hcxt +bt +∆= αIgTΘ(t)HIxt−k +ct +∆= at + bt +ht +∆= αIgTΘ(t + k)HIσ +f 1 +A,t(at) +∆= N(yt; at, ξ2) +f 2 +A,t(ct) +∆= N(yt; ct, ξ2) +f 3 +A,t(bt) +∆= N(yt; bt, ξ2) +fB,t (at, xt) +∆= δ(αcgTΘ(t)hcxt − at) +fD,t(at, bt, ct) +∆= δ(at + bt − ct) +fE,t (xt) +∆= 1 +4 +�4 +i=1 δ[xt − ej( π +2 i− π +4 )] +fK,t (ht, xt, bt+k) +∆= δ(htxt − bt+k) +fH,t(σ, ht) +∆= δ(αIgTΘ(t + k)HIσ − ht) +fG(σ, γ) +∆= � +M N(σm; 0, γ−1 +m ) +fF (γ) ≜ � +M Ga (γm; ϵ, η) +B. Factor Graph Representation for Joint Imaging and Com- +munication +Factor graphs and message passing are a powerful tool for +inference and estimation. In this subsection, a factor graph +model for (8) is established, based on which we can then +jointly estimate x and σ. To simplify the notations, we use +xt, yt and Θt to represent x(t), y(t) and Θ(t) respectively +and we also define some variables and functions in Table I. +With these abbreviations and definitions, (9), (10) and (12) - +(14) can be respectively rewritten as +p(x) = +L +� +t=1 +fE,t (xt) , +(15) +p(σ) = fG(σ, γ)fF (γ), +(16) +p(yt|xt) = p(yt|at)p(at|xt) += f 1 +A,t(at)fB,t(at, xt), t ≤ k +(17) +p (yt|xt, xt−k, σ) += p(yt|ct)p(ct|at, bt)p(at|xt)p(bt|xt−k, ht−k)p(ht−k|σ) += f 2 +A,t(ct)fD,t(at, bt, ct)fB,t (at, xt) +· fK,t−k (ht−k, xt−k, bt) fH,t−k(σ, ht−k), k < t ≤ L +(18) +and +p (yt|xt−k, σ) = p (yt|bt) p (bt|xt−k, ht−k) p(ht−k|σ) += f 3 +A,t(bt)fK,t−k (ht−k, xt−k, bt) fH,t−k(σ, ht−k), +L < k ≤ L + k. +(19) + +IEEE TRANSACTIONS ON SIGNAL PROCESSING , VOL. XX, NO. XX, 2022 +6 +Fig. 4: Factor graph representation for joint imaging and +communication problem as formulated by (20). +Then the joint distribution of x, σ, at, bt, ct, ht and γ condi- +tioned on y can be given by +p(x, σ, at, bt, ct, ht, γ|y) +∝ fE(x)fG(σ, γ)fF (γ)· +· +k +� +t=1 +f 1 +A,t(at)fB,t(at, xt) +· +L +� +t=k+1 +f 2 +A,t(ct)fD,t(at, bt, ct)fB,t(at, xt) +· fK,t−k(ht−k, xt−k, bt) · fH,t−k(σ, ht−k) +· +L+k +� +t=L+1 +f 3 +A,t(bt)fK,t−k(ht−k, xt−k, bt)fH,t−k(σ, ht−k). +(20) +Accordingly, the marginals of x and σ can be expressed by +M(x) = +� +∼{x} +p (x, σ, at, bt, ct, ht, γ | y) +(21) +and +M(σ) = +� +∼{σ} +p (x, σ, at, bt, ct, ht, γ | y) . +(22) +Based on the factorization in (20), we take k = 1 as an +example and establish the factor graph for joint communication +and imaging as shown in Fig. 4, where we drop the augments +of the functions for simplicity of notations. +It is noteworthy that though M(x) and M(σ) are given +in (21) and (22), it is difficult to maximize them due to the +high-dimensional integral. However, the factor graph in Fig.4 +provides us a possible way to find approximation marginal +functions of M(x) and M(σ), which are expected to allow +easy maximization. +C. Message Passing Based Echoes Decoupling Algorithm +Considering that the factor graph contains loops, so iterative +message passing is needed. We divide the graph into upper and +lower parts by the dotted line, where the lower part represents +the message update for x and the upper part represents the +message update for σ. To estimate x and σ jointly, the upper +and lower parts of Fig.4 are updated alternatively, and the +adaptive Sparse Bayesian learning (SBL) algorithm in [61] is +adopted to recover σ. +Based on the sum-product algorithm (SPA) [52], [53], the +belief (or marginal function) of xt can be calculated by the +product of all incoming messages to xt, i.e., +B(xt) = µfE,t→xt(xt) · µfB,t→xt(xt) · µfK,t→xt(xt). +(23) +where +µfE,t→xt = 1 +4 +4 +� +i=1 +δ[xt − ej( π +2 i− π +4 )]. +(24) +To obtain B(xt), we first need to derive messages in Fig. +4 from time index 1 to L + k successively to update +µfB,t→xt(xt). After that, we need to derive messages in +reverse order of time index, i.e., form time index L + k to +1, to update µfK,t→xt(xt). +1) When t ≤ k: Based on (1) and (12), we have +µf 1 +A,t→at(at) = N +� +at; mf 1 +A,t→at, W −1 +f 1 +A,t→at +� += N +� +at; yt, ξ2� +. +(25) +According to the structure of the factor graph, the message +µat→fB,t(at) is forward to fB,t, i.e., +µat→fB,t(at) = µf 1 +A,t→at(at) = N +� +at; yt, ξ2� +. +(26) +Based on SPA, we have +µfB,t→xt(xt) = +� +fB,t (at, xt) · µat→fB,t(at)dat. +(27) +Since µat→fB,t(at) is Gaussian message and fB,t (at, xt) +indicates a linear relationship between at and xt, µfB,t→xt(xt) +also has a Gaussian form, i.e., +µfB,t→xt(xt) = N +� +xt; mfB,t→xt, W −1 +fB,t→xt +� +, +(28) +where mfB,t→xt and WfB,t→xt are calculated by +WfB,t→xt = (αcgTΘthc)HWf 1 +A,t→at(αcgTΘthc), +(29) +and +mfB,t→xt = W −1 +fB,t→xt(αcgTΘthc)H(Wf 1 +A,t→atmf 1 +A,t→at). +(30) +The message outgoing from xt can be calculated by +µxt→fK,t(xt) = µfE,t→xt(xt) · µfB,t→xt(xt) ∝ +4 +� +i=1 +δ(xt − ej( π +2 i− π +4 )) · N +� +xt; mfB,t→xt, W −1 +fB,t→xt +� +(31) +Unfortunately, the message in (31) is no longer Gaussian, +which makes the subsequent message calculation intractable. +To this end, we approximate µxt→fK,t(xt) to a Gaussian +message ˜µxt→fK,t = N +� +xt; mxt→fK,t, Vxt→fK,t +� +by mini- + +H,1 +H,2 +H,L +h +SBL +K,L +X2 +b, +X +E,2 +E,L +E,1 +a2 +D,2 +/B,2 +a +D.3 +B,L +A,L+ +2 +A.1 +A,2 +A.,3 +t≤k +T> +L 0 and a sub-partition +µ = (µ1, . . . , µm) such that µ1 ≥ · · · ≥ µm > 0 with m < n and λi ≥ µi, for i = 1, ..., m. +The pair +(λ, µ) will be collectively referred to as a skew partition, and will typically be denoted λ/µ. The monomial +representation of G corresponding to λ/µ has basis vectors vi,j such that 1 ≤ i ≤ n and µi < j ≤ λi, where +we set µq := 0 for q > m. The action of x sends vi,j to vi+1,j and the action of y sends vi,j to vi,j+1, if such +basis elements exist, and otherwise sends them to 0. +Note that the actions of x and y commute. In order for the monomial representation associated to λ/µ to +exist (i.e., to actually be a G-representation), we require that for any i, there can only be at most 2s values +of j such that vi,j is a basis vector, and similarly with the roles of i and j reversed. +It is clear that, taking α(r, s) to be a graded Hopf algebra as in Definition 2.1.3, then a monomial +representation for α(r, s) is a graded representation, where vi,j is taken to be in degree (i, j). +Definition 2.2.2. Let V be a monomial representation of G. A monomial diagram is the diagram resulting +from drawing the grid box (i, j) if vi,j is a basis vector. Each grid box is called a cell. +In other words, the monomial diagram for the monomial representation corresponding to the skew partition +λ/µ is the same thing as the skew Young diagram of λ/µ. + +4 +GEORGE CAO AND KENT B. VASHAW +Definition 2.2.3. Let D be the monomial diagram for the monomial representation V . Then D is connected +if, for any choice of cells B1, B2, there exists a sequence of cells B1 = C1, C2, . . . , Cn = B2 for some n such +that Ci is edge-adjacent to Ci+1. +Example 2.2.4. If G is Z/4Z × Z/2Z, then the monomial representation corresponding to skew partition +(5, 4, 2, 2, 1, 1)/(3, 2) would have a basis vectors +v1,4, v1,5, v2,3, v2,4, v3,1, v3,2, v4,1, v4,2, v5,1, v6,1. +Its monomial diagram is shown in Figure 2a. If G were Z/2Z × Z/4Z, then the monomial diagram for the +representation corresponding to (4, 4, 2, 1)/(3, 1) is shown in Figure 2b. +Figure 1. Examples of monomial diagrams. +(a) +(b) +One may read the action of G on the monomial representation V directly from the monomial diagram +for V . There is a basis of V corresponding to the cells of the diagram. The action of x, in the diagram, is +moving to the right by one cell, if the cell exists, and 0 otherwise. The action of y, in the diagram, is moving +up by one cell, if the cell exists, and 0 otherwise. +A monomial representation is indecomposable if and only if its monomial diagram is connected. +In +Example 2.2.4, the representation given by Figure 2a is not indecomposable, since there are two disconnected +parts: the four white boxes in the first two columns are disconnected from the 6 white boxes in columns three +to six. This means that this representation can be written as a direct sum of a representation of dimension +4 and a representation of dimension 6. On the other hand, the monomial representations given by Figure 2b +is indecomposable. +The following lemmas establish that the dimensions of the indecomposable summands of V ⊗V ∗ and V ⊗n +are not dependent on orientation of the monomial diagram for V , but only the shape. +Lemma 2.2.5. Let V be a monomial representation. Let W be the monomial representation that results +from reflecting its monomial diagram about the line from bottom left to top right. Then, the dimensions of +the indecomposable summands of V ⊗ V ∗ are the same as the dimensions of the indecomposable summands +of W ⊗ W ∗ (up to permutation), and the dimensions of the indecomposable summands of V ⊗n are the same +as the dimensions of the indecomposable summands of W ⊗n (up to permutation). +Proof. Under the isomorphism α(r, s) ∼= α(r, s) which interchanges x and y, we have that V corresponds to +W, and vice versa. +□ +Lemma 2.2.6. Let V be a monomial representation. The monomial representation that results from rotating +its monomial diagram 180◦ gives a monomial representation isomorphic to V ∗. +Proof. Let V be a monomial representation with basis vectors v1, . . . , vn in the cells of its monomial diagram. +Let W be the monomial representation resulting from a 180◦ rotation of the monomial diagram of V . Let +the basis vectors of W be w1, . . . , wn such that they are the image of v1, . . . , vn, respectively, under this +rotation. +Consider ϕ : W → V ∗ defined by ϕ(wi)vj = 1 if i = j and 0 if i ̸= j. This is an isomorphism of vector +spaces, so to show that it is an isomorphism of representations we simply check that it respects the action +of G. We have +ϕ(ρW (x)wi)vj = +� +1 +if ρW (x)wi = wj +0 +otherwise, +(ρV ∗(x) · ϕ(wi))vj = ϕ(wi)(ρV (x)vj) = +� +1 +if ρV (x)vj = vi +0 +otherwise. + +ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS +5 +We have ρW (x)wi = wj if and only if wj is to the right of wi (in the monomial diagram of W). This is true +if and only if vj is to the left of vi, which is true if and only if ρV (x)vj = vi. Thus, +ϕ(ρW (x)wi)vj = (ρV ∗(x) · ϕ(wi))vj +for all i, j, so ϕ is a homomorphism of representations. Therefore, ϕ must be an isomorphism of representa- +tions, as desired. +□ +2.3. Computed data. Using the computer algebra system Magma [8], we can generate all possible mono- +mial diagrams of monomial representations V for a selected dimension of V , and compute the dimensions +of the indecomposable summands of V ⊗ V ∗. Table 1, Table 2, and Table 3 contain all monomial diagrams +for monomial representations of the specified dimension, along with the dimensions of the indecomposable +summands of V ⊗ V ∗. Monomial diagrams that are the same shape but different orientation are omitted. +All of these examples satisfy Conjecture 2.1.2. No pattern between the dimensions of the indecomposable +summands and the shape of the monomial diagram has been conjectured yet. +In fact, for each example in the table below, additional data was collected using the group tensor product, +and the dimensions of the indecomposable summands were the same as the same as using the tensor product +with the comultiplication structure. We formulate this conjecture: +Conjecture 2.3.1. Let V be an odd-dimensional indecomposable monomial representation of G = Z/2rZ × +Z/2sZ. Let ⊗ and (−)∗ denote the tensor product and dual of G-representations defined using the Hopf +algebra structure for α(r, s), and ⊗′ and (−)∨ denote the tensor product and dual of G-representations using +the standard Hopf algebra structure on kG. We conjecture that there are isomorphisms of G-representations +V ⊗ V ∗ ∼= V ⊗′ V ∨, V ⊗n ∼= V ⊗′n. +Table 1. Dimension 3 monomial representations. +Dimensions +Monomial Diagrams +[1, 4, 4] +Table 2. Dimension 5 monomial representations. +Dimensions +Monomial Diagrams +[1, 12, 12] +[1, 4, 4, 8, 8] +[1, 4, 4, 4, 4, 4, 4] + +6 +GEORGE CAO AND KENT B. VASHAW +Table 3. Dimension 7 monomial representations. +Dimensions +Monomial Diagrams +[1, 4, 4, 20, 20] +[1, 8, 8, 16, 16] +[1, 24, 24] +[1, 48] +[1, 8, 8, 8, 8, 8, 8] +[1, 4, 4, 4, 4, 8, 8, 8, 8] +[1, 4, 4, 4, 4, 16, 16] +[1, 4, 4, 40] +[1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4] + +ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS +7 +3. Benson’s tensor powers conjecture for monomial modules +In this section, we consider Conjecture 1.0.2 for α(r, s). +3.1. Basic results. We use the notation given in the introduction. Let V be an odd-dimensional monomial +module for α(r, s). +We assume that Conjecture 1.0.1 holds for α(r, s), so that V ⊗i has a unique odd- +dimensional indecomposable summand, which we denote by Vi. Recall that PV (n) will denote the dimension +of Vn. +We may also characterize Vi in the following recursive way: +Proposition 3.1.1. There is a unique odd-dimensional indecomposable summand of Vi ⊗ V , which is iso- +morphic to Vi+1. +Proof. We have V ⊗n = V ⊗(n−1) ⊗ V = (Vn−1 ⊕ W1 ⊕ · · · ⊕ Wm) ⊗ V , where dim Wi is even for all i. Then, +V ⊗n = (Vn−1 ⊗ V ) ⊕ (W1 ⊗ V ) ⊕ · · · ⊕ (Wm ⊗ V ). The only odd-dimensional indecomposable summand of +this is Vn, by assumption. Since dim(Vn−1 ⊗ V ) is odd, then Vn must be an indecomposable summand of +Vn−1 ⊗ V . +□ +For self-dual modules, it is straightforward to compute PV (n), as follows: +Proposition 3.1.2. If the monomial diagram of V is symmetric by rotation of 180◦, then Vodd = V and +Veven = k. In particular, +PV (n) = +� +dim V +if n odd +1 +if n even. +Proof. By Lemma 2.2.6, V is self-dual. Recall that the trivial representation k is a summand of +V ⊗ V ∗ ∼= V ⊗ V , so PV (2) = 1. Then V2 ⊗ V = k ⊗ V = V , so V3 = V . Thus PV (3) = dim V . By induction, +we find that Vodd = V and Veven = k. +□ +3.2. “Staircase” monomial representations. We next consider the monomial representations for α(1, 1). +We refer to these representations in the following way, based on their diagrams’ appearance. +Definition 3.2.1. Call the monomial representation given by the partitions +(m, m − 1, . . . , 1)/(m − 2, m − 3, . . . , 1) the m-staircase monomial representation. +From the classification of all indecomposable summands of Z/2Z × Z/2Z in [2, pp. 176], we know that +Ωm(k) are these odd-dimensional indecomposable representations, where Ωm(V ) is the m-th syzygy, defined +as the kernel of the m-th map in the projective resolution of V . We can calculate these syzygies and it can +be shown that the m-staircase monomial representation is Ω1−m(k). To examine tensor powers, we need two +lemmas. The following lemma is found in [3, Corollary 3.1.6] in the finite group case, and the same proof +holds for any finite group scheme, in particular α(r, s): +Lemma 3.2.2. If V and W are representations of kG for a finite group G, then Ω(V ) ⊗ W is isomorphic +to Ω(V ⊗ W), up to projective summands. +Also, we need the following lemma given in [18, Corollary 8.1.3]: +Lemma 3.2.3. If k is an algebraically closed field of characteristic 2 and 2 divides |G|, then all projective +representations of G have even dimension. +We can prove the following proposition about the tensor powers of staircase monomial representations: +Proposition 3.2.4. If V is the m-staircase representation, then Vn is the (mn − n + 1)-staircase monomial +representation. +Proof. We have that V = Ω1−m(k). Then V ⊗n ∼= (Ω1−m(k))⊗n. By Lemma 3.2.2, this is isomorphic, up +to projective summands, to Ω(1−m)n(k⊗n) = Ωn−mn(k). By Lemma 3.2.3, all projective representations +are even dimensional, so Vn, the odd-dimensional summand of V ⊗n, is also the odd-dimensional summand +of Ωn−mn(k). However, Ωn−mn(k) is the (mn − n + 1)-staircase monomial representation, which has odd- +dimension and is indecomposable. Thus Vn is the (mn − n + 1)-staircase monomial representation. +□ +It immediately follows that Conjecture 1.0.2 is satisfied in this case. + +8 +GEORGE CAO AND KENT B. VASHAW +Corollary 3.2.5. If V is the m-staircase module, then the function PV (n) is given by the linear polynomial +(2m − 2)n + 1. +Remark. It is shown in [4] that Conjecture 1.0.1 is true for G = Z/2Z × Z/2Z in general. +3.3. (4, 1) Monomial representation. Let V be the monomial representation of Z/2Z × Z/4Z (or α(1, 2)) +corresponding to the partition (4, 1). +Proposition 3.3.1. We have the following decomposition into indecomposable summands: +V2k ⊗ V = V2k+1 ⊕ F ⊕ · · · ⊕ F +� +�� +� +4k copies +and +V2k−1 ⊗ V = V2k ⊕ W ⊕ W ⊕ F ⊕ · · · ⊕ F +� +�� +� +4k−3 copies +, +where F is a free module of dimension 8 and W is dimension 4. +In order to prove this, we explicitly give the decomposition, via the following stronger lemma. +Lemma 3.3.2. We claim the following. +(1) The representation V2k is the direct sum of 1-dimensional homogeneous components V4k,2k and V2k−1+i,j +where j = 6k − 1 − 2i, . . ., 6k + 2 − 2i for i = 1, 3, 5, . . ., 2k − 1 and +j = 6k − 2i, . . . , 6k + 3 − 2i for i = 2, 4, 6, . . ., 2k. +(2) The representation V2k−1 is the direct sum of 1-dimensional homogeneous components V4k−2,2k−1 and +V2k−2+i,j where j = 6k − 3 − 2i, . . ., 6k − 2i for i = 1, 3, 5, . . ., 2k − 1 and +j = 6k − 4 − 2i, . . . , 6k − 1 − 2i for i = 2, 4, 6, . . ., 2k − 2. +(3) The representation V2k ⊗ V decomposes into the summands whose graded monomial diagrams are shown +in Table 4. +(4) The representation V2k−1 ⊗ V decomposes into the summands whose graded monomial diagrams are +shown in Table 5. +Table 4. Indecomposable summands of V2k ⊗ V . +V2k+1 summand: +Dimension 8 Family 1 summand: +v2k,6k ⊗ v1,1 +v2k,6k ⊗ v1,2 +v2k,6k ⊗ v1,3 +v2k,6k ⊗ v1,4 +V2k ⊗ v2,1 +v2k−1+i,6k−2i ⊗ v1,1 +v2k+i,6k−2i ⊗ v1,1 ++ v2k−1+i,6k−2i ⊗ v2,1 +v2k−1+i,6k+1−2i ⊗ v1,1 ++ v2k−1+i,6k−2i ⊗ v1,2 +v2k−1+i,6k+1−2i ⊗ v2,1 ++ v2k+i,6k−2i ⊗ v1,2 +v2k−1+i,6k+2−2i ⊗ v1,1 ++ v2k−1+i,6k−2i ⊗ v1,3 +v2k−1+i,6k+2−2i ⊗ v2,1 ++ v2k+i,6k−2i ⊗ v1,3 +v2k−1+i,6k+3−2i ⊗ v1,1 ++ v2k−1+i,6k+2−2i ⊗ v1,2 ++ v2k−1+i,6k+1−2i ⊗ v1,3 ++ v2k−1+i,6k−2i ⊗ v1,4 +v2k−1+i,6k+3−2i ⊗ v2,1 ++ v2k+i,6k−2i ⊗ v1,4 +for i = 2, 4, 6, . . ., 2k +Dimension 8 Family 2 summand: +Dimension 8 Family 3 summand: +Dimension 8 Family 4 summand: +v2k−1+i,6k−1−2i ⊗ v1,1 +v2k+i,6k−1−2i ⊗ v1,1 ++ v2k−1+i,6k−1−2i ⊗ v2,1 +v2k−1+i,6k−2i ⊗ v1,1 ++ v2k−1+i,6k−1−2i ⊗ v1,2 +v2k+i,6k−1−2i ⊗ v1,2 ++ v2k+i,6k−2i ⊗ v1,1 ++ v2k−1+i,6k−2i ⊗ v2,1 +v2k−1+i,6k+1−2i ⊗ v1,1 ++ v2k−1+i,6k−1−2i ⊗ v1,3 +v2k+i,6k+1−2i ⊗ v1,1 ++ v2k+i,6k−1−2i ⊗ v1,3 ++ v2k−1+i,6k+1−2i ⊗ v2,1 +v2k−1+i,6k+2−2i ⊗ v1,1 ++ v2k−1+i,6k+1−2i ⊗ v1,2 ++ v2k−1+i,6k−2i ⊗ v1,3 ++ v2k−1+i,6k−1−2i ⊗ v1,4 +v2k−1+i,6k+2−2i ⊗ v2,1 ++ v2k+i,6k+1−2i ⊗ v1,2 ++ v2k+i,6k−2i ⊗ v1,3 ++ v2k+i,6k−1−2i ⊗ v1,4 +v2k−1+i,6k−2i ⊗ v1,1 +v2k+i,6k−2i ⊗ v1,1 ++ v2k−1+i,6k−2i ⊗ v2,1 +v2k−1+i,6k+1−2i ⊗ v1,1 ++ v2k−1+i,6k−2i ⊗ v1,2 +v2k+i,6k+1−2i ⊗ v1,1 ++ v2k+i,6k−2i ⊗ v1,2 ++ v2k−1+i,6k+1−2i ⊗ v2,1 +v2k−1+i,6k+2−2i ⊗ v1,1 ++ v2k−1+i,6k−2i ⊗ v1,3 +v2k+i,6k−2i ⊗ v1,3 ++ v2k−1+i,6k+2−2i ⊗ v2,1 +v2k−1+i,6k+2−2i ⊗ v1,2 ++ v2k−1+i,6k+1−2i ⊗ v1,3 ++ v2k−1+i,6k−2i ⊗ v1,4 +v2k+i,6k+1−2i ⊗ v1,3 ++ v2k+i,6k−2i ⊗ v1,4 +v2k−1+i,6k+1−2i ⊗ v1,1 +v2k+i,6k+1−2i ⊗ v1,1 ++ v2k−1+i,6k+1−2i ⊗ v2,1 +v2k−1+i,6k+2−2i ⊗ v1,1 ++ v2k−1+i,6k+1−2i ⊗ v1,2 +v2k+i,6k+1−2i ⊗ v1,2 ++ v2k−1+i,6k+2−2i ⊗ v2,1 +v2k−1+i,6k+1−2i ⊗ v1,3 +v2k+i,6k+1−2i ⊗ v1,3 +v2k−1+i,6k+2−2i ⊗ v1,3 ++ v2k−1+i,6k+1−2i ⊗ v1,4 +v2k+i,6k+1−2i ⊗ v1,4 +for i = 1, 3, 5, . . ., 2k − 1 +for i = 1, 3, 5, . . ., 2k − 1 +for i = 1, 3, 5, . . ., 2k − 1 + +ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS +9 +Table 5. Indecomposable summands of V2k−1 ⊗ V . +V2k summand: +W1 summand: +W2 summand: +Dimension 8 Family 1 summand: +v2k−1,6k−4 ⊗ v1,1 +v2k−1,6k−4 ⊗ v1,2 ++ v2k−1,6k−3 ⊗ v1,1 +v2k−1,6k−4 ⊗ v1,3 ++ v2k−1,6k−2 ⊗ v1,1 +v2k−1,6k−2 ⊗ v1,2 ++ v2k−1,6k−3 ⊗ v1,3 ++ v2k−1,6k−4 ⊗ v1,4 +V2k−1 ⊗ v2,1 +v2k−1,6k−4 ⊗ v1,2 +v2k−1,6k−3 ⊗ v1,2 ++ v2k−1,6k−4 ⊗ v1,3 +v2k−1,6k−2 ⊗ v1,2 ++ v2k−1,6k−4 ⊗ v1,4 +v2k−1,6k−2 ⊗ v1,3 ++ v2k−1,6k−3 ⊗ v1,4 +v2k−1,6k−3 ⊗ v1,2 +v2k−1,6k−2 ⊗ v1,2 ++ v2k−1,6k−3 ⊗ v1,3 +v2k−1,6k−3 ⊗ v1,4 +v2k−1,6k−2 ⊗ v1,4 +v2k−2+i,6k−3−2i ⊗ v1,1 +v2k−1+i,6k−3−2i ⊗ v1,1 ++ v2k−2+i,6k−3−2i ⊗ v2,1 +v2k−2+i,6k−2−2i ⊗ v1,1 ++ v2k−2+i,6k−3−2i ⊗ v1,2 +v2k−2+i,6k−2−2i ⊗ v2,1 ++ v2k−1+i,6k−3−2i ⊗ v1,2 +v2k−2+i,6k−1−2i ⊗ v1,1 ++ v2k−2+i,6k−3−2i ⊗ v1,3 +v2k−2+i,6k−1−2i ⊗ v2,1 ++ v2k−1+i,6k−3−2i ⊗ v1,3 +v2k−2+i,6k−2i ⊗ v1,1 ++ v2k−2+i,6k−1−2i ⊗ v1,2 ++ v2k−2+i,6k−2−2i ⊗ v1,3 ++ v2k−2+i,6k−3−2i ⊗ v1,4 +v2k−2+i,6k−2i ⊗ v2,1 ++ v2k−1+i,6k−3−2i ⊗ v1,4 +for i = 1, 3, 5, . . ., 2k − 1 +Dimension 8 Family 2 summand: +Dimension 8 Family 3 summand: +Dimension 8 Family 4 summand: +v2k−2+i,6k−4−2i ⊗ v1,1 +v2k−1+i,6k−4−2i ⊗ v1,1 ++ v2k−2+i,6k−4−2i ⊗ v2,1 +v2k−2+i,6k−3−2i ⊗ v1,1 ++ v2k−2+i,6k−4−2i ⊗ v1,2 +v2k−1+i,6k−3−2i ⊗ v1,1 ++ v2k−1+i,6k−4−2i ⊗ v1,2 ++ v2k−2+i,6k−3−2i ⊗ v2,1 +v2k−2+i,6k−2−2i ⊗ v1,1 ++ v2k−2+i,6k−4−2i ⊗ v1,3 +v2k−1+i,6k−2−2i ⊗ v1,1 ++ v2k−1+i,6k−4−2i ⊗ v1,3 ++ v2k−2+i,6k−2−2i ⊗ v2,1 +v2k−2+i,6k−1−2i ⊗ v1,1 ++ v2k−2+i,6k−2−2i ⊗ v1,2 ++ v2k−2+i,6k−3−2i ⊗ v1,3 ++ v2k−2+i,6k−4−2i ⊗ v1,4 +v2k−2+i,6k−1−2i ⊗ v2,1 ++ v2k−1+i,6k−2−2i ⊗ v1,2 ++ v2k−1+i,6k−3−2i ⊗ v1,3 ++ v2k−1+i,6k−4−2i ⊗ v1,4 +v2k−2+i,6k−3−2i ⊗ v1,1 +v2k−1+i,6k−3−2i ⊗ v1,1 ++ v2k−2+i,6k−3−2i ⊗ v2,1 +v2k−2+i,6k−2−2i ⊗ v1,1 ++ v2k−2+i,6k−3−2i ⊗ v1,2 +v2k−1+i,6k−2−2i ⊗ v1,1 ++ v2k−1+i,6k−3−2i ⊗ v1,2 ++ v2k−2+i,6k−2−2i ⊗ v2,1 +v2k−2+i,6k−1−2i ⊗ v1,1 ++ v2k−2+i,6k−3−2i ⊗ v1,3 +v2k−1+i,6k−3−2i ⊗ v1,3 ++ v2k−2+i,6k−1−2i ⊗ v2,1 +v2k−2+i,6k−1−2i ⊗ v1,2 ++ v2k−2+i,6k−2−2i ⊗ v1,3 ++ v2k−2+i,6k−3−2i ⊗ v1,4 +v2k−1+i,6k−2−2i ⊗ v1,3 ++ v2k−1+i,6k−3−2i ⊗ v1,4 +v2k−2+i,6k−2−2i ⊗ v1,1 +v2k−1+i,6k−2−2i ⊗ v1,1 ++ v2k−2+i,6k−2−2i ⊗ v2,1 +v2k−2+i,6k−1−2i ⊗ v1,1 ++ v2k−2+i,6k−2−2i ⊗ v1,2 +v2k−1+i,6k−2−2i ⊗ v1,2 ++ v2k−2+i,6k−1−2i ⊗ v2,1 +v2k−2+i,6k−2−2i ⊗ v1,3 +v2k−1+i,6k−2−2i ⊗ v1,3 +v2k−2+i,6k−1−2i ⊗ v1,3 ++ v2k−2+i,6k−2−2i ⊗ v1,4 +v2k−1+i,6k−2−2i ⊗ v1,4 +for i = 2, 4, 6, . . ., 2k − 2 +for i = 2, 4, 6, . . ., 2k − 2 +for i = 2, 4, 6, . . ., 2k − 2 +Proof. We proceed by induction. It can be checked that (2) is true for k = 1. +For the inductive step, assume (2) is true. Consider the decomposition given in (4). First, we show that +each of these summands are subrepresentations. It can be checked in each case that the vectors chosen in +the diagrams above are chosen such that the action of x takes the displayed vector to the vector in the +box adjacent to the right (or 0 if that box does not exist) and the action of y takes the displayed vector to +the vector in the box adjacent above (or 0 if that box does not exist). Since each vector is a basis for the +homogeneous component it is in, then each diagram is closed under the action of x and y, and all summands +are subrepresentations. +Each of the claimed summands is indecomposable, since all of the diagrams are connected. +Finally, we must show that the direct sum of the indecomposable subrepresentations shown is the original +representation V2k−1 ⊗ V . First, we must know where the summands are located relative to each other. +Figure 3 shows the dimensions of the homogeneous components of V , V1 ⊗ V , V2 ⊗ V , V3 ⊗ V , and V4 ⊗ V , +respectively from left to right. The black outline is the location of Vi, the diamonds show the positions of +the bottom left cells of the dimension 8 free modules, and the circles show the positions of the bottom cells +of the dimension 4 summands, as shown in the key. This pattern is generalized to higher n. + +10 +GEORGE CAO AND KENT B. VASHAW +Figure 3. Positions of the indecomposable summands in the first five tensor powers. +1 +1 +1 +1 +1 +1 +2 +2 +2 +2 +1 +2 +3 +4 +3 +2 +1 +1 +2 +2 +2 +2 +1 +3 +4 +5 +4 +2 +1 +1 +2 +3 +4 +3 +2 +1 +1 +2 +2 +2 +2 +1 +3 +4 +5 +4 +2 +1 +1 +2 +3 +5 +4 +3 +2 +1 +2 +3 +4 +3 +2 +1 +1 +2 +2 +2 +2 +1 +3 +4 +5 +4 +2 +1 +1 +2 +3 +5 +4 +3 +2 +1 +3 +4 +5 +4 +2 +1 +1 +2 +3 +4 +3 +2 +1 +Key +W1 +W2 +Family 1 +Family 2 +Family 3 +Family 4 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +For every homogeneous component of V2k−1 ⊗ V , we can check that the vectors in the shown summands +span the vector space. For example, it can be checked that +{v2k−1,6k−2 ⊗ v1,2 + v2k−1,6k−3 ⊗ v1,3 + v2k−1,6k−4 ⊗ v1,4, +v2k−1,6k−2 ⊗ v1,2 + v2k−1,6k−3 ⊗ v1,3, +v2k−1,6k−2 ⊗ v1,2 + v2k−1,6k−4 ⊗ v1,4} +spans the homogeneous component of degree (2k, 6k). We do similar verifications for the other homogeneous +components, which is omitted from this proof. This means that (4) follows from (2). By a similar argument, +it can be shown that (3) follows from (1). Considering the degrees of the homogeneous components show +that (1) and (2) follow from (4) and (3), respectively. By induction, the lemma follows. +□ +Proposition 3.3.1 follows from this lemma, and as a corollary we observe that Conjecture 1.0.2 is satisfied +in this case. +Corollary 3.3.3. The function PV (n) is given by 4n + 1, a linear polynomial. +3.4. (4, 1) Monomial representation using syzygies. As in the previous section, let V be the monomial +representation given by the partition (4, 1) and let G = Z/2Z × Z/4Z. The monomial diagram of V and the +monomial diagram for the free G-module of rank 1 are given, respectively: +, +. +Since the free G-module of rank 1 is indecomposable, all projective modules are free modules. The syzygy +for V appears as the first module in the following short exact sequence: +0 +→ +→ +→ +→ +0. +Define N = Ω(V ). Then V = Ω−1(N). By Lemma 3.2.2, we have that V ⊗n ∼= (Ω−1(N))⊗n is isomorphic, +up to projective summands, to Ω−n(N ⊗n). By Lemma 3.2.3, all projective representations are even dimen- +sional, so Vn, the odd-dimensional summand of V ⊗n, is also the odd-dimensional summand of Ω−n(N ⊗n). + +ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS +11 +Thus, to find Vn, we must find the unique odd-dimensional indecomposable summand of N ⊗n and take n +cosyzygies. +We have that N is the monomial representation given by the partition (3). By Proposition 3.1.2, the +odd-dimensional indecomposable summand of N ⊗odd is isomorphic to N, and the odd-dimensional inde- +composable summand of N ⊗even is isomorphic to k. Thus computing Vn reduces to computing the unique +odd-dimensional summands of Ω−n(N) for odd n and Ω−n(k) for even n, which is a much simpler task. +This technique relies on the fact that Ω(V ) is a simpler representation than V . The same technique +works for partitions of the form (2m, 1), where we get PV (n) = 2mx + 1. However, for the representations +considered in Sections 3.5 and 3.6, taking syzygies does not simplify the picture, and this technique cannot +be applied. +3.5. (3, 1, 1) Monomial representation. Let V be the monomial representation of Z/4Z×Z/4Z (or α(2, 2)) +corresponding to the partition (3, 1, 1). +The syzygy method in Section 3.4 does not work for this monomial representation, since Ω(V ) is not easier +to analyze than V . However, we are still able to obtain an explicit decomposition, via a similar analysis +as Section 3.3, giving one of the first nontrivial examples where Benson’s tensor powers conjecture may be +verified. +Proposition 3.5.1. We have the following decomposition into indecomposable summands: +V2k−1 ⊗ V = V2k ⊕ W12 ⊕ · · · ⊕ W12 +� +�� +� +k copies +⊕ F ⊕ · · · ⊕ F +� +�� +� +3k−3 copies +⊕ W28 ⊕ · · · ⊕ W28 +� +�� +� +k−1 copies +, +V2k ⊗ V = V2k+1 ⊕ W20 ⊕ · · · ⊕ W20 +� +�� +� +2k copies +, +where dim W20 = 20, dim W12 = 12, dim W28 = 28, and F is a free module of dimension 16. +Again, we explicitly write the decomposition with a long but stronger lemma. +Lemma 3.5.2. We claim the following. +(1) The representation V2k+1 for k ≥ 1 is given by the graded monomial diagram in Table 6. +(2) The representation V2k for k ≥ 1 is given by the graded monomial diagram in Table 7. +(3) The representation V2k−1 ⊗ V decomposes into the following summands: V2k (given in Table 8); dimen- +sion 12 summands (given in Table 9); dimension 16 summands (given in Table 10); and dimension 28 +summands (given in Table 11). +(4) The representation V2k ⊗ V decomposes into the following summands: V2k+1 (given in Table 12) and +dimension 20 summands (given in Table 13). +Table 6. The pieces of V2k+1. These pieces show the nonzero actions of x and y, and they +overlap each other. They are arranged as in Figure 5 for V7. +(A, i) piece: +(B, i) piece: +vA,i +2k−1+2i,4k+1−2i +vA,i +2k−1+2i,4k+2−2i +vA,i +2k−1+2i,4k+3−2i +vA,i +2k−1+2i,4k+4−2i +vA,i +2k+2i,4k+1−2i +vA,i +2k+2i,4k+2−2i +vA,i +2k+2i,4k+3−2i +vA,i +2k+2i,4k+4−2i +vA,i +2k+1+2i,4k+1−2i +vA,i +2k+1+2i,4k+2−2i +vA,i +2k+1+2i,4k+3−2i +vA,i +2k+1+2i,4k+4−2i +vA,i +2k+2+2i,4k+1−2i +vA,i +2k+2+2i,4k+2−2i +vA,i +2k+2+2i,4k+3−2i +vB,i +2k−1+2i,4k+3−2i +vB,i +2k−1+2i,4k+4−2i +vB,i +2k−1+2i,4k+5−2i +vA,i−1 +2k−1+2i,4k+6−2i +vB,i +2k+2i,4k+3−2i +vB,i +2k+1+2i,4k+3−2i +vA,i +2k+2+2i,4k+3−2i +for i = 1, 2, . . . , k +for i = 2, 3, . . ., k +(B, 1) piece: +(B, k + 1) piece: +vB,1 +2k+1,4k+1 +vB,1 +2k+1,4k+2 +vB,1 +2k+1,4k+3 +vB,1 +2k+2,4k+1 +vB,1 +2k+3,4k+1 +vA,1 +2k+4,4k+1 +vB,k+1 +4k+1,2k+1 +vB,k+1 +4k+1,2k+2 +vB,k+1 +4k+1,2k+3 +vA,k +4k+1,2k+4 +vB,k+1 +4k+2,2k+1 +vB,k+1 +4k+3,2k+1 + +12 +GEORGE CAO AND KENT B. VASHAW +Figure 5. The location of the pieces for V7. The outlines show the (A, i) pieces, and the +shaded regions show the (B, i) pieces. +1 1 2 2 1 +1 1 2 1 +1 1 3 3 3 1 +1 1 3 2 1 +1 1 3 3 3 1 +1 1 3 2 +1 +1 +2 2 3 1 +2 +1 +1 +Table 7. The pieces of V2k. These pieces show the nonzero actions of x and y, and they +overlap each other. They are arranged as in Figure 6 for V6. +(A, i) piece: +(B, i) piece: +vi +2k−2+2i,4k−2i +vi +2k−2+2i,4k+1−2i +vi +2k−2+2i,4k+2−2i +vi +2k−2+2i,4k+3−2i +vi +2k−1+2i,4k−2i +vi +2k−1+2i,4k+1−2i +vi +2k−1+2i,4k+2−2i +vi +2k+2i,4k−2i +vi +2k+2i,4k+1−2i +vi +2k+2i,4k+2−2i +vi +2k+1+2i,4k−2i +v2k+2i,4k−2i +vi+1 +2k+2i,4k+1−2i +vi +2k+1+2i,4k−2i +for i = 1, 2, . . ., k +for i = 1, 2, . . . , k − 1 +(C, 1) piece: +(C, k) piece: +v2k,4k +v1 +2k,4k+1 +v4k,2k +vk +4k+1,2k +Figure 6. The location of the pieces for V6. The outlines show the (A, i) pieces, and the +shaded regions show the (B, i), (C, 1), and (C, k) pieces. +1 1 2 1 +1 1 1 +1 1 3 2 1 +1 1 2 +1 1 3 2 1 +1 1 2 +2 1 1 +1 + +ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS +13 +Table 8. The pieces of V2k. +(A, i) piece: +(B, i) piece: +vB,i +2k−3+2i,4k−1−2i ⊗ v1,1 ++ vA,i +2k−3+2i,4k−2−2i ⊗ v1,2 ++ vA,i−1 +2k−4+2i,4k−1−2i ⊗ v2,1 ++ vA,i +2k−3+2i,4k−3−2i ⊗ v1,3 ++ vA,i−1 +2k−5+2i,4k−1−2i ⊗ v3,1 +vB,i +2k−2+2i,4k−1−2i ⊗ v1,1 ++ vB,i +2k−3+2i,4k−1−2i ⊗ v2,1 ++ vA,i +2k−2+2i,4k−2−2i ⊗ v1,2 ++ vA,i−1 +2k−3+2i,4k−1−2i ⊗ v2,1 ++ vA,i +2k−2+2i,4k−3−2i ⊗ v1,3 +vB,i +2k−1+2i,4k−1−2i ⊗ v1,1 ++ vB,i +2k−3+2i,4k−1−2i ⊗ v3,1 ++ vA,i +2k−1+2i,4k−2−2i ⊗ v1,2 ++ vA,i−1 +2k−2+2i,4k−1−2i ⊗ v2,1 ++ vA,i +2k−1+2i,4k−3−2i ⊗ v1,3 ++ vA,i−1 +2k−3+2i,4k−1−2i ⊗ v3,1 +vA,i +2k+2i,4k−1−2i ⊗ v1,1 ++ vB,i +2k−1+2i,4k−1−2i ⊗ v2,1 ++ vB,i +2k−2+2i,4k−1−2i ⊗ v3,1 ++ vA,i +2k+2i,4k−2−2i ⊗ v1,2 ++ vA,i +2k+2i,4k−3−2i ⊗ v1,3 +vB,i +2k−3+2i,4k−2i ⊗ v1,1 ++ vB,i +2k−3+2i,4k−1−2i ⊗ v1,2 ++ vA,i +2k−3+2i,4k−1−2i ⊗ v1,2 ++ vA,i−1 +2k−4+2i,4k−2i ⊗ v2,1 ++ vA,i−1 +2k−5+2i,4k−2i ⊗ v3,1 +vB,i +2k−3+2i,4k−2i ⊗ v2,1 ++ vB,i +2k−2+2i,4k−1−2i ⊗ v1,2 ++ vA,i +2k−2+2i,4k−1−2i ⊗ v1,2 ++ vA,i−1 +2k−3+2i,4k−2i ⊗ v2,1 +vB,i +2k−3+2i,4k−2i ⊗ v3,1 ++ vB,i +2k−1+2i,4k−1−2i ⊗ v1,2 ++ vA,i +2k−1+2i,4k−1−2i ⊗ v1,2 ++ vA,i−1 +2k−2+2i,4k−2i ⊗ v2,1 ++ vA,i−1 +2k−3+2i,4k−2i ⊗ v3,1 +vB,i +2k−3+2i,4k+1−2i ⊗ v1,1 ++ vB,i +2k−3+2i,4k−1−2i ⊗ v1,3 ++ vA,i +2k−3+2i,4k−2i ⊗ v1,2 ++ vA,i−1 +2k−4+2i,4k+1−2i ⊗ v2,1 ++ vA,i +2k−3+2i,4k−1−2i ⊗ v1,3 ++ vA,i−1 +2k−5+2i,4k+1−2i ⊗ v3,1 +vB,i +2k−3+2i,4k+1−2i ⊗ v2,1 ++ vB,i +2k−2+2i,4k−1−2i ⊗ v1,3 ++ vA,i +2k−2+2i,4k−2i ⊗ v1,2 ++ vA,i−1 +2k−3+2i,4k+1−2i ⊗ v2,1 ++ vA,i +2k−2+2i,4k−1−2i ⊗ v1,3 +vB,i +2k−3+2i,4k+1−2i ⊗ v3,1 ++ vB,i +2k−1+2i,4k−1−2i ⊗ v1,3 ++ vA,i +2k−1+2i,4k−2i ⊗ v1,2 ++ vA,i−1 +2k−2+2i,4k+1−2i ⊗ v2,1 ++ vA,i +2k−1+2i,4k−1−2i ⊗ v1,3 ++ vA,i−1 +2k−3+2i,4k+1−2i ⊗ v3,1 +vA,i−1 +2k−3+2i,4k+2−2i ⊗ v1,1 ++ vB,i +2k−3+2i,4k+1−2i ⊗ v1,2 ++ vB,i +2k−3+2i,4k−2i ⊗ v1,3 ++ vA,i−1 +2k−4+2i,4k+2−2i ⊗ v2,1 ++ vA,i−1 +2k−5+2i,4k+2−2i ⊗ v3,1 +vA,i +2k−1+2i,4k−1−2i ⊗ v1,1 ++ vA,i +2k−1+2i,4k−2−2i ⊗ v1,2 ++ vB,i+1 +2k−1+2i,4k−2−2i ⊗ v1,2 ++ vA,i +2k−1+2i,4k−3−2i ⊗ v1,3 ++ vA,i +2k−2+2i,4k−1−2i ⊗ v2,1 ++ vB,i +2k−2+2i,4k−1−2i ⊗ v2,1 ++ vA,i +2k−3+2i,4k−1−2i ⊗ v3,1 +vA,i +2k+2i,4k−1−2i ⊗ v1,1 ++ vB,i +2k−1+2i,4k−1−2i ⊗ v2,1 ++ vB,i +2k−2+2i,4k−1−2i ⊗ v3,1 ++ vA,i +2k+2i,4k−2−2i ⊗ v1,2 ++ vA,i +2k+2i,4k−3−2i ⊗ v1,3 +vA,i +2k−1+2i,4k−2i ⊗ v1,1 ++ vB,i+1 +2k−1+2i,4k−1−2i ⊗ v1,2 ++ vB,i+1 +2k−1+2i,4k−2−2i ⊗ v1,3 ++ vA,i +2k−2+2i,4k−2i ⊗ v2,1 ++ vA,i +2k−3+2i,4k−2i ⊗ v3,1 +for i = 2, 3, . . . , k − 1 +for i = 1, 2, 3, . . ., k − 1 +(A, k) piece: +(C, k) piece: +vB,k +4k−3,2k−1 ⊗ v1,1 ++ vA,k−1 +4k−4,2k−1 ⊗ v2,1 ++ vA,k−1 +4k−5,2k−1 ⊗ v3,1 +vB,k +4k−2,2k−1 ⊗ v1,1 ++ vB,k +4k−3,2k−1 ⊗ v2,1 ++ vA,k−1 +4k−3,2k−1 ⊗ v2,1 +vB,k +4k−1,2k−1 ⊗ v1,1 ++ vB,k +4k−3,2k−1 ⊗ v3,1 ++ vA,k−1 +4k−2,2k−1 ⊗ v2,1 ++ vA,k−1 +4k−3,2k−1 ⊗ v3,1 +vB,k +4k−1,2k−1 ⊗ v2,1 ++ vB,k +4k−2,2k−1 ⊗ v3,1 +vB,k +4k−3,2k ⊗ v1,1 ++ vB,k +4k−3,2k−1 ⊗ v1,2 ++ vA,k−1 +4k−4,2k ⊗ v2,1 ++ vA,k−1 +4k−5,2k ⊗ v3,1 +vB,k +4k−2,2k−1 ⊗ v1,2 ++ vB,k +4k−3,2k ⊗ v2,1 ++ vA,k−1 +4k−3,2k ⊗ v2,1 +vB,k +4k−1,2k−1 ⊗ v1,2 ++ vB,k +4k−3,2k ⊗ v3,1 ++ vA,k−1 +4k−2,2k ⊗ v2,1 ++ vA,k−1 +4k−3,2k ⊗ v3,1 +vB,k +4k−3,2k+1 ⊗ v1,1 ++ vB,k +4k−3,2k−1 ⊗ v1,3 ++ vA,k−1 +4k−4,2k+1 ⊗ v2,1 ++ vA,k−1 +4k−5,2k+1 ⊗ v3,1 +vB,k +4k−2,2k−1 ⊗ v1,3 ++ vB,k +4k−3,2k+1 ⊗ v2,1 ++ vA,k−1 +4k−3,2k+1 ⊗ v2,1 +vB,k +4k−1,2k−1 ⊗ v1,3 ++ vB,k +4k−3,2k+1 ⊗ v3,1 ++ vA,k−1 +4k−2,2k+1 ⊗ v2,1 ++ vA,k−1 +4k−3,2k+1 ⊗ v3,1 +vA,k−1 +4k−3,2k+2 ⊗ v1,1 ++ vB,k +4k−3,2k+1 ⊗ v1,2 ++ vB,k +4k−3,2k ⊗ v1,3 ++ vA,k−1 +4k−4,2k+2 ⊗ v2,1 ++ vA,k−1 +4k−5,2k+2 ⊗ v3,1 +vB,k +4k−2,2k−1 ⊗ v2,1 +vB,k +4k−1,2k−1 ⊗ v2,1 ++ vB,k +4k−2,2k−1 ⊗ v3,1 + +14 +GEORGE CAO AND KENT B. VASHAW +Table 8 (continued). +(A, 1) piece: +(C, 1) piece: +vB,1 +2k−1,4k−3 ⊗ v1,1 ++ vA,1 +2k−1,4k−4 ⊗ v1,2 ++ vA,1 +2k−1,4k−5 ⊗ v1,3 +vB,1 +2k,4k−3 ⊗ v1,1 ++ vB,1 +2k−1,4k−3 ⊗ v2,1 ++ vA,1 +2k,4k−4 ⊗ v1,2 ++ vA,1 +2k,4k−5 ⊗ v1,3 +vB,1 +2k+1,4k−3 ⊗ v1,1 ++ vB,1 +2k−1,4k−3 ⊗ v3,1 ++ vA,1 +2k+1,4k−4 ⊗ v1,2 ++ vA,1 +2k+1,4k−5 ⊗ v1,3 +vA,1 +2k+2,4k−3 ⊗ v1,1 ++ vB,1 +2k+1,4k−3 ⊗ v2,1 ++ vB,1 +2k,4k−3 ⊗ v3,1 ++ vA,1 +2k+2,4k−4 ⊗ v1,2 ++ vA,1 +2k+2,4k−5 ⊗ v1,3 +vB,1 +2k−1,4k−2 ⊗ v1,1 ++ vB,1 +2k−1,4k−3 ⊗ v1,2 ++ vA,1 +2k−1,4k−3 ⊗ v1,2 +vB,1 +2k−1,4k−2 ⊗ v2,1 ++ vB,1 +2k,4k−3 ⊗ v1,2 ++ vA,1 +2k,4k−3 ⊗ v1,2 +vB,1 +2k−1,4k−2 ⊗ v3,1 ++ vB,1 +2k+1,4k−3 ⊗ v1,2 ++ vA,1 +2k+1,4k−3 ⊗ v1,2 +vB,1 +2k−1,4k−1 ⊗ v1,1 ++ vB,1 +2k−1,4k−3 ⊗ v1,3 ++ vA,1 +2k−1,4k−2 ⊗ v1,2 ++ vA,1 +2k−1,4k−3 ⊗ v1,3 +vB,1 +2k−1,4k−1 ⊗ v2,1 ++ vB,1 +2k,4k−3 ⊗ v1,3 ++ vA,1 +2k,4k−2 ⊗ v1,2 ++ vA,1 +2k,4k−3 ⊗ v1,3 +vB,1 +2k−1,4k−1 ⊗ v3,1 ++ vB,1 +2k+1,4k−3 ⊗ v1,3 ++ vA,1 +2k+1,4k−2 ⊗ v1,2 ++ vA,1 +2k+1,4k−3 ⊗ v1,3 +vB,1 +2k−1,4k−1 ⊗ v1,2 ++ vB,1 +2k−1,4k−2 ⊗ v1,3 +vB,1 +2k−1,4k−2 ⊗ v1,2 +vB,1 +2k−1,4k−1 ⊗ v1,2 ++ vB,1 +2k−1,4k−2 ⊗ v1,3 +Table 9. The dimension 12 summands. Note that when i = 1, k, some of the vectors do +not exist. This construction works with those nonexistent vectors removed. +Dimension 12 summand: +vA,i +2k−2+2i,4k−2−2i ⊗ v1,2 ++ vA,i +2k−2+2i,4k−3−2i ⊗ v1,3 ++ vB,i +2k−2+2i,4k−2−2i ⊗ v1,1 +vA,i +2k−1+2i,4k−2−2i ⊗ v1,2 ++ vA,i +2k−1+2i,4k−3−2i ⊗ v1,3 ++ vB,i +2k−1+2i,4k−2−2i ⊗ v1,1 ++ vB,i +2k−2+2i,4k−2−2i ⊗ v2,1 +vA,i +2k+2i,4k−2−2i ⊗ v1,2 ++ vA,i +2k+2i,4k−3−2i ⊗ v1,3 ++ vA,i +2k+2i,4k−2−2i ⊗ v1,1 ++ vB,i +2k−2+2i,4k−2−2i ⊗ v3,1 +vA,i +2k+2i,4k−2−2i ⊗ v2,1 ++ vB,i +2k−1+2i,4k−2−2i ⊗ v3,1 +vA,i +2k−3+2i,4k−1−2i ⊗ v1,2 ++ vB,i +2k−3+2i,4k−1−2i ⊗ v1,2 +vA,i +2k−2+2i,4k−1−2i ⊗ v1,2 ++ vB,i +2k−2+2i,4k−1−2i ⊗ v1,2 +vA,i +2k−1+2i,4k−1−2i ⊗ v1,2 ++ vB,i +2k−1+2i,4k−1−2i ⊗ v1,2 +vA,i +2k−3+2i,4k−2i ⊗ v1,2 ++ vA,i +2k−3+2i,4k−1−2i ⊗ v1,3 ++ vB,i +2k−3+2i,4k−2i ⊗ v1,2 ++ vB,i +2k−3+2i,4k−1−2i ⊗ v1,3 +vA,i +2k−2+2i,4k−2i ⊗ v1,2 ++ vA,i +2k−2+2i,4k−1−2i ⊗ v1,3 ++ vB,i +2k−2+2i,4k−1−2i ⊗ v1,3 +vA,i +2k−1+2i,4k−2i ⊗ v1,2 ++ vA,i +2k−1+2i,4k−1−2i ⊗ v1,3 ++ vB,i +2k−1+2i,4k−1−2i ⊗ v1,3 +vB,i +2k−3+2i,4k+1−2i ⊗ v1,2 +vA,i−1 +2k−3+2i,4k+2−2i ⊗ v1,2 ++ vB,i +2k−3+2i,4k+1−2i ⊗ v1,3 +for i = 1, 2, . . . , k + +ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS +15 +Table 10. The dimension 16 summands, which are all free modules. +There are three +families of them, which differ in only how they are expressed below. +Dimension 16 Family 1 summand: +vA,i +2k−3+2i,4k−3−2i ⊗ v1,1 +vA,i +2k−2+2i,4k−3−2i ⊗ v1,1 ++ vA,i +2k−3+2i,4k−3−2i ⊗ v2,1 +vA,i +2k−1+2i,4k−3−2i ⊗ v1,1 ++ vA,i +2k−3+2i,4k−3−2i ⊗ v3,1 +vA,i +2k+2i,4k−3+2i ⊗ v1,1 ++ vA,i +2k−1+2i,4k−3−2i ⊗ v2,1 ++ vA,i +2k−2+2i,4k−3−2i ⊗ v3,1 +vA,i +2k−3+2i,4k−2−2i ⊗ v1,1 ++ vA,i +2k−3+2i,4k−3−2i ⊗ v1,2 +vA,i +2k−2+2i,4k−2−2i ⊗ v1,1 ++ vA,i +2k−3+2i,4k−2−2i ⊗ v2,1 ++ vA,i +2k−2+2i,4k−3−2i ⊗ v1,2 +vA,i +2k−1+2i,4k−2−2i ⊗ v1,1 ++ vA,i +2k−3+2i,4k−2−2i ⊗ v3,1 ++ vA,i +2k−1+2i,4k−3−2i ⊗ v1,2 +vA,i +2k+2i,4k−2−2i ⊗ v1,1 ++ vA,i +2k−1+2i,4k−2−2i ⊗ v2,1 ++ vA,i +2k−2+2i,4k−2−2i ⊗ v3,1 ++ vA,i +2k+2i,4k−3−2i ⊗ v1,2 +vA,i +2k−3+2i,4k−1−2i ⊗ v1,1 ++ vA,i +2k−3+2i,4k−3−2i ⊗ v1,3 +vA,i +2k−2+2i,4k−1−2i ⊗ v1,1 ++ vA,i +2k−3+2i,4k−1−2i ⊗ v2,1 ++ vA,i +2k−2+2i,4k−3−2i ⊗ v1,3 +vA,i +2k−1+2i,4k−1−2i ⊗ v1,1 ++ vA,i +2k−3+2i,4k−1−2i ⊗ v3,1 ++ vA,i +2k−1+2i,4k−3−2i ⊗ v1,3 +vA,i +2k+2i,4k−1−2i ⊗ v1,1 ++ vA,i +2k−1+2i,4k−1−2i ⊗ v2,1 ++ vA,i +2k−2+2i,4k−1−2i ⊗ v3,1 ++ vA,i +2k+2i,4k−3−2i ⊗ v1,3 +vA,i +2k−3+2i,4k−2i ⊗ v1,1 ++ vA,i +2k−3+2i,4k−1−2i ⊗ v1,2 ++ vA,i +2k−3+2i,4k−2−2i ⊗ v1,3 +vA,i +2k−2+2i,4k−2i ⊗ v1,1 ++ vA,i +2k−3+2i,4k−2i ⊗ v2,1 ++ vA,i +2k−2+2i,4k−1−2i ⊗ v1,2 ++ vA,i +2k−2+2i,4k−2−2i ⊗ v1,3 +vA,i +2k−1+2i,4k−2i ⊗ v1,1 ++ vA,i +2k−3+2i,4k−2i ⊗ v3,1 ++ vA,i +2k−1+2i,4k−1−2i ⊗ v1,2 ++ vA,i +2k−1+2i,4k−2−2i ⊗ v1,3 +vA,i +2k−1+2i,4k−2i ⊗ v2,1 ++ vA,i +2k−2+2i,4k−2i ⊗ v3,1 ++ vA,i +2k+2i,4k−1−2i ⊗ v1,2 ++ vA,i +2k+2i,4k−2−2i ⊗ v1,3 +for i = 1, 2, . . . , k − 1 +Dimension 16 Family 2 summand: +vA,i +2k−3+2i,4k−3−2i ⊗ v2,1 +vA,i +2k−2+2i,4k−3−2i ⊗ v2,1 ++ vA,i +2k−3+2i,4k−3−2i ⊗ v3,1 +vA,i +2k−1+2i,4k−3−2i ⊗ v2,1 +vA,i +2k+2i,4k−3−2i ⊗ v2,1 ++ vA,i +2k−1+2i,4k−3−2i ⊗ v3,1 +vA,i +2k−3+2i,4k−2−2i ⊗ v2,1 +vA,i +2k−2+2i,4k−2−2i ⊗ v2,1 ++ vA,i +2k−3+2i,4k−2−2i ⊗ v3,1 +vA,i +2k−1+2i,4k−2−2i ⊗ v2,1 +vA,i +2k+2i,4k−2−2i ⊗ v2,1 ++ vA,i +2k−1+2i,4k−2−2i ⊗ v3,1 +vA,i +2k−3+2i,4k−1−2i ⊗ v2,1 +vA,i +2k−2+2i,4k−1−2i ⊗ v2,1 ++ vA,i +2k−3+2i,4k−1−2i ⊗ v3,1 +vA,i +2k−1+2i,4k−1−2i ⊗ v2,1 +vA,i +2k+2i,4k−1−2i ⊗ v2,1 ++ vA,i +2k−1+2i,4k−1−2i ⊗ v3,1 +vA,i +2k−3+2i,4k−2i ⊗ v2,1 +vA,i +2k−2+2i,4k−2i ⊗ v2,1 ++ vA,i +2k−3+2i,4k−2i ⊗ v3,1 +vA,i +2k−1+2i,4k−2i ⊗ v2,1 +vA,i +2k−1+2i,4k−2i ⊗ v3,1 +for i = 1, 2, . . . , k − 1 +Dimension 16 Family 3 summand: +vA,i +2k−3+2i,4k−3−2i ⊗ v1,2 +vA,i +2k−2+2i,4k−3−2i ⊗ v1,2 +vA,i +2k−1+2i,4k−3−2i ⊗ v1,2 +vA,i +2k+2i,4k−3−2i ⊗ v1,2 +vA,i +2k−3+2i,4k−2−2i ⊗ v1,2 ++ vA,i +2k−3+2i,4k−3−2i ⊗ v1,3 +vA,i +2k−2+2i,4k−2−2i ⊗ v1,2 ++ vA,i +2k−2+2i,4k−3−2i ⊗ v1,3 +vA,i +2k−1+2i,4k−2−2i ⊗ v1,2 ++ vA,i +2k−1+2i,4k−3−2i ⊗ v1,3 +vA,i +2k+2i,4k−2−2i ⊗ v1,2 ++ vA,i +2k+2i,4k−3−2i ⊗ v1,3 +vA,i +2k−3+2i,4k−1−2i ⊗ v1,2 +vA,i +2k−2+2i,4k−1−2i ⊗ v1,2 +vA,i +2k−1+2i,4k−1−2i ⊗ v1,2 +vA,i +2k+2i,4k−1−2i ⊗ v1,2 +vA,i +2k−3+2i,4k−2i ⊗ v1,2 ++ vA,i +2k−3+2i,4k−1−2i ⊗ v1,3 +vA,i +2k−2+2i,4k−2i ⊗ v1,2 ++ vA,i +2k−2+2i,4k−1−2i ⊗ v1,3 +vA,i +2k−1+2i,4k−2i ⊗ v1,2 ++ vA,i +2k−1+2i,4k−1−2i ⊗ v1,3 +vA,i +2k+2i,4k−1−2i ⊗ v1,3 +for i = 1, 2, . . . , k − 1 + +16 +GEORGE CAO AND KENT B. VASHAW +Table 11. The dimension 28 summands, each of which is written in three pieces. These +pieces show the nonzero actions of x and y and overlap each other. They are arranged as in +Figure 7. Note that for i = k − 1, some of the vectors do not exist. This construction works +with those nonexistent vectors removed. +Dimension 28 summand: +vA,i +2k−3+2i,4k−2−2i ⊗ v1,2 +vA,i +2k−2+2i,4k−2−2i ⊗ v1,2 +vA,i +2k−1+2i,4k−2−2i ⊗ v1,2 +vA,i +2k+2i,4k−2−2i ⊗ v1,2 +vA,i +2k−3+2i,4k−1−2i ⊗ v1,2 ++ vA,i +2k−3+2i,4k−2−2i ⊗ v1,3 +vA,i +2k−2+2i,4k−1−2i ⊗ v1,2 ++ vA,i +2k−2+2i,4k−2−2i ⊗ v1,3 +vA,i +2k−1+2i,4k−1−2i ⊗ v1,2 ++ vA,i +2k−1+2i,4k−2−2i ⊗ v1,3 +vA,i +2k+2i,4k−1−2i ⊗ v1,2 ++ vA,i +2k+2i,4k−2−2i ⊗ v1,3 +vA,i +2k−3+2i,4k−2i ⊗ v1,2 +vA,i +2k−2+2i,4k−2i ⊗ v1,2 +vA,i +2k−1+2i,4k−2i ⊗ v1,2 +vA,i +2k−3+2i,4k−2i ⊗ v1,3 +vA,i +2k−2+2i,4k−2i ⊗ v1,3 +vA,i +2k−1+2i,4k−2i ⊗ v1,3 +vA,i +2k−1+2i,4k−3−2i ⊗ v1,1 ++ vB,i+1 +2k−1+2i,4k−3−2i ⊗ v1,1 +vA,i +2k+2i,4k−3−2i ⊗ v1,1 ++ vA,i +2k−1+2i,4k−3−2i ⊗ v2,1 ++ vB,i+1 +2k+2i,4k−3−2i ⊗ v1,1 ++ vB,i+1 +2k−1+2i,4k−3−2i ⊗ v2,1 +vA,i +2k−1+2i,4k−3−2i ⊗ v3,1 ++ vB,i+1 +2k+1+2i,4k−3−2i ⊗ v1,1 ++ vB,i+1 +2k−1+2i,4k−3−2i ⊗ v3,1 +vA,i +2k+2i,4k−3−2i ⊗ v3,1 ++ vA,i+1 +2k+2+2i,4k−3−2i ⊗ v1,1 ++ vB,i+1 +2k+1+2i,4k−3−2i ⊗ v2,1 ++ vB,i+1 +2k+2i,4k−3−2i ⊗ v3,1 +vA,i +2k−1+2i,4k−2−2i ⊗ v1,1 ++ vA,i +2k−1+2i,4k−3−2i ⊗ v1,2 ++ vB,i+1 +2k−1+2i,4k−2−2i ⊗ v1,1 ++ vB,i+1 +2k−1+2i,4k−3−2i ⊗ v1,2 +vA,i +2k+2i,4k−2−2i ⊗ v1,1 ++ vA,i +2k−1+2i,4k−2−2i ⊗ v2,1 ++ vA,i +2k+2i,4k−3−2i ⊗ v1,2 ++ vB,i+1 +2k−1+2i,4k−2−2i ⊗ v2,1 ++ vB,i+1 +2k+2i,4k−3−2i ⊗ v1,2 +vA,i +2k+2i,4k−3−2i ⊗ v3,1 ++ vB,i+1 +2k−1+2i,4k−2−2i ⊗ v3,1 ++ vB,i+1 +2k+1+2i,4k−3−2i ⊗ v1,2 +vA,i +2k+1+2i,4k−3−2i ⊗ v3,1 ++ vA,i+1 +2k+2+2i,4k−3−2i ⊗ v1,2 +vA,i +2k−1+2i,4k−1−2i ⊗ v1,1 ++ vA,i +2k−1+2i,4k−3−2i ⊗ v1,3 ++ vB,i+1 +2k−1+2i,4k−1−2i ⊗ v1,1 ++ vB,i+1 +2k−1+2i,4k−3−2i ⊗ v1,3 +vA,i +2k+2i,4k−1−2i ⊗ v1,1 ++ vA,i +2k−1+2i,4k−1−2i ⊗ v2,1 ++ vA,i +2k+2i,4k−3−2i ⊗ v1,3 ++ vB,i+1 +2k−1+2i,4k−1−2i ⊗ v2,1 ++ vB,i+1 +2k+2i,4k−3−2i ⊗ v1,3 +vA,i +2k−1+2i,4k−1−2i ⊗ v3,1 ++ vB,i+1 +2k−1+2i,4k−1−2i ⊗ v3,1 ++ vB,i+1 +2k+1+2i,4k−3−2i ⊗ v1,3 +vA,i +2k+2i,4k−1−2i ⊗ v3,1 ++ vA,i+1 +2k+2+2i,4k−3−2i ⊗ v1,3 +vA,i +2k−1+2i,4k−1−2i ⊗ v1,2 ++ vA,i +2k−1+2i,4k−2−2i ⊗ v1,3 ++ vB,i+1 +2k−1+2i,4k−1−2i ⊗ v1,2 ++ vB,i+1 +2k−1+2i,4k−2−2i ⊗ v1,3 +vA,i +2k+2i,4k−1−2i ⊗ v1,2 ++ vA,i +2k+2i,4k−2−2i ⊗ v1,3 +vA,i +2k−1+2i,4k−2−2i ⊗ v1,2 ++ vB,i+1 +2k−1+2i,4k−2−2i ⊗ v1,2 +vA,i +2k+2i,4k−2−2i ⊗ v1,2 +vA,i +2k−1+2i,4k−1−2i ⊗ v1,2 ++ vA,i +2k−1+2i,4k−2−2i ⊗ v1,3 ++ vB,i+1 +2k−1+2i,4k−1−2i ⊗ v1,2 ++ vB,i+1 +2k−1+2i,4k−2−2i ⊗ v1,3 +vA,i +2k+2i,4k−1−2i ⊗ v1,2 ++ vA,i +2k+2i,4k−2−2i ⊗ v1,3 +for i = 1, 2, . . . , k − 1 + +ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS +17 +Figure 7. The location of the pieces for the dimension 28 summand. The outlines show +the positions of the two pieces with 14 cells, and the shaded region shows the position of +the piece with 4 cells. +1 1 1 1 +1 1 1 1 +1 1 3 2 1 1 +1 1 2 1 +1 1 1 +1 1 1 +Table 12. The pieces of V2k+1. +(A, i) piece: +vi +2k−2+2i,4k−2i ⊗ v1,1 +vi +2k−1+2i,4k−2i ⊗ v1,1 ++ vi +2k−2+2i,4k−2i ⊗ v2,1 +vi +2k+2i,4k−2i ⊗ v1,1 ++ vi +2k−2+2i,4k−2i ⊗ v3,1 +vi +2k+1+2i,4k−2i ⊗ v1,1 ++ vi +2k+2i,4k−2i ⊗ v2,1 ++ vi +2k−1+2i,4k−2i ⊗ v3,1 +vi +2k−2+2i,4k+1−2i ⊗ v1,1 ++ vi +2k−2+2i,4k−2i ⊗ v1,2 +vi +2k−1+2i,4k+1−2i ⊗ v1,1 ++ vi +2k−1+2i,4k−2i ⊗ v1,2 ++ vi +2k−2+2i,4k+1−2i ⊗ v2,1 +vi +2k+2i,4k+1−2i ⊗ v1,1 ++ vi +2k+2i,4k−2i ⊗ v1,2 ++ vi +2k−2+2i,4k+1−2i ⊗ v3,1 +vi +2k+1+2i,4k−2i ⊗ v1,2 ++ vi +2k+2i,4k+1−2i ⊗ v2,1 ++ vi +2k−1+2i,4k+1−2i ⊗ v3,1 +vi +2k−2+2i,4k+2−2i ⊗ v1,1 ++ vi +2k−2+2i,4k−2i ⊗ v1,3 +vi +2k−1+2i,4k+2−2i ⊗ v1,1 ++ vi +2k−2+2i,4k+2−2i ⊗ v2,1 ++ vi +2k−1+2i,4k−2i ⊗ v1,3 +vi +2k+2i,4k+2−2i ⊗ v1,1 ++ vi +2k−2+2i,4k+2−2i ⊗ v3,1 ++ vi +2k+2i,4k−2i ⊗ v1,3 +vi +2k+2i,4k+2−2i ⊗ v2,1 ++ vi +2k−1+2i,4k+2−2i ⊗ v3,1 ++ vi +2k+1+2i,4k−2i ⊗ v1,3 +vi +2k−2+2i,4k+3−2i ⊗ v1,1 ++ vi +2k−2+2i,4k+2−2i ⊗ v1,2 ++ vi +2k−2+2i,4k+1−2i ⊗ v1,3 +vi +2k−2+2i,4k+3−2i ⊗ v2,1 ++ vi +2k−1+2i,4k+2−2i ⊗ v1,2 ++ vi +2k−1+2i,4k+1−2i ⊗ v1,3 +vi +2k−2+2i,4k+3−2i ⊗ v3,1 ++ vi +2k+2i,4k+2−2i ⊗ v1,2 ++ vi +2k+2i,4k+1−2i ⊗ v1,3 +for i = 1, 2, . . . , k +(B, 1) piece: +(B, k + 1) piece: +v1 +2k,4k−2 ⊗ v1,3 ++ v1 +2k,4k−1 ⊗ v1,2 ++ v1 +2k,4k ⊗ v1,1 ++ v2k,4k ⊗ v1,1 +v1 +2k+1,4k−2 ⊗ v1,3 ++ v1 +2k+1,4k−1 ⊗ v1,2 ++ v1 +2k+1,4k ⊗ v1,1 ++ v1 +2k,4k ⊗ v2,1 ++ v2k,4k ⊗ v2,1 +v1 +2k+2,4k−2 ⊗ v1,3 ++ v1 +2k+2,4k−1 ⊗ v1,2 ++ v1 +2k+2,4k ⊗ v1,1 ++ v1 +2k,4k ⊗ v3,1 ++ v2k,4k ⊗ v3,1 +v1 +2k+3,4k−2 ⊗ v1,3 ++ v1 +2k+2,4k ⊗ v2,1 ++ v1 +2k+1,4k ⊗ v3,1 +v2k,4k ⊗ v1,2 +v1 +2k,4k+1 ⊗ v1,2 ++ v2k,4k ⊗ v1,3 +vk +4k−2,2k ⊗ v3,1 ++ vk +4k−1,2k ⊗ v2,1 ++ vk +4k,2k ⊗ v1,1 ++ v4k,2k ⊗ v1,1 +v4k,2k ⊗ v2,1 +vk +4k+1,2k ⊗ v2,1 ++ v4k,2k ⊗ v3,1 +vk +4k−2,2k+1 ⊗ v3,1 ++ vk +4k−1,2k+1 ⊗ v2,1 ++ vk +4k,2k+1 ⊗ v1,1 ++ vk +4k,2k ⊗ v1,2 ++ v4k,2k ⊗ v1,2 +vk +4k−2,2k+2 ⊗ v3,1 ++ vk +4k−1,2k+2 ⊗ v2,1 ++ vk +4k,2k+2 ⊗ v1,1 ++ vk +4k,2k ⊗ v1,3 ++ v4k,2k ⊗ v1,3 +vk +4k,2k+1 ⊗ v1,2 ++ vk +4k,2k+1 ⊗ v1,3 ++ vk +4k−2,2k+3 ⊗ v3,1 + +18 +GEORGE CAO AND KENT B. VASHAW +Table 12 (continued). +(B, i) piece: +vi−1 +2k−2+2i,4k+2−2i ⊗ v1,1 ++ vi +2k−2+2i,4k+2−2i ⊗ v1,1 ++ v2k−2+2i,4k+2−2i ⊗ v1,1 ++ vi−1 +2k−3+2i,4k+2−2i ⊗ v2,1 ++ vi +2k−2+2i,4k+1−2i ⊗ v1,2 ++ vi−1 +2k−4+2i,4k+2−2i ⊗ v3,1 ++ vi +2k−2+2i,4k−2i ⊗ v1,3 +vi +2k−1+2i,4k+2−2i ⊗ v1,1 ++ vi +2k−2+2i,4k+2−2i ⊗ v2,1 ++ v2k−2+2i,4k+2−2i ⊗ v2,1 ++ vi +2k−1+2i,4k+1−2i ⊗ v1,2 ++ vi +2k−1+2i,4k−2i ⊗ v1,3 +vi +2k+2i,4k+2−2i ⊗ v1,1 ++ vi +2k−2+2i,4k+2−2i ⊗ v3,1 ++ vi−1 +2k−1+2i,4k+2−2i ⊗ v2,1 ++ v2k−2+2i,4k+2−2i ⊗ v3,1 ++ vi +2k+2i,4k+1−2i ⊗ v1,2 ++ vi +2k+2i,4k−2i ⊗ v1,3 +vi +2k+2i,4k+2−2i ⊗ v2,1 ++ vi +2k−1+2i,4k+2−2i ⊗ v3,1 ++ vi +2k+1+2i,4k−2i ⊗ v1,3 +vi−1 +2k−2+2i,4k+3−2i ⊗ v1,1 ++ vi−1 +2k−2+2i,4k+2−2i ⊗ v1,2 ++ v2k−2+2i,4k+2−2i ⊗ v1,2 ++ vi−1 +2k−3+2i,4k+3−2i ⊗ v2,1 ++ vi−1 +2k−4+2i,4k+3−2i ⊗ v3,1 +vi−1 +2k−2+2i,4k+4−2i ⊗ v1,1 ++ vi−1 +2k−2+2i,4k+2−2i ⊗ v1,3 ++ vi +2k−2+2i,4k+3−2i ⊗ v1,2 ++ v2k−2+2i,4k+2−2i ⊗ v1,3 ++ vi−1 +2k−3+2i,4k+4−2i ⊗ v2,1 ++ vi−1 +2k−4+2i,4k+4−2i ⊗ v3,1 +vi−1 +2k−4+2i,4k+5−2i ⊗ v3,1 ++ vi−1 +2k−2+2i,4k+4−2i ⊗ v1,2 ++ vi−1 +2k−2+2i,4k+3−2i ⊗ v1,3 +for i = 2, 3, . . . , k +Table 13. The dimension 20 summands, which can be split into two orientations, each +orientation being characterized in two pieces which overlap at the indicated equality. The +pieces are arranged as in Figure 8. +Vertical orientation: +vi +2k+2i,4k−2i ⊗ v2,1 ++ vi +2k−1+2i,4k−2i ⊗ v3,1 +vi +2k+1+2i,4k−2i ⊗ v2,1 +vi +2k+1+2i,4k−2i ⊗ v3,1 +vi +2k−2+2i,4k+1−2i ⊗ v1,1 +vi +2k−1+2i,4k+1−2i ⊗ v1,1 ++ vi +2k−2+2i,4k+1−2i ⊗ v2,1 +vi +2k+2i,4k+1−2i ⊗ v1,1 ++ vi +2k−2+2i,4k+1−2i ⊗ v3,1 +vi +2k+2i,4k+1−2i ⊗ v2,1 ++ vi +2k−1+2i,4k+1−2i ⊗ v3,1 +vi +2k−2+2i,4k+2−2i ⊗ v1,1 ++ vi +2k−2+2i,4k+1−2i ⊗ v1,2 +vi +2k−1+2i,4k+2−2i ⊗ v1,1 ++ vi +2k−1+2i,4k+1−2i ⊗ v1,2 ++ vi +2k−2+2i,4k+2−2i ⊗ v2,1 +vi +2k+2i,4k+2−2i ⊗ v1,1 ++ vi +2k+2i,4k+1−2i ⊗ v1,2 ++ vi +2k−2+2i,4k+2−2i ⊗ v3,1 +vi +2k+2i,4k+2−2i ⊗ v2,1 ++ vi +2k−1+2i,4k+2−2i ⊗ v3,1 +vi +2k−2+2i,4k+3−2i ⊗ v1,1 ++ vi +2k−2+2i,4k+1−2i ⊗ v1,3 +vi +2k−2+2i,4k+3−2i ⊗ v2,1 ++ vi +2k−1+2i,4k+1−2i ⊗ v1,3 +vi +2k−2+2i,4k+3−2i ⊗ v3,1 ++ vi +2k+2i,4k+1−2i ⊗ v1,3 +vi +2k−2+2i,4k+3−2i ⊗ v1,2 ++ vi +2k−2+2i,4k+2−2i ⊗ v1,3 +vi +2k−1+2i,4k+2−2i ⊗ v1,3 +vi +2k+2i,4k+2−2i ⊗ v1,3 +vi +2k−1+2i,4k−2i ⊗ v2,1 += += += +vi +2k−1+2i,4k+1−2i ⊗ v2,1 += +vi +2k−1+2i,4k+2−2i ⊗ v2,1 += +for i = 1, 2, . . . , k + +ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS +19 +Table 13 (continued). +Horizontal orientation: +vi +2k−1+2i,4k−2i ⊗ v1,1 +vi +2k+2i,4k−2i ⊗ v1,1 ++ vi +2k−1+2i,4k−2i ⊗ v2,1 +vi +2k+1+2i,4k−2i ⊗ v1,1 ++ vi +2k−1+2i,4k−2i ⊗ v3,1 +vi +2k+1+2i,4k−2i ⊗ v2,1 ++ vi +2k+2i,4k−2i ⊗ v3,1 +vi +2k−1+2i,4k+1−2i ⊗ v1,1 ++ vi +2k−1+2i,4k−2i ⊗ v1,2 +vi +2k+2i,4k+1−2i ⊗ v1,1 ++ vi +2k−1+2i,4k+1−2i ⊗ v2,1 ++ vi +2k+2i,4k−2i ⊗ v1,2 +vi +2k+1+2i,4k−2i ⊗ v1,2 ++ vi +2k−1+2i,4k+1−2i ⊗ v3,1 +vi +2k+2i,4k+1−2i ⊗ v3,1 +vi +2k−1+2i,4k+2−2i ⊗ v1,1 ++ vi +2k−1+2i,4k−2i ⊗ v1,3 +vi +2k+2i,4k+2−2i ⊗ v1,1 ++ vi +2k−1+2i,4k+2−2i ⊗ v2,1 ++ vi +2k+2i,4k−2i ⊗ v1,3 +vi +2k+1+2i,4k−2i ⊗ v1,3 ++ vi +2k−1+2i,4k+2−2i ⊗ v3,1 +vi +2k+2i,4k+2−2i ⊗ v3,1 +vi +2k−2+2i,4k+2−2i ⊗ v1,2 ++ vi +2k−2+2i,4k+1−2i ⊗ v1,3 +vi +2k−1+2i,4k+2−2i ⊗ v1,2 ++ vi +2k−1+2i,4k+1−2i ⊗ v1,3 +vi +2k+2i,4k+2−2i ⊗ v1,2 ++ vi +2k+2i,4k+1−2i ⊗ v1,3 +vi +2k−2+2i,4k+3−2i ⊗ v1,2 +vi +2k−2+2i,4k+3−2i ⊗ v1,3 +vi +2k−2+2i,4k+1−2i ⊗ v1,2 +vi +2k−1+2i,4k+1−2i ⊗ v1,2 +vi +2k+2i,4k+1−2i ⊗ v1,2 += += += += += +for i = 1, 2, . . . , k +Figure 8. The location of the pieces for the dimension 20 summands. The outlines show +the positions of the pieces with 17 cells, and the shaded region shows the position of the +piece with 4 cells. +1 1 1 1 +1 1 2 1 +1 1 2 1 +1 1 1 +1 1 1 +1 1 1 1 +1 1 1 1 +1 2 2 1 1 +1 1 1 +1 +1 +Now, the proof is very similar to the (4, 1) partition case, which was Lemma 3.3.2. +Proof. Similar to the proof of Lemma 3.3.2, the explicit constructions show that the summands are both +subrepresentations and indecomposable. The positions of the summands of V2k−1 ⊗ V are summarized in +Figure 9, and the positions of the summands of V2k ⊗ V are summarized in Figure 11 below. The black +outline is the location of Vi within the tensor product. +It can be checked that these summands span the tensor product. This is very similar to Lemma 3.3.2, +and we have omitted the proof. +□ +As a corollary, we observe that Conjecture 1.0.2 again holds in this case; this time, the function PV (n) is +not polynomial, but is still quasi-polynomial. +Corollary 3.5.3. The function PV (n) is a quasi-polynomial with period 2, given by 10n − 5 for odd n and +6n + 1 for even n. + +20 +GEORGE CAO AND KENT B. VASHAW +Figure 9. Positions of the indecomposable summands in V7 ⊗ V . +1 +2 +4 +5 +5 +3 +1 +2 +3 +6 +6 +4 +1 +1 +2 +7 +9 13 10 5 +1 +2 +3 +9 10 11 5 +1 +1 +2 +7 +9 15 12 8 +2 +2 +3 +9 10 12 6 +2 +4 +6 13 11 8 +2 +5 +6 10 5 +2 +5 +4 +5 +1 +3 +1 +1 +1 +Key +Dim. 12 +Family 1 +Family 2 +Family 3 +Dim. 28 +1 1 1 1 +1 1 1 +1 1 1 +1 +1 +1 1 1 1 +1 1 1 1 +1 1 1 1 +1 1 1 1 +1 1 1 1 +1 1 1 1 +1 1 1 1 +1 1 1 1 +1 1 1 1 +1 1 1 1 +1 1 1 1 +1 1 1 1 +1 1 1 1 +1 1 1 1 +1 1 3 2 1 1 +1 1 2 1 +1 1 1 +1 1 1 +Figure 11. Positions of the indecomposable summands in V8 ⊗ V . +1 +2 +4 +4 +3 +1 +2 +3 +5 +3 +1 +1 +2 +7 +8 +9 +4 +1 +2 +3 +8 +6 +4 +1 +2 +7 +8 11 5 +2 +2 +3 +8 +6 +5 +1 +2 +7 +8 11 5 +2 +2 +3 +8 +6 +5 +4 +5 +9 +4 +2 +4 +3 +4 +3 +1 +1 +1 +Key +Vertical +Horizontal +1 1 1 1 +1 1 2 1 +1 1 2 1 +1 1 1 +1 1 1 +1 1 1 1 +1 1 1 1 +1 2 2 1 1 +1 1 1 +1 +1 +3.6. (4, 2)/(1) Monomial representation. Let V be the monomial representation of Z/2Z × Z/4Z (or +α(1, 2)) corresponding to the skew partition (4, 2)/(1). This is another example where the syzygy technique +outlined in Section 3.4 is not applicable. +Proposition 3.6.1. We have the following decomposition into indecomposable summands: +V2k−1 ⊗ V = V2k ⊕ F ⊕ · · · ⊕ F +� +�� +� +3k−3 copies +⊕ W12 ⊕ · · · ⊕ W12 +� +�� +� +2k−1 copies +, +V2k ⊗ V = V2k+1 ⊕ F ⊕ · · · ⊕ F +� +�� +� +3k copies +⊕ W12 ⊕ · · · ⊕ W12 +� +�� +� +2k copies +, +where dim W12 = 12 and F is a free module of dimension 8. +As before, we write the decomposition with a stronger lemma. +Lemma 3.6.2. We claim the following. +(1) The representation V2k+1 for k ≥ 1 is given by the graded monomial diagram in Table 14. +(2) The representation V2k for k ≥ 1 is given by the graded monomial diagram in Table 15. +(3) The representation V2k−1 ⊗V decomposes into the following summands: V2k (given in Table 16); dimen- +sion 8 summands (given in Table 17); and dimension 12 summands (given in Table 18). +(4) The representation V2k ⊗V decomposes into the following summands: V2k+1 (given in Table 19); dimen- +sion 8 summands (given in Table 20); and dimension 12 summands (given in Table 21). + +ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS +21 +Table 14. The pieces of V2k+1. These pieces show the nonzero actions of x and y, and +they overlap each other. They are arranged as in Figure 13 for V5. +(A, 1) piece: +(A, i) piece: +vA,1 +4k+2,2k+1 +vA,1 +4k+1,2k+2 +vA,1 +4k+2,2k+2 +vA,1 +4k+1,2k+3 +vA,1 +4k+1,2k+4 +vA,1 +4k+1,2k+5 +vA,i +4k+4−2i,2k−3+4i +vA,i−1 +4k+5−2i,2k−3+4i +vA,i +4k+3−2i,2k−2+4i +vA,i +4k+4−2i,2k−2+4i +vA,i +4k+3−2i,2k−1+4i +vA,i +4k+3−2i,2k+4i +vA,i +4k+3−2i,2k+1+4i +for i = 2, 3, . . ., k +(B, i) piece: +(A, k + 1) piece: +vB,i +4k+2−2i,2k−1+4i +vB,i +4k+3−2i,2k−1+4i +vB,i +4k+2−2i,2k+4i +vB,i +4k+3−2i,2k+4i +vB,i +4k+2−2i,2k+1+4i +vA,i +4k+3−2i,2k+1+4i +vB,i +4k+2−2i,2k+2+4i +vA,k+1 +2k+2,6k+1 +vA,k +2k+3,6k+1 +vA,k+1 +2k+1,6k+2 +vA,k+1 +2k+2,6k+2 +vA,k+1 +2k+1,6k+3 +vA,k+1 +2k+1,6k+4 +for i = 1, 2, . . ., k +Figure 13. The location of the pieces for V5. The outlines show the (B, i) pieces, and the +shaded regions show the (A, i) pieces. +1 +1 +1 +2 +2 +1 +1 +1 +2 +2 +1 +2 +2 +1 +1 +1 +2 +2 +1 +1 +1 +Table 15. The pieces of V2k. These pieces show the nonzero actions of x and y, and they +overlap each other. They are arranged as in Figure 14 for V6. +(A, 1) piece: +(A, i) piece: +vA,1 +4k,2k +vA,1 +4k−1,2k+1 +vA,1 +4k,2k+1 +vA,1 +4k−1,2k+2 +vA,1 +4k−1,2k+3 +vA,1 +4k−1,2k+4 +vA,i +4k+2−2i,2k−4+4i +vA,i−1 +4k+3−2i,2k−4+4i +vA,i +4k+1−2i,2k−3+4i +vA,i +4k+2−2i,2k−3+4i +vA,i +4k+1−2i,2k−2+4i +vA,i +4k+1−2i,2k−1+4i +vA,i +4k+1−2i,2k+4i +for i = 2, 3, . . ., k +C piece: +(B, i) piece: +v2k,6k +vB,k +2k,6k+1 +vB,i +4k−2i,2k−2+4i +vB,i +4k+1−2i,2k−2+4i +vB,i +4k−2i,2k−1+4i +vB,i +4k+1−2i,2k−1+4i +vB,i +4k−2i,2k+4i +vA,i +4k+1−2i,2k+4i +vB,i +4k−2i,2k+1+4i +for i = 1, 2, . . ., k + +22 +GEORGE CAO AND KENT B. VASHAW +Figure 14. The location of the pieces for V6. The outlines show the (B, i) pieces and the +shaded regions show the (A, i) and C pieces. +1 +1 +1 +2 +2 +1 +1 +1 +2 +2 +1 +2 +2 +1 +1 +1 +2 +2 +1 +2 +2 +1 +1 +1 +2 +1 +Table 16. The pieces of V2k. Note that in pieces (A, k) and (B, k), some of the vectors do +not exist. This construction works with those nonexistent vectors removed. +(A, i) piece: +(B, i) piece: +vA,i−1 +4k+1−2i,2k−6+4i ⊗ v1,2 ++ vA,i−1 +4k+1−2i,2k−7+4i ⊗ v1,3 ++ vB,i−1 +4k+1−2i,2k−7+4i ⊗ v1,3 ++ vA,i−1 +4k+1−2i,2k−8+4i ⊗ v1,4 ++ vA,i +4k−2i,2k−5+4i ⊗ v2,1 +vA,i−1 +4k+1−2i,2k−6+4i ⊗ v2,2 ++ vA,i−1 +4k+2−2i,2k−8+4i ⊗ v1,4 ++ vA,i−1 +4k+1−2i,2k−5+4i ⊗ v2,1 +vA,i +4k−2i,2k−5+4i ⊗ v1,2 ++ vB,i−1 +4k−2i,2k−6+4i ⊗ v1,3 ++ vB,i−1 +4k−2i,2k−7+4i ⊗ v1,4 ++ vA,i +4k−1−2i,2k−4+4i ⊗ v2,1 +vA,i−1 +4k+1−2i,2k−5+4i ⊗ v1,2 ++ vB,i−1 +4k+1−2i,2k−6+4i ⊗ v1,3 ++ vB,i−1 +4k+1−2i,2k−7+4i ⊗ v1,4 ++ vA,i +4k−2i,2k−4+4i ⊗ v2,1 ++ vA,i +4k−2i,2k−5+4i ⊗ v2,2 +vA,i +4k−2i,2k−4+4i ⊗ v1,2 ++ vA,i +4k−2i,2k−5+4i ⊗ v1,3 ++ vB,i−1 +4k−2i,2k−5+4i ⊗ v1,3 ++ vA,i +4k−1−2i,2k−3+4i ⊗ v2,1 ++ vA,i +4k−1−2i,2k−4+4i ⊗ v2,2 +vA,i +4k−2i,2k−5+4i ⊗ v1,4 ++ vB,i−1 +4k−2i,2k−4+4i ⊗ v1,3 ++ vB,i−1 +4k−2i,2k−5+4i ⊗ v1,4 ++ vA,i +4k−1−2i,2k−2+4i ⊗ v2,1 +vA,i +4k−2i,2k−4+4i ⊗ v1,4 ++ vA,i +4k−1−2i,2k−1+4i ⊗ v2,1 ++ vA,i +4k−1−2i,2k−2+4i ⊗ v2,2 +vA,i +4k−1−2i,2k−4+4i ⊗ v1,2 ++ vB,i +4k−2−2i,2k−3+4i ⊗ v2,1 +vA,i +4k−2i,2k−4+4i ⊗ v1,2 ++ vA,i +4k−1−2i,2k−4+4i ⊗ v2,2 ++ vB,i +4k−1−2i,2k−3+4i ⊗ v2,1 +vA,i +4k−1−2i,2k−3+4i ⊗ v1,2 ++ vA,i +4k−1−2i,2k−4+4i ⊗ v1,3 ++ vB,i +4k−2−2i,2k−2+4i ⊗ v2,1 ++ vB,i +4k−2−2i,2k−3+4i ⊗ v2,2 +vA,i +4k−2i,2k−4+4i ⊗ v1,3 ++ vA,i +4k−1−2i,2k−3+4i ⊗ v2,2 ++ vB,i +4k−1−2i,2k−2+4i ⊗ v2,1 ++ vB,i +4k−1−2i,2k−3+4i ⊗ v2,2 +vA,i +4k−1−2i,2k−2+4i ⊗ v1,2 ++ vA,i +4k−1−2i,2k−4+4i ⊗ v1,4 ++ vB,i +4k−2−2i,2k−1+4i ⊗ v2,1 +vA,i +4k−2i,2k−4+4i ⊗ v1,4 ++ vA,i +4k−1−2i,2k−1+4i ⊗ v2,1 ++ vA,i +4k−1−2i,2k−2+4i ⊗ v2,2 +vA,i +4k−1−2i,2k−1+4i ⊗ v1,2 ++ vA,i +4k−1−2i,2k−2+4i ⊗ v1,3 ++ vA,i +4k−1−2i,2k−3+4i ⊗ v1,4 ++ vB,i +4k−2−2i,2k+4i ⊗ v2,1 ++ vB,i +4k−2−2i,2k−1+4i ⊗ v2,2 +for i = 2, 3, . . ., k +for i = 1, 2, . . . , k + +ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS +23 +Table 16 (continued). +(A, 1) piece: +C piece: +vA,1 +4k−2,2k−1 ⊗ v2,1 +vA,1 +4k−3,2k ⊗ v2,1 ++ vA,1 +4k−2,2k−1 ⊗ v1,2 +vA,1 +4k−2,2k ⊗ v2,1 ++ vA,1 +4k−2,2k−1 ⊗ v2,2 +vA,1 +4k−3,2k+1 ⊗ v2,1 ++ vA,1 +4k−3,2k ⊗ v2,2 ++ vA,1 +4k−2,2k ⊗ v1,2 ++ vA,1 +4k−2,2k−1 ⊗ v1,3 +vA,1 +4k−3,2k+2 ⊗ v2,1 ++ vA,1 +4k−2,2k−1 ⊗ v1,4 +vA,1 +4k−3,2k+3 ⊗ v2,1 ++ vA,1 +4k−3,2k+2 ⊗ v2,2 ++ vA,1 +4k−2,2k ⊗ v1,4 +vA,k +2k−1,6k−3 ⊗ v1,3 +vA,k +2k−1,6k−2 ⊗ v1,3 ++ vA,k +2k−1,6k−3 ⊗ v1,4 +Table 17. The dimension 8 summands, which are all free modules. There are three families +of them, which differ in only how they are expressed below. +Dimension 8 Family 1 summand: +Dimension 8 Family 2 summand: +Dimension 8 Family 3 summand: +vB,i +4k−2−2i,2k−3+4i ⊗ v1,2 +vB,i +4k−1−2i,2k−3+4i ⊗ v1,2 ++ vB,i +4k−2−2i,2k−3+4i ⊗ v2,2 +vB,i +4k−2−2i,2k−2+4i ⊗ v1,2 ++ vB,i +4k−2−2i,2k−3+4i ⊗ v1,3 +vB,i +4k−1−2i,2k−2+4i ⊗ v1,2 ++ vB,i +4k−1−2i,2k−3+4i ⊗ v1,3 ++ vB,i +4k−2−2i,2k−2+4i ⊗ v2,2 +vB,i +4k−2−2i,2k−1+4i ⊗ v1,2 ++ vB,i +4k−2−2i,2k−3+4i ⊗ v1,4 +vA,i +4k−1−2i,2k−1+4i ⊗ v1,2 ++ vB,i +4k−1−2i,2k−3+4i ⊗ v1,4 ++ vB,i +4k−2−2i,2k−1+4i ⊗ v2,2 +vB,i +4k−2−2i,2k+4i ⊗ v1,2 ++ vB,i +4k−2−2i,2k−1+4i ⊗ v1,3 ++ vB,i +4k−2−2i,2k−2+4i ⊗ v1,4 +vA,i +4k−1−2i,2k−1+4i ⊗ v1,3 ++ vB,i +4k−1−2i,2k−2+4i ⊗ v1,4 ++ vB,i +4k−2−2i,2k+4i ⊗ v2,2 +vB,i +4k−2−2i,2k−2+4i ⊗ v1,2 +vB,i +4k−1−2i,2k−2+4i ⊗ v1,2 ++ vB,i +4k−2−2i,2k−2+4i ⊗ v2,2 +vB,i +4k−2−2i,2k−1+4i ⊗ v1,2 ++ vB,i +4k−2−2i,2k−2+4i ⊗ v1,3 +vA,i +4k−1−2i,2k−1+4i ⊗ v1,2 ++ vB,i +4k−1−2i,2k−2+4i ⊗ v1,3 ++ vB,i +4k−2−2i,2k−1+4i ⊗ v2,2 +vB,i +4k−2−2i,2k+4i ⊗ v1,2 ++ vB,i +4k−2−2i,2k−2+4i ⊗ v1,4 +vB,i +4k−1−2i,2k−2+4i ⊗ v1,4 ++ vB,i +4k−2−2i,2k+4i ⊗ v2,2 +vB,i +4k−2−2i,2k+4i ⊗ v1,3 ++ vB,i +4k−2−2i,2k−1+4i ⊗ v1,4 +vB,i +4k−1−2i,2k−1+4i ⊗ v1,4 +vA,i +4k−1−2i,2k−4+4i ⊗ v1,2 +vA,i +4k−2i,2k−4+4i ⊗ v1,2 ++ vA,i +4k−1−2i,2k−4+4i ⊗ v2,2 +vA,i +4k−1−2i,2k−3+4i ⊗ v1,2 ++ vA,i +4k−1−2i,2k−4+4i ⊗ v1,3 +vA,i +4k−2i,2k−4+4i ⊗ v1,3 ++ vA,i +4k−1−2i,2k−3+4i ⊗ v2,2 +vA,i +4k−1−2i,2k−2+4i ⊗ v1,2 ++ vA,i +4k−1−2i,2k−4+4i ⊗ v1,4 +vA,i +4k−2i,2k−4+4i ⊗ v1,4 ++ vA,i +4k−1−2i,2k−2+4i ⊗ v2,2 +vA,i +4k−1−2i,2k−1+4i ⊗ v1,2 ++ vA,i +4k−1−2i,2k−2+4i ⊗ v1,3 ++ vA,i +4k−1−2i,2k−3+4i ⊗ v1,4 +vA,i +4k−1−2i,2k−1+4i ⊗ v2,2 +for i = 1, 2, . . . , k − 1 +for i = 1, 2, . . . , k − 1 +for i = 1, 2, . . ., k − 1 +Figure 15. The location of the pieces for the dimension 12 summands. The outlines show +the positions of the piece with 9 cells, and the shaded region shows the position of the +overlapping piece with 8 cells. +1 +1 +1 +2 +2 +1 +1 +1 +1 +1 + +24 +GEORGE CAO AND KENT B. VASHAW +Table 18. The dimension 12 summands, which can be split into two families (which differ +in only how they are expressed). Each family is characterized in two pieces which overlap at +the indicated equality. The pieces are arranged as in Figure 15. In the dimension 12 Family +1 summand for i = k, some of the vectors do not exist. This construction works with those +nonexistent vectors removed. +Dimension 12 Family 1 summand: +Dimension 12 Family 2 summand: +vA,i +4k−1−2i,2k−4+4i ⊗ v2,1 +vA,i +4k−2i,2k−4+4i ⊗ v2,1 +vA,i +4k−1−2i,2k−3+4i ⊗ v2,1 ++ vA,i +4k−1−2i,2k−4+4i ⊗ v2,2 +vA,i +4k−2i,2k−4+4i ⊗ v2,2 +vA,i +4k−1−2i,2k−2+4i ⊗ v2,1 +vA,i +4k−1−2i,2k−2+4i ⊗ v1,2 ++ vA,i +4k−1−2i,2k−3+4i ⊗ v1,3 ++ vB,i +4k−2−2i,2k−1+4i ⊗ v2,1 +vA,i +4k−1−2i,2k−1+4i ⊗ v2,1 ++ vA,i +4k−1−2i,2k−2+4i ⊗ v2,2 +vA,i +4k−1−2i,2k−1+4i ⊗ v1,2 ++ vA,i +4k−1−2i,2k−3+4i ⊗ v1,4 ++ vB,i +4k−2−2i,2k+4i ⊗ v2,1 ++ vB,i +4k−2−2i,2k−1+4i ⊗ v2,2 +vA,i +4k−1−2i,2k−1+4i ⊗ v1,3 ++ vA,i +4k−1−2i,2k−2+4i ⊗ v1,4 +vB,i +4k−2−2i,2k−2+4i ⊗ v2,1 +vB,i +4k−1−2i,2k−2+4i ⊗ v2,1 +vB,i +4k−2−2i,2k−1+4i ⊗ v2,1 ++ vB,i +4k−2−2i,2k−2+4i ⊗ v2,2 +vA,i +4k−1−2i,2k−1+4i ⊗ v2,1 ++ vB,i +4k−1−2i,2k−2+4i ⊗ v2,2 +vB,i +4k−2−2i,2k+4i ⊗ v2,1 +vB,i +4k−2−2i,2k+4i ⊗ v1,2 ++ vB,i +4k−2−2i,2k−1+4i ⊗ v1,3 ++ vA,i+1 +4k−2−2i,2k+4i ⊗ v1,2 ++ vA,i+1 +4k−2−2i,2k−1+4i ⊗ v1,3 ++ vA,i+1 +4k−3−2i,2k+1+4i ⊗ v2,1 ++ vA,i+1 +4k−3−2i,2k+4i ⊗ v2,2 +vB,i +4k−2−2i,2k+4i ⊗ v2,2 +vB,i +4k−2−2i,2k−1+4i ⊗ v1,4 ++ vA,i+1 +4k−2−2i,2k−1+4i ⊗ v1,4 ++ vA,i+1 +4k−3−2i,2k+2+4i ⊗ v2,1 +vB,i +4k−2−2i,2k+4i ⊗ v1,4 ++ vA,i+1 +4k−2−2i,2k+4i ⊗ v1,4 ++ vA,i+1 +4k−3−2i,2k+2+4i ⊗ v2,2 +vA,i +4k−1−2i,2k−4+4i ⊗ v2,2 ++ vB,i +4k−1−2i,2k−3+4i ⊗ v2,1 += +vA,i +4k−1−2i,2k−3+4i ⊗ v1,2 ++ vB,i +4k−2−2i,2k−2+4i ⊗ v2,1 ++ vB,i +4k−2−2i,2k−3+4i ⊗ v2,2 +vA,i +4k−1−2i,2k−3+4i ⊗ v2,2 ++ vB,i +4k−1−2i,2k−2+4i ⊗ v2,1 ++ vB,i +4k−1−2i,2k−3+4i ⊗ v2,2 += += += += +vA,i+1 +4k−2−2i,2k−1+4i ⊗ v2,1 ++ vB,i +4k−2−2i,2k−2+4i ⊗ v2,2 += +vB,i +4k−2−2i,2k−1+4i ⊗ v1,2 ++ vA,i+1 +4k−2−2i,2k−1+4i ⊗ v1,2 ++ vA,i+1 +4k−3−2i,2k+4i ⊗ v2,1 +vA,i+1 +4k−2−2i,2k+4i ⊗ v2,1 ++ vA,i+1 +4k−2−2i,2k−1+4i ⊗ v2,2 ++ vB,i +4k−2−2i,2k−1+4i ⊗ v2,2 += += += += +for i = 1, 2, . . ., k +for i = 1, 2, . . ., k − 1 + +ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS +25 +Table 19. The pieces of V2k+1. +(A, i) piece: +(B, i) piece: +vA,i−1 +4k+3−2i,2k−5+4i ⊗ v1,2 ++ vA,i−1 +4k+3−2i,2k−6+4i ⊗ v1,3 ++ vB,i−1 +4k+3−2i,2k−6+4i ⊗ v1,3 ++ vA,i−1 +4k+3−2i,2k−7+4i ⊗ v1,4 ++ vA,i +4k+2−2i,2k−4+4i ⊗ v2,1 +vA,i−1 +4k+3−2i,2k−5+4i ⊗ v2,2 ++ vA,i−1 +4k+4−2i,2k−7+4i ⊗ v1,4 ++ vA,i−1 +4k+3−2i,2k−4+4i ⊗ v2,1 +vA,i +4k+2−2i,2k−4+4i ⊗ v1,2 ++ vB,i−1 +4k+2−2i,2k−5+4i ⊗ v1,3 ++ vB,i−1 +4k+2−2i,2k−6+4i ⊗ v1,4 ++ vA,i +4k+1−2i,2k−3+4i ⊗ v2,1 +vA,i−1 +4k+3−2i,2k−4+4i ⊗ v1,2 ++ vB,i−1 +4k+3−2i,2k−5+4i ⊗ v1,3 ++ vB,i−1 +4k+3−2i,2k−6+4i ⊗ v1,4 ++ vA,i +4k+2−2i,2k−3+4i ⊗ v2,1 ++ vA,i +4k+2−2i,2k−4+4i ⊗ v2,2 +vA,i +4k+2−2i,2k−3+4i ⊗ v1,2 ++ vA,i +4k+2−2i,2k−4+4i ⊗ v1,3 ++ vB,i−1 +4k+2−2i,2k−4+4i ⊗ v1,3 ++ vA,i +4k+1−2i,2k−2+4i ⊗ v2,1 ++ vA,i +4k+1−2i,2k−3+4i ⊗ v2,2 +vA,i +4k+2−2i,2k−4+4i ⊗ v1,4 ++ vB,i−1 +4k+2−2i,2k−3+4i ⊗ v1,3 ++ vB,i−1 +4k+2−2i,2k−4+4i ⊗ v1,4 ++ vA,i +4k+1−2i,2k−1+4i ⊗ v2,1 +vA,i +4k+2−2i,2k−3+4i ⊗ v1,4 ++ vA,i +4k+1−2i,2k+4i ⊗ v2,1 ++ vA,i +4k+1−2i,2k−1+4i ⊗ v2,2 +vA,i +4k+1−2i,2k−3+4i ⊗ v1,2 ++ vB,i +4k−2i,2k−2+4i ⊗ v2,1 +vA,i +4k+2−2i,2k−3+4i ⊗ v1,2 ++ vA,i +4k+1−2i,2k−3+4i ⊗ v2,2 ++ vB,i +4k+1−2i,2k−2+4i ⊗ v2,1 +vA,i +4k+1−2i,2k−2+4i ⊗ v1,2 ++ vA,i +4k+1−2i,2k−3+4i ⊗ v1,3 ++ vB,i +4k−2i,2k−1+4i ⊗ v2,1 ++ vB,i +4k−2i,2k−2+4i ⊗ v2,2 +vA,i +4k+2−2i,2k−3+4i ⊗ v1,3 ++ vA,i +4k+1−2i,2k−2+4i ⊗ v2,2 ++ vB,i +4k+1−2i,2k−1+4i ⊗ v2,1 ++ vB,i +4k+1−2i,2k−2+4i ⊗ v2,2 +vA,i +4k+1−2i,2k−1+4i ⊗ v1,2 ++ vA,i +4k+1−2i,2k−3+4i ⊗ v1,4 ++ vB,i +4k−2i,2k+4i ⊗ v2,1 +vA,i +4k+2−2i,2k−3+4i ⊗ v1,4 ++ vA,i +4k+1−2i,2k+4i ⊗ v2,1 ++ vA,i +4k+1−2i,2k−1+4i ⊗ v2,2 +vA,i +4k+1−2i,2k+4i ⊗ v1,2 ++ vA,i +4k+1−2i,2k−1+4i ⊗ v1,3 ++ vA,i +4k+1−2i,2k−2+4i ⊗ v1,4 ++ vB,i +4k−2i,2k+1+4i ⊗ v2,1 ++ vB,i +4k−2i,2k+4i ⊗ v2,2 +for i = 2, 3, . . ., k +for i = 1, 2, . . . , k +(A, k + 1) piece: +(A, 1) piece: +vA,k +2k+1,6k−1 ⊗ v1,2 ++ vA,k +2k+1,6k−2 ⊗ v1,3 ++ vB,k +2k+1,6k−2 ⊗ v1,3 ++ vA,k +2k+1,6k−3 ⊗ v1,4 ++ vB,k +2k,6k ⊗ v2,1 ++ v2k,6k ⊗ v2,1 +vA,k +2k+1,6k−1 ⊗ v2,2 ++ vA,k +2k+2,6k−3 ⊗ v1,4 ++ vA,k +2k+1,6k ⊗ v2,1 +vB,k +2k,6k ⊗ v1,2 ++ v2k,6k ⊗ v1,2 ++ vB,k +2k,6k−1 ⊗ v1,3 ++ vB,k +2k,6k−2 ⊗ v1,4 +vA,k +2k+1,6k ⊗ v1,2 ++ vB,k +2k,6k ⊗ v2,2 ++ v2k,6k ⊗ v2,2 ++ vB,k +2k+1,6k−1 ⊗ v1,3 ++ vB,k +2k+1,6k−2 ⊗ v1,4 +v2k,6k ⊗ v1,3 +vB,k +2k,6k+1 ⊗ v1,3 ++ v2k,6k ⊗ v1,4 +vA,1 +4k,2k ⊗ v2,1 +vA,1 +4k−1,2k+1 ⊗ v2,1 ++ vA,1 +4k,2k ⊗ v1,2 +vA,1 +4k,2k+1 ⊗ v2,1 ++ vA,1 +4k,2k ⊗ v2,2 +vA,1 +4k−1,2k+2 ⊗ v2,1 ++ vA,1 +4k−1,2k+1 ⊗ v2,2 ++ vA,1 +4k,2k+1 ⊗ v1,2 ++ vA,1 +4k,2k ⊗ v1,3 +vA,1 +4k−1,2k+3 ⊗ v2,1 ++ vA,1 +4k,2k ⊗ v1,4 +vA,1 +4k−1,2k+4 ⊗ v2,1 ++ vA,1 +4k−1,2k+3 ⊗ v2,2 ++ vA,1 +4k,2k+1 ⊗ v1,4 + +26 +GEORGE CAO AND KENT B. VASHAW +Table 20. The dimension 8 summands, which are all free modules. There are three families +of them. In fact, these three families are the same as in Table 17, except with different indices +for the first tensorand of every basis vector. +Dimension 8 Family 1 summand: +Dimension 8 Family 2 summand: +Dimension 8 Family 3 summand: +vB,i +4k−2i,2k−2+4i ⊗ v1,2 +vB,i +4k+1−2i,2k−2+4i ⊗ v1,2 ++ vB,i +4k−2i,2k−2+4i ⊗ v2,2 +vB,i +4k−2i,2k−1+4i ⊗ v1,2 ++ vB,i +4k−2i,2k−2+4i ⊗ v1,3 +vB,i +4k+1−2i,2k−1+4i ⊗ v1,2 ++ vB,i +4k+1−2i,2k−2+4i ⊗ v1,3 ++ vB,i +4k−2i,2k−1+4i ⊗ v2,2 +vB,i +4k−2i,2k+4i ⊗ v1,2 ++ vB,i +4k−2i,2k−2+4i ⊗ v1,4 +vA,i +4k+1−2i,2k+4i ⊗ v1,2 ++ vB,i +4k+1−2i,2k−2+4i ⊗ v1,4 ++ vB,i +4k−2i,2k+4i ⊗ v2,2 +vB,i +4k−2i,2k+1+4i ⊗ v1,2 ++ vB,i +4k−2i,2k+4i ⊗ v1,3 ++ vB,i +4k−2i,2k−1+4i ⊗ v1,4 +vA,i +4k+1−2i,2k+4i ⊗ v1,3 ++ vB,i +4k+1−2i,2k−1+4i ⊗ v1,4 ++ vB,i +4k−2i,2k+1+4i ⊗ v2,2 +vB,i +4k−2i,2k−1+4i ⊗ v1,2 +vB,i +4k+1−2i,2k−1+4i ⊗ v1,2 ++ vB,i +4k−2i,2k−1+4i ⊗ v2,2 +vB,i +4k−2i,2k+4i ⊗ v1,2 ++ vB,i +4k−2i,2k−1+4i ⊗ v1,3 +vA,i +4k+1−2i,2k+4i ⊗ v1,2 ++ vB,i +4k+1−2i,2k−1+4i ⊗ v1,3 ++ vB,i +4k−2i,2k+4i ⊗ v2,2 +vB,i +4k−2i,2k+1+4i ⊗ v1,2 ++ vB,i +4k−2i,2k−1+4i ⊗ v1,4 +vB,i +4k+1−2i,2k−1+4i ⊗ v1,4 ++ vB,i +4k−2i,2k+1+4i ⊗ v2,2 +vB,i +4k−2i,2k+1+4i ⊗ v1,3 ++ vB,i +4k−2i,2k+4i ⊗ v1,4 +vB,i +4k+1−2i,2k+4i ⊗ v1,4 +vA,i +4k+1−2i,2k−3+4i ⊗ v1,2 +vA,i +4k+2−2i,2k−3+4i ⊗ v1,2 ++ vA,i +4k+1−2i,2k−3+4i ⊗ v2,2 +vA,i +4k+1−2i,2k−2+4i ⊗ v1,2 ++ vA,i +4k+1−2i,2k−3+4i ⊗ v1,3 +vA,i +4k+2−2i,2k−3+4i ⊗ v1,3 ++ vA,i +4k+1−2i,2k−2+4i ⊗ v2,2 +vA,i +4k+1−2i,2k−1+4i ⊗ v1,2 ++ vA,i +4k+1−2i,2k−3+4i ⊗ v1,4 +vA,i +4k+2−2i,2k−3+4i ⊗ v1,4 ++ vA,i +4k+1−2i,2k−1+4i ⊗ v2,2 +vA,i +4k+1−2i,2k+4i ⊗ v1,2 ++ vA,i +4k+1−2i,2k−1+4i ⊗ v1,3 ++ vA,i +4k+1−2i,2k−2+4i ⊗ v1,4 +vA,i +4k+1−2i,2k+4i ⊗ v2,2 +for i = 1, 2, . . ., k +for i = 1, 2, . . . , k +for i = 1, 2, . . . , k +Table 21. The dimension 12 summands. They are the same as in Table 18, except with +different indicies for the first tensorand of every basis vector and that there is an additional +one of these summands. The pieces are arranged in the same way as Figure 15. +Additional Dimension 12 summand: +vB,k +2k,6k−1 ⊗ v2,1 +vB,k +2k+1,6k−1 ⊗ v2,1 +vB,k +2k,6k ⊗ v2,1 ++ vB,k +2k,6k−1 ⊗ v2,2 +vA,k +2k+1,6k ⊗ v2,1 ++ vB,k +2k+1,6k−1 ⊗ v2,2 +vB,k +2k,6k+1 ⊗ v2,1 +vB,k +2k,6k+1 ⊗ v1,2 ++ v2k,6k ⊗ v1,3 +vB,k +2k,6k+1 ⊗ v2,2 +v2k,6k ⊗ v1,4 +vB,k +2k,6k+1 ⊗ v1,4 +vB,k +2k,6k−1 ⊗ v2,2 ++ vB,k +2k,6k ⊗ v2,1 ++ v2k,6k ⊗ v2,1 += +v2k,6k ⊗ v1,2 +v2k,6k ⊗ v2,2 += += += += + +ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS +27 +Table 21 (continued). +Dimension 12 Family 1 summand: +Dimension 12 Family 2 summand: +vA,i +4k+1−2i,2k−3+4i ⊗ v2,1 +vA,i +4k+2−2i,2k−3+4i ⊗ v2,1 +vA,i +4k+1−2i,2k−2+4i ⊗ v2,1 ++ vA,i +4k+1−2i,2k−3+4i ⊗ v2,2 +vA,i +4k+2−2i,2k−3+4i ⊗ v2,2 +vA,i +4k+1−2i,2k−1+4i ⊗ v2,1 +vA,i +4k+1−2i,2k−1+4i ⊗ v1,2 ++ vA,i +4k+1−2i,2k−2+4i ⊗ v1,3 ++ vB,i +4k−2i,2k+4i ⊗ v2,1 +vA,i +4k+1−2i,2k+4i ⊗ v2,1 ++ vA,i +4k+1−2i,2k−1+4i ⊗ v2,2 +vA,i +4k+1−2i,2k+4i ⊗ v1,2 ++ vA,i +4k+1−2i,2k−2+4i ⊗ v1,4 ++ vB,i +4k−2i,2k+1+4i ⊗ v2,1 ++ vB,i +4k−2i,2k+4i ⊗ v2,2 +vA,i +4k+1−2i,2k+4i ⊗ v1,3 ++ vA,i +4k+1−2i,2k−1+4i ⊗ v1,4 +vB,i +4k−2i,2k−1+4i ⊗ v2,1 +vB,i +4k+1−2i,2k−1+4i ⊗ v2,1 +vB,i +4k−2i,2k+4i ⊗ v2,1 ++ vB,i +4k−2i,2k−1+4i ⊗ v2,2 +vA,i +4k+1−2i,2k+4i ⊗ v2,1 ++ vB,i +4k+1−2i,2k−1+4i ⊗ v2,2 +vB,i +4k−2i,2k+1+4i ⊗ v2,1 +vB,i +4k−2i,2k+1+4i ⊗ v1,2 ++ vB,i +4k−2i,2k+4i ⊗ v1,3 ++ vA,i+1 +4k−2i,2k+1+4i ⊗ v1,2 ++ vA,i+1 +4k−2i,2k+4i ⊗ v1,3 ++ vA,i+1 +4k−1−2i,2k+2+4i ⊗ v2,1 ++ vA,i+1 +4k−1−2i,2k+1+4i ⊗ v2,2 +vB,i +4k−2i,2k+1+4i ⊗ v2,2 +vB,i +4k−2i,2k+4i ⊗ v1,4 ++ vA,i+1 +4k−2i,2k+4i ⊗ v1,4 ++ vA,i+1 +4k−1−2i,2k+3+4i ⊗ v2,1 +vB,i +4k−2i,2k+1+4i ⊗ v1,4 ++ vA,i+1 +4k−2i,2k+1+4i ⊗ v1,4 ++ vA,i+1 +4k−1−2i,2k+3+4i ⊗ v2,2 +vA,i +4k+1−2i,2k−3+4i ⊗ v2,2 ++ vB,i +4k+1−2i,2k−2+4i ⊗ v2,1 += +vA,i +4k+1−2i,2k−2+4i ⊗ v1,2 ++ vB,i +4k−2i,2k−1+4i ⊗ v2,1 ++ vB,i +4k−2i,2k−2+4i ⊗ v2,2 +vA,i +4k+1−2i,2k−2+4i ⊗ v2,2 ++ vB,i +4k+1−2i,2k−1+4i ⊗ v2,1 ++ vB,i +4k+1−2i,2k−2+4i ⊗ v2,2 += += += += +vA,i+1 +4k−2i,2k+4i ⊗ v2,1 ++ vB,i +4k−2i,2k−1+4i ⊗ v2,2 += +vB,i +4k−2i,2k+4i ⊗ v1,2 ++ vA,i+1 +4k−2i,2k+4i ⊗ v1,2 ++ vA,i+1 +4k−1−2i,2k+1+4i ⊗ v2,1 +vA,i+1 +4k−2i,2k+1+4i ⊗ v2,1 ++ vA,i+1 +4k−2i,2k+4i ⊗ v2,2 ++ vB,i +4k−2i,2k+4i ⊗ v2,2 += += += += +for i = 1, 2, . . ., k +for i = 1, 2, . . ., k − 1 +Proof. Similar to the proof of Lemma 3.3.2, the explicit constructions show that the summands are both +subrepresentations and indecomposable. The positions of the summands are summarized in Figure 16, where +the black outline indicates the position of Vi within the tensor product. +Again, it can be checked that these summands span the respective tensor products. This is very similar +to Lemma 3.3.2, and we have omitted the proof. +□ +This yet again verifies that Conjecture 1.0.2 holds for this monomial representation: +Corollary 3.6.3. The function PV (n) is a quasi-polynomial with period 2, given by 6n − 1 for odd n and +6n + 1 for even n. +We close this section by noting that the computations involved in the theorem above resolve an analogue of +a question asked by Benson and Symonds. We first recall some terminology. A G-module M is called Omega- +algebraic if there exist finitely many modules M1, ..., Mm such that every non-projective indecomposable +summand of M ⊗n, for all n, is equal to Ωi(Mj) for some i ∈ Z and j = 1, ..., m ([7, Definition 14.1]). Omega- +algebraic modules, and certain generalizations, have been used in analyzing the dimension of the largest + +28 +GEORGE CAO AND KENT B. VASHAW +Figure 16. Positions of the indecomposable summands in V6 ⊗V (left) and V7 ⊗V (right). +1 +2 +2 +2 +5 +1 +1 +5 +6 +2 +8 +4 +5 +9 +1 +2 +8 +5 +1 +5 +8 +1 +2 +8 +5 +5 +9 +1 +2 +8 +5 +1 +5 +8 +1 +2 +8 +5 +4 +8 +1 +4 +4 +3 +1 +1 +Key +Dim. 8 +Family 1 +Family 2 +Additional +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +2 +2 +1 +1 +1 +1 +1 +1 +1 +1 +2 +2 +1 +1 +1 +1 +1 +1 +1 +1 +2 +2 +1 +1 +1 +1 +1 +1 +2 +2 +2 +5 +1 +1 +5 +6 +2 +8 +4 +5 +9 +1 +2 +8 +5 +1 +5 +8 +1 +2 +8 +5 +5 +9 +1 +2 +8 +5 +1 +5 +8 +1 +2 +8 +5 +5 +9 +1 +1 +7 +5 +2 +6 +1 +3 +3 +2 +1 +non-projective summand of tensor powers [9]. In [7], Benson and Symonds posed the following question (and +provided evidence that its answer was negative): +Question 3.6.4. Must every faithful G-module be Omega-algebraic? +Lemma 3.6.2 shows that the analogue of Benson and Symonds’ question for graded α(r, s)-modules is +false. +Lemma 3.6.5. The (4, 2)/(1) monomial representation is a faithful G-module which is not Omega-algebraic +in the category of graded α(1, 2)-modules. +Proof. Let V be the monomial representation corresponding to the partition (4, 2)/(1). +Note that V is +faithful (as a G-module). It suffices to show that if i ̸= j and i, j > 1, then Vi is not any syzygy Ωk(Vj). +This is equivalent to showing that if i ̸= j, then Vi is not any cosyzygy Ω−k(Vj). +Since all projective modules are free modules in this case, the minimal projective cover of Vj is a direct +sum of many copies of the free module of rank 1. By Lemma 3.6.2, we know that for positive integer j, +the graded representation Vj has homogeneous components with degrees (2j, j), (2j, j + 1), (2j − 1, j + 1), +and (2j − 1, j + 4) that are dimension 1 and homogeneous components with degrees (2j − 1, j + 2) and +(2j − 1, j + 3) that are dimension 2. By the same lemma, we know that there are no other homogeneous +components with degree (2j, n) for n ̸= j, j + 1 and no homogeneous components with degree (2j − 1, n) for +n ̸= j + 1, . . . , j + 4. Also, we know that there are no other homogeneous components with degree (m, n) +with m > 2j or n < j. See Figure 19a for the graded diagram. +This means that the minimal projective cover of Vj has copies of the graded free module of rank 1 +which has the homogeneous component in its top right corner as degrees (2j, j + 1), (2j − 1, j + 3), and +(2j −1, j +4). This implies that the cosyzygy of Vj has homogeneous components with degrees (2j −1, j −2), +(2j, j−2), (2j, j−1), (2j−1, j−1) that are dimension 1 and homogeneous components with degrees (2j−1, j) +and (2j − 1, j + 1) that are dimension 2. Also, we know that there are no other homogeneous components +with degree (2j, n) for some other n, no other homogeneous components with degree (2j − 1, n) for some +other n, no homogeneous components with degree (m, n) for m > 2j or n < j − 2. See Figure 19b for the +graded diagram. + +ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS +29 +Figure 18. The bottom right of the graded diagram of Vj and cosyzygies, displaying the +dimensions of the homogeneous components. +(a) The rightmost columns of Vj. +1 +1 +1 +2 +2 +1 +... +(b) The rightmost columns of Ω−1(Vj). +1 +1 +1 +1 +2 +2 +... +The formation of the rightmost two columns of Ω−1(Vj) is invariant to cosyzygies. Specifically, if Ω−k(Vj) +has its rightmost two columns with the relative locations and dimensions as in Figure 19b, then let the +degrees be (m, n), (m + 1, n), (m, n + 1), and (m + 1, n + 1) for the dimension 1 homogeneous components +and (m, n + 2) and (m, n + 3) for the dimension 2 homogeneous components. Then, the minimum projective +cover has copies of the rank 1 free module, with the top right corners of these free modules having degrees +(m + 1, n + 1) and (m, n + 3), with the latter having multiplicity 2. Then, the cosyzygy Ω−k−1(Vj) has +homogeneous components with degrees (m, n − 2), (m + 1, n − 2), (m, n − 1), and (m + 1, n − 1) that are +dimension 1 and has homogeneous components with degrees (m, n) and (m, n+1) that are dimension 2. This +is the exact same relative locations and dimensions as the homogeneous components shown in Figure 19b, +so this is invariant. +Hence, all Ω−k(Vj) for k > 0 have rightmost columns as in Figure 19b. However, none of the Vi have +these rightmost columns by Lemma 3.6.2. Therefore, Vi ̸∼= Ω−k(Vj) for k > 0, as desired. +□ +4. Further questions and conjectures +The data on tensor powers computed with Magma prompts the following questions, which we observe +hold in many cases: +Question 4.0.1. If V is an odd-dimensional indecomposable monomial representation, then we have that +(dim V )n ≡ PV (n) (mod 4). In particular, dim V2k ≡ 1 (mod 4) and dim V2k+1 ≡ dim V (mod 4). +Question 4.0.2. Let V be an odd-dimensional indecomposable monomial representation. Then, V ⊗n is the +direct sum of an odd-dimensional indecomposable representation (with dimension PV (n)) and indecomposable +representations with dimensions divisible by 4. +Table 22 summarizes the monomial modules that we have computational evidence for the polynomial +or quasi-polynomial (computational evidence defined by having at least n + 2 data points for a degree n +polynomial guess). The notation [f(x), g(x)] is defined as a quasi-polynomial that is f(x) when x is odd and +g(x) when x is even. Note that all quasi-polynomials in the table are either of period 1 or 2. +The monomial representations are not listed in any important order. +The ones not listed are either +proved previously (180◦-symmetric; (2m, 1); staircase; (3, 1, 1); or (4, 2)/(1) partitions) or have been difficult +to compute sufficient data for. Also, note that some monomial representations are Vi’s for smaller monomial +partitions. For example, the (5, 4, 1)/(1) monomial representation is V2 for the monomial representation +V := (4, 1), so if PV (n) = 4n + 1 then PV2(n) = 8n + 1, as supported by computation in the table below. +Also, note that there are cases of monomial representations in which PV (n) is not linear or quasi-linear. +For example, the monomial representation given by the partition (6, 1) is conjectured (with computational +evidence), but is not yet proven, to have PV (n) = 2n2 + 4n + 1. + +30 +GEORGE CAO AND KENT B. VASHAW +Table 22. Monomial representations and conjectured polynomials from computational ev- +idence. +Partition +Computed Quasi-polynomial +(3, 2) +[10x − 5, 6x + 1] +(6, 1) +2x2 + 4x + 1 +(5, 1, 1) +[18x − 11, 10x + 1] +(4, 3) +[4x + 3, 4x + 1] +(4, 1, 1, 1) +[8x − 1, 8x + 1] +(6, 2)/(1) +[10x − 3, 10x + 1] +(5, 3)/(1) +[12x − 5, 12x − 7] +(5, 2, 1)/(1) +6x + 1 +(4, 3, 1)/(1) +[12x − 4, 12x + 1] +(4, 2, 1, 1)/(1) +[20x − 13, 12x + 1] +(6, 3)/(2) +[4x + 3, 8x + 1] +(5, 3, 1)/(2) +[8x − 1, 12x + 1] +(5, 2, 2)/(1, 1) +[10x − 3, 10x + 1] +(4, 4, 1)/(2) +[12x − 5, 12x − 7] +(4, 3, 2)/(2) +[14x − 7, 10x + 1] +(4, 3, 2)/(1, 1) +38x − 31 +(5, 4, 1)/(3) +2x2 + 4x + 1 +(5, 3, 2)/(2, 1) +[10x − 3, 10x + 1] +(7, 1, 1) +[20x − 11, 12x + 1] +(5, 4) +[20x − 11, 12x + 1] +(4, 4, 1) +[12x − 3, 12x + 1] +(8, 2)/(1) +[12x − 3, 12x + 1] +(7, 3)/(1) +12x2 − 4x + 1 +(7, 2, 1)/(1) +2x2 + 6x + 1 +(6, 4)/(1) +48x − 39 +(5, 4, 1)/(1) +8x + 1 +(5, 2, 1, 1, 1)/(1) +8x + 1 +(4, 4, 1, 1)/(1) +[12x − 3, 12x + 1] +(7, 3, 1)/(2) +[18x − 9, 18x + 1] +(6, 4, 1)/(2) +16x − 7 +(6, 3, 2)/(2) +[14x − 5, 10x + 1] +(6, 3, 1, 1)/(2) +[18x − 9, 14x + 1] +(5, 5, 1)/(2) +8x2 + 1 +(5, 4, 2)/(2) +16x − 7 +(5, 4, 2)/(1, 1) +8x2 + 1 +(5, 3, 1, 1, 1)/(2) +[10x − 1, 18x + 1] +(8, 4)/(3) +[12x − 3, 12x + 1] +(7, 4, 1)/(3) +[8x + 1, 12x + 1] +(7, 3, 2)/(2, 1) +[16x − 7, 16x + 1] +(6, 3, 2, 1)/(2, 1) +8x + 1 +(5, 5, 2)/(2, 1) +8x + 1 +(5, 4, 3)/(3) +8x + 1 +(4, 4, 3, 1)/(3) +56x − 47 +(7, 5, 1)/(4) +[12x − 3, 16x + 1] +(7, 4, 2)/(3, 1) +[10x − 1, 10x + 1] +(6, 4, 2, 1)/(3, 1) +[10x − 1, 18x + 1] +(6, 3, 3, 1)/(2, 2) +64x − 55 +(5, 5, 2, 1)/(3, 1) +[14x − 5, 14x − 7] + +ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS +31 +(5, 5, 1, 1, 1)/(4) +[16x−, 16x + 1] +(5, 3, 3, 2)/(2, 2) +[14x − 5, 10x + 1] +(7, 6, 1)/(5) +4x2 + 4x + 1 +(7, 5, 2)/(4, 1) +[12x − 3, 12x + 1] +(6, 5, 2, 1)/(4, 1) +2x2 + 6x + 1 +(6, 4, 3, 1)/(3, 2) +[12x − 3, 16x + 1] +(5, 5, 3, 1)/(4, 1) +16x − 7 +(6, 4, 3, 2)/(3, 2, 1) +[14x − 5, 14x + 1] +(6, 5, 4, 1)/(4, 3) +2x2 + 6x + 1 +(6, 5, 3, 2)/(4, 2, 1) +[12x − 3, 12x + 1] +(5, 5, 2, 2, 2)/(4, 1, 1, 1) +4x2 + 4x + 1 +5. Acknowledgements +We would like to thank Pavel Etingof for suggesting this project and giving valuable advice along the +way. We are also grateful to Dave Benson and Peter Symonds for helpful discussions. This research was +conducted while the first author was a participant in the MIT PRIMES-USA program, which we thank +for making this research opportunity possible. Research of K. B. Vashaw was partially supported by NSF +Postdoctoral Fellowship DMS-2103272. +References +[1] J. L. Alperin. Local Representation Theory: Modular representations as an introduction to the local representation theory +of finite groups. Cambridge Studies in Advanced Mathematics, 11. Cambridge University Press, Cambridge, 1986. x+178 pp. +[2] D. Benson. Modular Representation Theory: New Trends and Methods. Springer-Verlag Berlin Heidelberg, 1984. +[3] D. Benson. Representations and Cohomology I: Basic Representation Theory of Finite Groups and Associate Algebras. +Cambridge University Press, Cambridge, 1998. +[4] D. Benson. Some conjectures and their consequences for tensor products of modules over a finite p-group. J. Algebra, 558 +(2020), 24–42, DOI 10.1016/j.jalgebra.2019.10.012. +[5] D. Benson. Modular representation theory and commutative Banach algebras. arXiv:2008.13155. To appear in Mem. Amer. +Math. Soc. +[6] D. J. Benson and J. F. Carlson. Nilpotent elements in the Green ring. J. Algebra 104 (1986), 329–350. +[7] D. Benson and P. Symonds. The non-projective part of the tensor powers of a module. J. London Math. Soc., 101 (2019), +no. 2, 828–856, DOI 10.1112/jlms.12288. +[8] W. Bosma and J. Cannon. Handbook of Magma Functions, Magma Computer Algebra, Sydney, 1996. +[9] A. Chirvasitu, T. Hudson and A. Upadhyay, Recursive sequences attached to modular representations of finite groups. J. +Algebra 602 (2022), 599–636. +[10] K. Coulembier, P. Etingof, and V. Ostrik. On Frobenius exact symmetric tensor categories. arXiv:2107.02372. +[11] K. Coulembier, V. Ostrik, and D. Tubbenhauer. Growth rates of the number of indecomposable summands in tensor powers. +arXiv:2301.00885. +[12] P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, E. Yudovina. Introduction to Representation Theory +with historical interludes by Slava Gerovitch. Student Mathematical Library, 59. American Mathematical Society, Providence, +RI, 2011. viii+228 pp. +[13] P. Etingof and A. S. Kannan. Lectures on symmetric tensor categories. arXiv:2103.04878. +[14] P. Etingof and V. Ostrik, On semisimplification of tensor categories, Representation theory and algebraic geometry—a +conference celebrating the birthdays of Sasha Beilinson and Victor Ginzburg, 3–35, Trends Math., Birkh¨auser/Springer, Cham, +2022. +[15] J. C. Jantzen. Representations of algebraic groups. Mathematical Surveys and Monographs, 107. American Mathematical +Society, Providence, RI, 2003. +[16] S. Montgomery. Hopf algebras and their actions on rings. CBMS Regional Conference Series in Mathematics, 82. Published +for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, +RI, 1993. xiv+238 pp. +[17] W. C. Waterhouse. Introduction to Affine Group Schemes. Springer-Verlag New York, 1979. +[18] P. Webb. A Course in Finite Group Representation Theory. Cambridge University Press, Cambridge, 2016. +Montgomery High School, Skillman, NJ 08558, U.S.A. +Email address: george.cao@mtsdstudent.us +Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. +Email address: kentv@mit.edu + diff --git a/O9E3T4oBgHgl3EQfCQlj/content/tmp_files/load_file.txt b/O9E3T4oBgHgl3EQfCQlj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4da5428a4a83259aa5238431476bc7bdea104230 --- /dev/null +++ b/O9E3T4oBgHgl3EQfCQlj/content/tmp_files/load_file.txt @@ -0,0 +1,4253 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf,len=4252 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='04274v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='RT] 11 Jan 2023 ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS GEORGE CAO AND KENT B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' VASHAW Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Dave Benson conjectured in 2020 that if G is a finite 2-group and V is an odd-dimensional indecomposable representation of G over an algebraically closed field k of characteristic 2, then the only odd-dimensional indecomposable summand of V ⊗V ∗ is the trivial representation k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' This would imply that a tensor power of an odd-dimensional indecomposable representation of G over k has a unique odd-dimensional summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Benson has further conjectured that, given such a representation V , the function sending a positive integer n to the dimension of the unique odd-dimensional indecomposable summand of V ⊗n is quasi-polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We examine this conjecture for monomial modules, a class of graded representations for the group Z/2rZ×Z/2sZ which correspond to skew Young diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We prove the tensor powers conjecture for several modules, giving some of the first nontrivial cases where this conjecture has been verified, and we give conjectural quasi-polynomials for a broad range of monomial modules based on computational evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Introduction It is well-known that the representation theory of finite groups bifurcates, depending on the characteristic of the field that the representations are taken over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' When the field k is characteristic 0, or more generally when the characteristic does not divide the order of the finite group G, then all representations of G over k are semisimple, and a full understanding of the tensor products of representations for G can be obtained by finitely many computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' On the other hand, when the characteristic of k divides the order of G, it is often no longer possible, in many cases, to compute all the indecomposable representations, and even very basic questions about the decompositions of tensor products of G-representations remain a mystery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Several of these open questions will be the primary focus of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Let k be an algebraically closed field of characteristic p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Based on patterns observed from a significant body of computational evidence, Dave Benson has proposed the following conjecture [4, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1]: Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 (Benson).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Let p = 2 and G a finite 2-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' If V is an odd-dimensional indecomposable representation of kG, then V ⊗V ∗ is a direct sum of k and indecomposable representations whose dimensions are divisible by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The weaker version of this conjecture, in which the indecomposable representations other than k have even dimension, is also unproven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' It is well-known that a summand isomorphic to k always exists in the decomposition for V ⊗ V ∗ [1, Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4], and indeed has multiplicity 1 [6, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' If true, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 (or its weaker analogue) would also imply that any tensor power V ⊗n of V has a unique odd-dimensional indecomposable summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Denote this summand by Vn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Benson introduced the function PV : n �→ dim(Vn) in [5, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' He made the following conjecture: Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2 (Benson).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The function PV (n) is quasi-polynomial, that is, there exist polynomials f0, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' , fm−1 such that PV (n) = fi(n) if n ≡ i (mod m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' This conjecture is closely related to work of Benson and Symonds [7], where a similar conjecture was made for the dimension of the largest non-projective summand of a tensor power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' In the case where V is not necessarily taken to be indecomposable, questions related to the number of indecomposable summands of V ⊗n of dimension coprime to p have been recently approached via the theory of symmetric tensor categories, see [10, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3] and [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Benson’s conjectures may be formulated in a tensor-categorical context via the notion of semisimplification of a tensor category [14], although such a formulation does not play a role in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1 2 GEORGE CAO AND KENT B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' VASHAW Benson’s tensor powers conjecture (Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2) is unknown even in the simplest nontrivial exam- ples [13, Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We contribute by analyzing the tensor powers conjecture for a class of represen- tations called monomial representations, which correspond to skew Young diagrams, for a certain group scheme α(r, s), which is isomorphic (as an algebra, although not as a coalgebra) to the group algebra of Z/2rZ × Z/2sZ over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' In particular, we prove the following: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3 (See Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3, and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Let k be an algebraically closed field of characteristic 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (1) Let V be the monomial representation for α(1, 2) corresponding to the Young diagram .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Then V ⊗n has a unique odd-dimensional indecomposable summand, and the dimension is given by 4n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (2) Let V be the monomial representation for α(2, 2) corresponding to the Young diagram .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Then V ⊗n has a unique odd-dimensional indecomposable summand, and the dimension is given by 10n − 5 for odd n and 6n + 1 for even n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (3) Let V be the monomial representation for α(1, 2) corresponding to the skew Young diagram .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Then V ⊗n has a unique odd-dimensional indecomposable summand, and the dimension is given by 6n − 1 for odd n and 6n + 1 for even n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We accomplish this by explicitly constructing the unique odd-dimensional indecomposable summand Vn of V ⊗n, by decomposing the tensor product V ⊗ Vn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' It is enough to consider this tensor product, as opposed to the full V ⊗n, since the tensor product of an even-dimensional representation with any other representation breaks down into even-dimensional summands [4, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We also provide a list of quasi-polynonomials which we conjecture, based on computational evidence, to give the function PV (n) for a number of monomial representations V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' This list includes quasi-polynomials of period 1 and 2, and the polynomials themselves are either linear or quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' In Section 2, we give background on the group schemes α(r, s) and the monomial representations which are the main object of study in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' In Section 3, we prove Benson’s tensor powers conjecture (Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2) for several monomial representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' There are no computations of this sort in the literature, and they give the first known examples where the tensor powers conjecture is verified in a way which does not follow from general homological algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' In the process, we give an example which answers an analogue of a question asked by Benson and Symonds in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Lastly, in Section 4, we introduce several additional questions and conjectures based on computational evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The group scheme α(r, s) and monomial representations Throughout this paper, unless otherwise stated, k denotes an algebraically closed field of characteristic 2, and ⊗ will mean ⊗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The group scheme α(r, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The tensor product and duality operations for group representations (see [12, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4]) are given by a canonical structure on the group algebra kG of a cocommutative Hopf algebra, or affine group scheme [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' In this section, we define an alternative group scheme structure on the group algebra kG, which is used to define all tensor products of representations over the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The benefit of using this alternate group scheme structure is that the tensor product of the representations we consider will respect a certain natural grading of these representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' However, based on computational ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS 3 evidence, we do conjecture that for the representations in this paper, the use of this alternative group scheme structure does not effect the dimensions of the indecomposable summands of the tensor powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Let G := Z/2rZ × Z/2sZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Then G has a minimal generating set {g, h}, where g and h commute and g2r = 1 = h2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The group algebra kG is the formal span of elements of G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' by abuse of notation, we let g and h again refer to the corresponding vectors in kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Set x := g + 1 and y := h + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Both x and y are nilpotent, since x2r = (g + 1)2r = g2r + 12r = 1 + 1 = 0, and similarly for y, using the fact that k is characteristic 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Since g and h generate kG as an algebra, then to define a (unital) algebra map from kG, it suffices to specify its value on x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Define α(r, s) as the Hopf algebra which is isomorphic to kG as algebras, but where we use the comultiplication structure α(r, s) → α(r, s) ⊗ α(r, s) defined by x �→ x ⊗ 1 + 1 ⊗ x, y �→ y ⊗ 1 + 1 ⊗ y, and antipode which is the identity map on both x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' This corresponds to a generalization of the affine group scheme αp given in [17, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1] and is a special case of a restricted enveloping algebra of a Lie algebra, as in [15, Section I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' For the remainder of the paper, unless stated otherwise, the tensor product and dual of representations will be defined by the Hopf algebra structure given in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The analogues of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 and Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2 hold for the group scheme α(r, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Lastly, we define a Z2-grading on α(r, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Let x and y be the elements of α(r, s) as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Define x and y to have degrees (1, 0) and (0, 1) ∈ Z2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' It is straightforward to observe that not only does this give α(r, s) the structure of a graded algebra, but also the structure of a graded coalgebra, and in fact that of a graded Hopf algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Indeed, this is one of the primary motivations for using the group scheme α(r, s) as opposed to kG for G = Z/2rZ × Z/2sZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Recall that if H is a graded Hopf algebra and V and W are graded modules for H, then V ∗ and V ⊗ W are also graded modules for H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Monomial representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Fix two positive integers r and s, and denote G := Z/2rZ × Z/2sZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We denote the elements x and y of kG (or, what is the same, α(r, s)) as in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' For the remainder of the paper, we concentrate on Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2 for a specific class of representations of G, which we define now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Choose a partition λ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' , λn) such that λ1 ≥ · · · ≥ λn > 0 and a sub-partition µ = (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' , µm) such that µ1 ≥ · · · ≥ µm > 0 with m < n and λi ≥ µi, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The pair (λ, µ) will be collectively referred to as a skew partition, and will typically be denoted λ/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The monomial representation of G corresponding to λ/µ has basis vectors vi,j such that 1 ≤ i ≤ n and µi < j ≤ λi, where we set µq := 0 for q > m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The action of x sends vi,j to vi+1,j and the action of y sends vi,j to vi,j+1, if such basis elements exist, and otherwise sends them to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Note that the actions of x and y commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' In order for the monomial representation associated to λ/µ to exist (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', to actually be a G-representation), we require that for any i, there can only be at most 2s values of j such that vi,j is a basis vector, and similarly with the roles of i and j reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' It is clear that, taking α(r, s) to be a graded Hopf algebra as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3, then a monomial representation for α(r, s) is a graded representation, where vi,j is taken to be in degree (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Let V be a monomial representation of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' A monomial diagram is the diagram resulting from drawing the grid box (i, j) if vi,j is a basis vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Each grid box is called a cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' In other words, the monomial diagram for the monomial representation corresponding to the skew partition λ/µ is the same thing as the skew Young diagram of λ/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 4 GEORGE CAO AND KENT B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' VASHAW Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Let D be the monomial diagram for the monomial representation V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Then D is connected if, for any choice of cells B1, B2, there exists a sequence of cells B1 = C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' , Cn = B2 for some n such that Ci is edge-adjacent to Ci+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' If G is Z/4Z × Z/2Z, then the monomial representation corresponding to skew partition (5, 4, 2, 2, 1, 1)/(3, 2) would have a basis vectors v1,4, v1,5, v2,3, v2,4, v3,1, v3,2, v4,1, v4,2, v5,1, v6,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Its monomial diagram is shown in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' If G were Z/2Z × Z/4Z, then the monomial diagram for the representation corresponding to (4, 4, 2, 1)/(3, 1) is shown in Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Examples of monomial diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (a) (b) One may read the action of G on the monomial representation V directly from the monomial diagram for V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' There is a basis of V corresponding to the cells of the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The action of x, in the diagram, is moving to the right by one cell, if the cell exists, and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The action of y, in the diagram, is moving up by one cell, if the cell exists, and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' A monomial representation is indecomposable if and only if its monomial diagram is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' In Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4, the representation given by Figure 2a is not indecomposable, since there are two disconnected parts: the four white boxes in the first two columns are disconnected from the 6 white boxes in columns three to six.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' This means that this representation can be written as a direct sum of a representation of dimension 4 and a representation of dimension 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' On the other hand, the monomial representations given by Figure 2b is indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The following lemmas establish that the dimensions of the indecomposable summands of V ⊗V ∗ and V ⊗n are not dependent on orientation of the monomial diagram for V , but only the shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Let V be a monomial representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Let W be the monomial representation that results from reflecting its monomial diagram about the line from bottom left to top right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Then, the dimensions of the indecomposable summands of V ⊗ V ∗ are the same as the dimensions of the indecomposable summands of W ⊗ W ∗ (up to permutation), and the dimensions of the indecomposable summands of V ⊗n are the same as the dimensions of the indecomposable summands of W ⊗n (up to permutation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Under the isomorphism α(r, s) ∼= α(r, s) which interchanges x and y, we have that V corresponds to W, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Let V be a monomial representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The monomial representation that results from rotating its monomial diagram 180◦ gives a monomial representation isomorphic to V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Let V be a monomial representation with basis vectors v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' , vn in the cells of its monomial diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Let W be the monomial representation resulting from a 180◦ rotation of the monomial diagram of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Let the basis vectors of W be w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' , wn such that they are the image of v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' , vn, respectively, under this rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Consider ϕ : W → V ∗ defined by ϕ(wi)vj = 1 if i = j and 0 if i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' This is an isomorphism of vector spaces, so to show that it is an isomorphism of representations we simply check that it respects the action of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We have ϕ(ρW (x)wi)vj = � 1 if ρW (x)wi = wj 0 otherwise, (ρV ∗(x) · ϕ(wi))vj = ϕ(wi)(ρV (x)vj) = � 1 if ρV (x)vj = vi 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS 5 We have ρW (x)wi = wj if and only if wj is to the right of wi (in the monomial diagram of W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' This is true if and only if vj is to the left of vi, which is true if and only if ρV (x)vj = vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Thus, ϕ(ρW (x)wi)vj = (ρV ∗(x) · ϕ(wi))vj for all i, j, so ϕ is a homomorphism of representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Therefore, ϕ must be an isomorphism of representa- tions, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Computed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Using the computer algebra system Magma [8], we can generate all possible mono- mial diagrams of monomial representations V for a selected dimension of V , and compute the dimensions of the indecomposable summands of V ⊗ V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Table 1, Table 2, and Table 3 contain all monomial diagrams for monomial representations of the specified dimension, along with the dimensions of the indecomposable summands of V ⊗ V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Monomial diagrams that are the same shape but different orientation are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' All of these examples satisfy Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' No pattern between the dimensions of the indecomposable summands and the shape of the monomial diagram has been conjectured yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' In fact, for each example in the table below, additional data was collected using the group tensor product, and the dimensions of the indecomposable summands were the same as the same as using the tensor product with the comultiplication structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We formulate this conjecture: Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Let V be an odd-dimensional indecomposable monomial representation of G = Z/2rZ × Z/2sZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Let ⊗ and (−)∗ denote the tensor product and dual of G-representations defined using the Hopf algebra structure for α(r, s), and ⊗′ and (−)∨ denote the tensor product and dual of G-representations using the standard Hopf algebra structure on kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We conjecture that there are isomorphisms of G-representations V ⊗ V ∗ ∼= V ⊗′ V ∨, V ⊗n ∼= V ⊗′n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Dimension 3 monomial representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Dimensions Monomial Diagrams [1, 4, 4] Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Dimension 5 monomial representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Dimensions Monomial Diagrams [1, 12, 12] [1, 4, 4, 8, 8] [1, 4, 4, 4, 4, 4, 4] 6 GEORGE CAO AND KENT B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' VASHAW Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Dimension 7 monomial representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Dimensions Monomial Diagrams [1, 4, 4, 20, 20] [1, 8, 8, 16, 16] [1, 24, 24] [1, 48] [1, 8, 8, 8, 8, 8, 8] [1, 4, 4, 4, 4, 8, 8, 8, 8] [1, 4, 4, 4, 4, 16, 16] [1, 4, 4, 40] [1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4] ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Benson’s tensor powers conjecture for monomial modules In this section, we consider Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2 for α(r, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Basic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We use the notation given in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Let V be an odd-dimensional monomial module for α(r, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We assume that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 holds for α(r, s), so that V ⊗i has a unique odd- dimensional indecomposable summand, which we denote by Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Recall that PV (n) will denote the dimension of Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We may also characterize Vi in the following recursive way: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' There is a unique odd-dimensional indecomposable summand of Vi ⊗ V , which is iso- morphic to Vi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We have V ⊗n = V ⊗(n−1) ⊗ V = (Vn−1 ⊕ W1 ⊕ · · · ⊕ Wm) ⊗ V , where dim Wi is even for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Then, V ⊗n = (Vn−1 ⊗ V ) ⊕ (W1 ⊗ V ) ⊕ · · · ⊕ (Wm ⊗ V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The only odd-dimensional indecomposable summand of this is Vn, by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Since dim(Vn−1 ⊗ V ) is odd, then Vn must be an indecomposable summand of Vn−1 ⊗ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' □ For self-dual modules, it is straightforward to compute PV (n), as follows: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' If the monomial diagram of V is symmetric by rotation of 180◦, then Vodd = V and Veven = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' In particular, PV (n) = � dim V if n odd 1 if n even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6, V is self-dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Recall that the trivial representation k is a summand of V ⊗ V ∗ ∼= V ⊗ V , so PV (2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Then V2 ⊗ V = k ⊗ V = V , so V3 = V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Thus PV (3) = dim V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' By induction, we find that Vodd = V and Veven = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' “Staircase” monomial representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We next consider the monomial representations for α(1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We refer to these representations in the following way, based on their diagrams’ appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Call the monomial representation given by the partitions (m, m − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' , 1)/(m − 2, m − 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' , 1) the m-staircase monomial representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' From the classification of all indecomposable summands of Z/2Z × Z/2Z in [2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 176], we know that Ωm(k) are these odd-dimensional indecomposable representations, where Ωm(V ) is the m-th syzygy, defined as the kernel of the m-th map in the projective resolution of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We can calculate these syzygies and it can be shown that the m-staircase monomial representation is Ω1−m(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' To examine tensor powers, we need two lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The following lemma is found in [3, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6] in the finite group case, and the same proof holds for any finite group scheme, in particular α(r, s): Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' If V and W are representations of kG for a finite group G, then Ω(V ) ⊗ W is isomorphic to Ω(V ⊗ W), up to projective summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Also, we need the following lemma given in [18, Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3]: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' If k is an algebraically closed field of characteristic 2 and 2 divides |G|, then all projective representations of G have even dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We can prove the following proposition about the tensor powers of staircase monomial representations: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' If V is the m-staircase representation, then Vn is the (mn − n + 1)-staircase monomial representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We have that V = Ω1−m(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Then V ⊗n ∼= (Ω1−m(k))⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2, this is isomorphic, up to projective summands, to Ω(1−m)n(k⊗n) = Ωn−mn(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3, all projective representations are even dimensional, so Vn, the odd-dimensional summand of V ⊗n, is also the odd-dimensional summand of Ωn−mn(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' However, Ωn−mn(k) is the (mn − n + 1)-staircase monomial representation, which has odd- dimension and is indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Thus Vn is the (mn − n + 1)-staircase monomial representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' □ It immediately follows that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2 is satisfied in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 8 GEORGE CAO AND KENT B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' VASHAW Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' If V is the m-staircase module, then the function PV (n) is given by the linear polynomial (2m − 2)n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' It is shown in [4] that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 is true for G = Z/2Z × Z/2Z in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (4, 1) Monomial representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Let V be the monomial representation of Z/2Z × Z/4Z (or α(1, 2)) corresponding to the partition (4, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We have the following decomposition into indecomposable summands: V2k ⊗ V = V2k+1 ⊕ F ⊕ · · · ⊕ F � �� � 4k copies and V2k−1 ⊗ V = V2k ⊕ W ⊕ W ⊕ F ⊕ · · · ⊕ F � �� � 4k−3 copies , where F is a free module of dimension 8 and W is dimension 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' In order to prove this, we explicitly give the decomposition, via the following stronger lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We claim the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (1) The representation V2k is the direct sum of 1-dimensional homogeneous components V4k,2k and V2k−1+i,j where j = 6k − 1 − 2i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', 6k + 2 − 2i for i = 1, 3, 5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', 2k − 1 and j = 6k − 2i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' , 6k + 3 − 2i for i = 2, 4, 6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (2) The representation V2k−1 is the direct sum of 1-dimensional homogeneous components V4k−2,2k−1 and V2k−2+i,j where j = 6k − 3 − 2i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', 6k − 2i for i = 1, 3, 5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', 2k − 1 and j = 6k − 4 − 2i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' , 6k − 1 − 2i for i = 2, 4, 6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', 2k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (3) The representation V2k ⊗ V decomposes into the summands whose graded monomial diagrams are shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (4) The representation V2k−1 ⊗ V decomposes into the summands whose graded monomial diagrams are shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Indecomposable summands of V2k ⊗ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' V2k+1 summand: Dimension 8 Family 1 summand: v2k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6k ⊗ v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 v2k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6k ⊗ v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2 v2k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6k ⊗ v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3 v2k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6k ⊗ v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4 V2k ⊗ v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 v2k−1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6k−2i ⊗ v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 v2k+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6k−2i ⊗ v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 + v2k−1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6k−2i ⊗ v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 v2k−1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6k+1−2i ⊗ v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 + v2k−1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6k−2i ⊗ v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2 v2k−1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6k+1−2i ⊗ v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 + v2k+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6k−2i ⊗ v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2 v2k−1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6k+2−2i ⊗ v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 + v2k−1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6k−2i ⊗ v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3 v2k−1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6k+2−2i ⊗ v2,' metadata={'source': 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+page_content='3 + v2k−2+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6k−2−2i ⊗ v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4 v2k−1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6k−2−2i ⊗ v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4 for i = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', 2k − 2 for i = 2, 4, 6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', 2k − 2 for i = 2, 4, 6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', 2k − 2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We proceed by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' It can be checked that (2) is true for k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' For the inductive step, assume (2) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Consider the decomposition given in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' First, we show that each of these summands are subrepresentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' It can be checked in each case that the vectors chosen in the diagrams above are chosen such that the action of x takes the displayed vector to the vector in the box adjacent to the right (or 0 if that box does not exist) and the action of y takes the displayed vector to the vector in the box adjacent above (or 0 if that box does not exist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Since each vector is a basis for the homogeneous component it is in, then each diagram is closed under the action of x and y, and all summands are subrepresentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Each of the claimed summands is indecomposable, since all of the diagrams are connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Finally, we must show that the direct sum of the indecomposable subrepresentations shown is the original representation V2k−1 ⊗ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' First, we must know where the summands are located relative to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Figure 3 shows the dimensions of the homogeneous components of V , V1 ⊗ V , V2 ⊗ V , V3 ⊗ V , and V4 ⊗ V , respectively from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The black outline is the location of Vi, the diamonds show the positions of the bottom left cells of the dimension 8 free modules, and the circles show the positions of the bottom cells of the dimension 4 summands, as shown in the key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' This pattern is generalized to higher n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 10 GEORGE CAO AND KENT B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' VASHAW Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Positions of the indecomposable summands in the first five tensor powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='For every homogeneous component of V2k−1 ⊗ V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' we can check that the vectors in the shown summands span the vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' For example, it can be checked that {v2k−1,6k−2 ⊗ v1,2 + v2k−1,6k−3 ⊗ v1,3 + v2k−1,6k−4 ⊗ v1,4, v2k−1,6k−2 ⊗ v1,2 + v2k−1,6k−3 ⊗ v1,3, v2k−1,6k−2 ⊗ v1,2 + v2k−1,6k−4 ⊗ v1,4} spans the homogeneous component of degree (2k, 6k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We do similar verifications for the other homogeneous components, which is omitted from this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' This means that (4) follows from (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' By a similar argument, it can be shown that (3) follows from (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Considering the degrees of the homogeneous components show that (1) and (2) follow from (4) and (3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' By induction, the lemma follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1 follows from this lemma, and as a corollary we observe that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2 is satisfied in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The function PV (n) is given by 4n + 1, a linear polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (4, 1) Monomial representation using syzygies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' As in the previous section, let V be the monomial representation given by the partition (4, 1) and let G = Z/2Z × Z/4Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The monomial diagram of V and the monomial diagram for the free G-module of rank 1 are given, respectively: , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Since the free G-module of rank 1 is indecomposable, all projective modules are free modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The syzygy for V appears as the first module in the following short exact sequence: 0 → → → → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Define N = Ω(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Then V = Ω−1(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2, we have that V ⊗n ∼= (Ω−1(N))⊗n is isomorphic, up to projective summands, to Ω−n(N ⊗n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3, all projective representations are even dimen- sional, so Vn, the odd-dimensional summand of V ⊗n, is also the odd-dimensional summand of Ω−n(N ⊗n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS 11 Thus, to find Vn, we must find the unique odd-dimensional indecomposable summand of N ⊗n and take n cosyzygies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We have that N is the monomial representation given by the partition (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2, the odd-dimensional indecomposable summand of N ⊗odd is isomorphic to N, and the odd-dimensional inde- composable summand of N ⊗even is isomorphic to k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Thus computing Vn reduces to computing the unique odd-dimensional summands of Ω−n(N) for odd n and Ω−n(k) for even n, which is a much simpler task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' This technique relies on the fact that Ω(V ) is a simpler representation than V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The same technique works for partitions of the form (2m, 1), where we get PV (n) = 2mx + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' However, for the representations considered in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6, taking syzygies does not simplify the picture, and this technique cannot be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (3, 1, 1) Monomial representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Let V be the monomial representation of Z/4Z×Z/4Z (or α(2, 2)) corresponding to the partition (3, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The syzygy method in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4 does not work for this monomial representation, since Ω(V ) is not easier to analyze than V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' However, we are still able to obtain an explicit decomposition, via a similar analysis as Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3, giving one of the first nontrivial examples where Benson’s tensor powers conjecture may be verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We have the following decomposition into indecomposable summands: V2k−1 ⊗ V = V2k ⊕ W12 ⊕ · · · ⊕ W12 � �� � k copies ⊕ F ⊕ · · · ⊕ F � �� � 3k−3 copies ⊕ W28 ⊕ · · · ⊕ W28 � �� � k−1 copies , V2k ⊗ V = V2k+1 ⊕ W20 ⊕ · · · ⊕ W20 � �� � 2k copies , where dim W20 = 20, dim W12 = 12, dim W28 = 28, and F is a free module of dimension 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Again, we explicitly write the decomposition with a long but stronger lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We claim the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (1) The representation V2k+1 for k ≥ 1 is given by the graded monomial diagram in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (2) The representation V2k for k ≥ 1 is given by the graded monomial diagram in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (3) The representation V2k−1 ⊗ V decomposes into the following summands: V2k (given in Table 8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' dimen- sion 12 summands (given in Table 9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' dimension 16 summands (given in Table 10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' and dimension 28 summands (given in Table 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (4) The representation V2k ⊗ V decomposes into the following summands: V2k+1 (given in Table 12) and dimension 20 summands (given in Table 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The pieces of V2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' These pieces show the nonzero actions of x and y, and they overlap each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' They are arranged as in Figure 5 for V7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' i) piece: (B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' i) piece: vA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='i 2k−1+2i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4k+1−2i vA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='i 2k−1+2i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4k+2−2i vA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='i 2k−1+2i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4k+3−2i vA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='i 2k−1+2i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4k+4−2i vA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='i 2k+2i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4k+1−2i vA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='i 2k+2i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4k+2−2i vA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='i 2k+2i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4k+3−2i vA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='i 2k+2i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4k+4−2i vA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='i 2k+1+2i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4k+3−2i vB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='i 2k−1+2i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4k+4−2i vB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='i 2k−1+2i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4k+5−2i vA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='i−1 2k−1+2i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4k+6−2i vB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='i 2k+2i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4k+3−2i vB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='i 2k+1+2i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4k+3−2i vA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='i 2k+2+2i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4k+3−2i for i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' , k for i = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', k (B, 1) piece: (B, k + 1) piece: vB,1 2k+1,4k+1 vB,1 2k+1,4k+2 vB,1 2k+1,4k+3 vB,1 2k+2,4k+1 vB,1 2k+3,4k+1 vA,1 2k+4,4k+1 vB,k+1 4k+1,2k+1 vB,k+1 4k+1,2k+2 vB,k+1 4k+1,2k+3 vA,k 4k+1,2k+4 vB,k+1 4k+2,2k+1 vB,k+1 4k+3,2k+1 12 GEORGE CAO AND KENT B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' VASHAW Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The location of the pieces for V7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The outlines show the (A, i) pieces, and the shaded regions show the (B, i) pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1 1 2 2 1 1 1 2 1 1 1 3 3 3 1 1 1 3 2 1 1 1 3 3 3 1 1 1 3 2 1 1 2 2 3 1 2 1 1 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The pieces of V2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' These pieces show the nonzero actions of x and y, and they overlap each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' They are arranged as in Figure 6 for V6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (A, i) piece: (B, i) piece: vi 2k−2+2i,4k−2i vi 2k−2+2i,4k+1−2i vi 2k−2+2i,4k+2−2i vi 2k−2+2i,4k+3−2i vi 2k−1+2i,4k−2i vi 2k−1+2i,4k+1−2i vi 2k−1+2i,4k+2−2i vi 2k+2i,4k−2i vi 2k+2i,4k+1−2i vi 2k+2i,4k+2−2i vi 2k+1+2i,4k−2i v2k+2i,4k−2i vi+1 2k+2i,4k+1−2i vi 2k+1+2i,4k−2i for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', k for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' , k − 1 (C, 1) piece: (C, k) piece: v2k,4k v1 2k,4k+1 v4k,2k vk 4k+1,2k Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The location of the pieces for V6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The outlines show the (A, i) pieces, and the shaded regions show the (B, i), (C, 1), and (C, k) pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1 1 2 1 1 1 1 1 1 3 2 1 1 1 2 1 1 3 2 1 1 1 2 2 1 1 1 ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS 13 Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The pieces of V2k.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' , k − 1 16 GEORGE CAO AND KENT B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' VASHAW Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The dimension 28 summands, each of which is written in three pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' These pieces show the nonzero actions of x and y and overlap each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' They are arranged as in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Note that for i = k − 1, some of the vectors do not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' This construction works with those nonexistent vectors removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Dimension 28 summand: vA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='i 2k−3+2i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4k−2−2i ⊗ v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4k+1−2i ⊗ v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2 vi 2k+2i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4k+1−2i ⊗ v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2 = = = = = for i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' , k Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The location of the pieces for the dimension 20 summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The outlines show the positions of the pieces with 17 cells, and the shaded region shows the position of the piece with 4 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1 1 1 1 1 1 2 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 Now, the proof is very similar to the (4, 1) partition case, which was Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Similar to the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2, the explicit constructions show that the summands are both subrepresentations and indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The positions of the summands of V2k−1 ⊗ V are summarized in Figure 9, and the positions of the summands of V2k ⊗ V are summarized in Figure 11 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The black outline is the location of Vi within the tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' It can be checked that these summands span the tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' This is very similar to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2, and we have omitted the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' □ As a corollary, we observe that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2 again holds in this case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' this time, the function PV (n) is not polynomial, but is still quasi-polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The function PV (n) is a quasi-polynomial with period 2, given by 10n − 5 for odd n and 6n + 1 for even n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 20 GEORGE CAO AND KENT B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' VASHAW Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Positions of the indecomposable summands in V7 ⊗ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1 2 4 5 5 3 1 2 3 6 6 4 1 1 2 7 9 13 10 5 1 2 3 9 10 11 5 1 1 2 7 9 15 12 8 2 2 3 9 10 12 6 2 4 6 13 11 8 2 5 6 10 5 2 5 4 5 1 3 1 1 1 Key Dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 12 Family 1 Family 2 Family 3 Dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 28 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 2 1 1 1 1 2 1 1 1 1 1 1 1 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Positions of the indecomposable summands in V8 ⊗ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1 2 4 4 3 1 2 3 5 3 1 1 2 7 8 9 4 1 2 3 8 6 4 1 2 7 8 11 5 2 2 3 8 6 5 1 2 7 8 11 5 2 2 3 8 6 5 4 5 9 4 2 4 3 4 3 1 1 1 Key Vertical Horizontal 1 1 1 1 1 1 2 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (4, 2)/(1) Monomial representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Let V be the monomial representation of Z/2Z × Z/4Z (or α(1, 2)) corresponding to the skew partition (4, 2)/(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' This is another example where the syzygy technique outlined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4 is not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We have the following decomposition into indecomposable summands: V2k−1 ⊗ V = V2k ⊕ F ⊕ · · · ⊕ F � �� � 3k−3 copies ⊕ W12 ⊕ · · · ⊕ W12 � �� � 2k−1 copies , V2k ⊗ V = V2k+1 ⊕ F ⊕ · · · ⊕ F � �� � 3k copies ⊕ W12 ⊕ · · · ⊕ W12 � �� � 2k copies , where dim W12 = 12 and F is a free module of dimension 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' As before, we write the decomposition with a stronger lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We claim the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (1) The representation V2k+1 for k ≥ 1 is given by the graded monomial diagram in Table 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (2) The representation V2k for k ≥ 1 is given by the graded monomial diagram in Table 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (3) The representation V2k−1 ⊗V decomposes into the following summands: V2k (given in Table 16);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' dimen- sion 8 summands (given in Table 17);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' and dimension 12 summands (given in Table 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (4) The representation V2k ⊗V decomposes into the following summands: V2k+1 (given in Table 19);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' dimen- sion 8 summands (given in Table 20);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' and dimension 12 summands (given in Table 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS 21 Table 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The pieces of V2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' These pieces show the nonzero actions of x and y, and they overlap each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' They are arranged as in Figure 13 for V5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (A, 1) piece: (A, i) piece: vA,1 4k+2,2k+1 vA,1 4k+1,2k+2 vA,1 4k+2,2k+2 vA,1 4k+1,2k+3 vA,1 4k+1,2k+4 vA,1 4k+1,2k+5 vA,i 4k+4−2i,2k−3+4i vA,i−1 4k+5−2i,2k−3+4i vA,i 4k+3−2i,2k−2+4i vA,i 4k+4−2i,2k−2+4i vA,i 4k+3−2i,2k−1+4i vA,i 4k+3−2i,2k+4i vA,i 4k+3−2i,2k+1+4i for i = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', k (B, i) piece: (A, k + 1) piece: vB,i 4k+2−2i,2k−1+4i vB,i 4k+3−2i,2k−1+4i vB,i 4k+2−2i,2k+4i vB,i 4k+3−2i,2k+4i vB,i 4k+2−2i,2k+1+4i vA,i 4k+3−2i,2k+1+4i vB,i 4k+2−2i,2k+2+4i vA,k+1 2k+2,6k+1 vA,k 2k+3,6k+1 vA,k+1 2k+1,6k+2 vA,k+1 2k+2,6k+2 vA,k+1 2k+1,6k+3 vA,k+1 2k+1,6k+4 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', k Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The location of the pieces for V5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The outlines show the (B, i) pieces, and the shaded regions show the (A, i) pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1 1 1 2 2 1 1 1 2 2 1 2 2 1 1 1 2 2 1 1 1 Table 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The pieces of V2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' These pieces show the nonzero actions of x and y, and they overlap each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' They are arranged as in Figure 14 for V6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (A, 1) piece: (A, i) piece: vA,1 4k,2k vA,1 4k−1,2k+1 vA,1 4k,2k+1 vA,1 4k−1,2k+2 vA,1 4k−1,2k+3 vA,1 4k−1,2k+4 vA,i 4k+2−2i,2k−4+4i vA,i−1 4k+3−2i,2k−4+4i vA,i 4k+1−2i,2k−3+4i vA,i 4k+2−2i,2k−3+4i vA,i 4k+1−2i,2k−2+4i vA,i 4k+1−2i,2k−1+4i vA,i 4k+1−2i,2k+4i for i = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', k C piece: (B, i) piece: v2k,6k vB,k 2k,6k+1 vB,i 4k−2i,2k−2+4i vB,i 4k+1−2i,2k−2+4i vB,i 4k−2i,2k−1+4i vB,i 4k+1−2i,2k−1+4i vB,i 4k−2i,2k+4i vA,i 4k+1−2i,2k+4i vB,i 4k−2i,2k+1+4i for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', k 22 GEORGE CAO AND KENT B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' VASHAW Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The location of the pieces for V6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The outlines show the (B, i) pieces and the shaded regions show the (A, i) and C pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1 1 1 2 2 1 1 1 2 2 1 2 2 1 1 1 2 2 1 2 2 1 1 1 2 1 Table 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The pieces of V2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Note that in pieces (A, k) and (B, k), some of the vectors do not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' This construction works with those nonexistent vectors removed.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' , k ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS 23 Table 16 (continued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (A, 1) piece: C piece: vA,1 4k−2,2k−1 ⊗ v2,1 vA,1 4k−3,2k ⊗ v2,1 + vA,1 4k−2,2k−1 ⊗ v1,2 vA,1 4k−2,2k ⊗ v2,1 + vA,1 4k−2,2k−1 ⊗ v2,2 vA,1 4k−3,2k+1 ⊗ v2,1 + vA,1 4k−3,2k ⊗ v2,2 + vA,1 4k−2,2k ⊗ v1,2 + vA,1 4k−2,2k−1 ⊗ v1,3 vA,1 4k−3,2k+2 ⊗ v2,1 + vA,1 4k−2,2k−1 ⊗ v1,4 vA,1 4k−3,2k+3 ⊗ v2,1 + vA,1 4k−3,2k+2 ⊗ v2,2 + vA,1 4k−2,2k ⊗ v1,4 vA,k 2k−1,6k−3 ⊗ v1,3 vA,k 2k−1,6k−2 ⊗ v1,3 + vA,k 2k−1,6k−3 ⊗ v1,4 Table 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The dimension 8 summands, which are all free modules.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2 for i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' , k − 1 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' , k − 1 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', k − 1 Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The location of the pieces for the dimension 12 summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The outlines show the positions of the piece with 9 cells, and the shaded region shows the position of the overlapping piece with 8 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1 1 1 2 2 1 1 1 1 1 24 GEORGE CAO AND KENT B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' VASHAW Table 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The dimension 12 summands, which can be split into two families (which differ in only how they are expressed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Each family is characterized in two pieces which overlap at the indicated equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The pieces are arranged as in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' In the dimension 12 Family 1 summand for i = k, some of the vectors do not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' This construction works with those nonexistent vectors removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Dimension 12 Family 1 summand: Dimension 12 Family 2 summand: vA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='i 4k−1−2i,' 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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' , k for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' , k Table 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The dimension 12 summands.' metadata={'source': 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for i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', k for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', k − 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Similar to the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2, the explicit constructions show that the summands are both subrepresentations and indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The positions of the summands are summarized in Figure 16, where the black outline indicates the position of Vi within the tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Again, it can be checked that these summands span the respective tensor products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' This is very similar to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2, and we have omitted the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' □ This yet again verifies that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2 holds for this monomial representation: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The function PV (n) is a quasi-polynomial with period 2, given by 6n − 1 for odd n and 6n + 1 for even n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We close this section by noting that the computations involved in the theorem above resolve an analogue of a question asked by Benson and Symonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We first recall some terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' A G-module M is called Omega- algebraic if there exist finitely many modules M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', Mm such that every non-projective indecomposable summand of M ⊗n, for all n, is equal to Ωi(Mj) for some i ∈ Z and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', m ([7, Definition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Omega- algebraic modules, and certain generalizations, have been used in analyzing the dimension of the largest 28 GEORGE CAO AND KENT B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' VASHAW Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Positions of the indecomposable summands in V6 ⊗V (left) and V7 ⊗V (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1 2 2 2 5 1 1 5 6 2 8 4 5 9 1 2 8 5 1 5 8 1 2 8 5 5 9 1 2 8 5 1 5 8 1 2 8 5 4 8 1 4 4 3 1 1 Key Dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 8 Family 1 Family 2 Additional 1 1 1 1 1 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 1 2 2 1 1 1 1 1 1 2 2 2 5 1 1 5 6 2 8 4 5 9 1 2 8 5 1 5 8 1 2 8 5 5 9 1 2 8 5 1 5 8 1 2 8 5 5 9 1 1 7 5 2 6 1 3 3 2 1 non-projective summand of tensor powers [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' In [7], Benson and Symonds posed the following question (and provided evidence that its answer was negative): Question 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Must every faithful G-module be Omega-algebraic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2 shows that the analogue of Benson and Symonds’ question for graded α(r, s)-modules is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The (4, 2)/(1) monomial representation is a faithful G-module which is not Omega-algebraic in the category of graded α(1, 2)-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Let V be the monomial representation corresponding to the partition (4, 2)/(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Note that V is faithful (as a G-module).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' It suffices to show that if i ̸= j and i, j > 1, then Vi is not any syzygy Ωk(Vj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' This is equivalent to showing that if i ̸= j, then Vi is not any cosyzygy Ω−k(Vj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Since all projective modules are free modules in this case, the minimal projective cover of Vj is a direct sum of many copies of the free module of rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2, we know that for positive integer j, the graded representation Vj has homogeneous components with degrees (2j, j), (2j, j + 1), (2j − 1, j + 1), and (2j − 1, j + 4) that are dimension 1 and homogeneous components with degrees (2j − 1, j + 2) and (2j − 1, j + 3) that are dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' By the same lemma, we know that there are no other homogeneous components with degree (2j, n) for n ̸= j, j + 1 and no homogeneous components with degree (2j − 1, n) for n ̸= j + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' , j + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Also, we know that there are no other homogeneous components with degree (m, n) with m > 2j or n < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' See Figure 19a for the graded diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' This means that the minimal projective cover of Vj has copies of the graded free module of rank 1 which has the homogeneous component in its top right corner as degrees (2j, j + 1), (2j − 1, j + 3), and (2j −1, j +4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' This implies that the cosyzygy of Vj has homogeneous components with degrees (2j −1, j −2), (2j, j−2), (2j, j−1), (2j−1, j−1) that are dimension 1 and homogeneous components with degrees (2j−1, j) and (2j − 1, j + 1) that are dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Also, we know that there are no other homogeneous components with degree (2j, n) for some other n, no other homogeneous components with degree (2j − 1, n) for some other n, no homogeneous components with degree (m, n) for m > 2j or n < j − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' See Figure 19b for the graded diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' ON THE DECOMPOSITION OF TENSOR PRODUCTS OF MONOMIAL MODULES FOR FINITE 2-GROUPS 29 Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The bottom right of the graded diagram of Vj and cosyzygies, displaying the dimensions of the homogeneous components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (a) The rightmost columns of Vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1 1 1 2 2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (b) The rightmost columns of Ω−1(Vj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1 1 1 1 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The formation of the rightmost two columns of Ω−1(Vj) is invariant to cosyzygies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Specifically, if Ω−k(Vj) has its rightmost two columns with the relative locations and dimensions as in Figure 19b, then let the degrees be (m, n), (m + 1, n), (m, n + 1), and (m + 1, n + 1) for the dimension 1 homogeneous components and (m, n + 2) and (m, n + 3) for the dimension 2 homogeneous components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Then, the minimum projective cover has copies of the rank 1 free module, with the top right corners of these free modules having degrees (m + 1, n + 1) and (m, n + 3), with the latter having multiplicity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Then, the cosyzygy Ω−k−1(Vj) has homogeneous components with degrees (m, n − 2), (m + 1, n − 2), (m, n − 1), and (m + 1, n − 1) that are dimension 1 and has homogeneous components with degrees (m, n) and (m, n+1) that are dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' This is the exact same relative locations and dimensions as the homogeneous components shown in Figure 19b, so this is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Hence, all Ω−k(Vj) for k > 0 have rightmost columns as in Figure 19b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' However, none of the Vi have these rightmost columns by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Therefore, Vi ̸∼= Ω−k(Vj) for k > 0, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Further questions and conjectures The data on tensor powers computed with Magma prompts the following questions, which we observe hold in many cases: Question 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' If V is an odd-dimensional indecomposable monomial representation, then we have that (dim V )n ≡ PV (n) (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' In particular, dim V2k ≡ 1 (mod 4) and dim V2k+1 ≡ dim V (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Question 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Let V be an odd-dimensional indecomposable monomial representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Then, V ⊗n is the direct sum of an odd-dimensional indecomposable representation (with dimension PV (n)) and indecomposable representations with dimensions divisible by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Table 22 summarizes the monomial modules that we have computational evidence for the polynomial or quasi-polynomial (computational evidence defined by having at least n + 2 data points for a degree n polynomial guess).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The notation [f(x), g(x)] is defined as a quasi-polynomial that is f(x) when x is odd and g(x) when x is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Note that all quasi-polynomials in the table are either of period 1 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The monomial representations are not listed in any important order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The ones not listed are either proved previously (180◦-symmetric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (2m, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' staircase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' (3, 1, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' or (4, 2)/(1) partitions) or have been difficult to compute sufficient data for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Also, note that some monomial representations are Vi’s for smaller monomial partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' For example, the (5, 4, 1)/(1) monomial representation is V2 for the monomial representation V := (4, 1), so if PV (n) = 4n + 1 then PV2(n) = 8n + 1, as supported by computation in the table below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Also, note that there are cases of monomial representations in which PV (n) is not linear or quasi-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' For example, the monomial representation given by the partition (6, 1) is conjectured (with computational evidence), but is not yet proven, to have PV (n) = 2n2 + 4n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 30 GEORGE CAO AND KENT B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' VASHAW Table 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Monomial representations and conjectured polynomials from computational ev- idence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Partition Computed Quasi-polynomial (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 2) [10x − 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 6x + 1] (6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1) 2x2 + 4x + 1 (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1) [18x − 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 10x + 1] (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 3) [4x + 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 4x + 1] (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1) [8x − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 8x + 1] (6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 2)/(1) [10x − 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 10x + 1] (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 3)/(1) [12x − 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 12x − 7] (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1)/(1) 6x + 1 (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1)/(1) [12x − 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 12x + 1] (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1)/(1) [20x − 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 12x + 1] (6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 3)/(2) [4x + 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 8x + 1] (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 3,' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1) [10x − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 10x + 1] (6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1)/(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1) [10x − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 18x + 1] 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 2)/(4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1) [12x − 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 12x + 1] (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 2)/(4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 1) 4x2 + 4x + 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Acknowledgements We would like to thank Pavel Etingof for suggesting this project and giving valuable advice along the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' We are also grateful to Dave Benson and Peter Symonds for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' This research was conducted while the first author was a participant in the MIT PRIMES-USA program, which we thank for making this research opportunity possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Research of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Vashaw was partially supported by NSF Postdoctoral Fellowship DMS-2103272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Alperin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Local Representation Theory: Modular representations as an introduction to the local representation theory of finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Cambridge Studies in Advanced Mathematics, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Cambridge University Press, Cambridge, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' x+178 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Benson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Modular Representation Theory: New Trends and Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Springer-Verlag Berlin Heidelberg, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Benson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Representations and Cohomology I: Basic Representation Theory of Finite Groups and Associate Algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Cambridge University Press, Cambridge, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' [4] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Benson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Some conjectures and their consequences for tensor products of modules over a finite p-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Algebra, 558 (2020), 24–42, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='jalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' [5] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Benson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Modular representation theory and commutative Banach algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='13155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' To appear in Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Benson and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Carlson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Nilpotent elements in the Green ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Algebra 104 (1986), 329–350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' [7] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Benson and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Symonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' The non-projective part of the tensor powers of a module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', 101 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' 2, 828–856, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='1112/jlms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='12288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' [8] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Bosma and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Cannon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Handbook of Magma Functions, Magma Computer Algebra, Sydney, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Chirvasitu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Hudson and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Upadhyay, Recursive sequences attached to modular representations of finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Algebra 602 (2022), 599–636.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' [10] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Coulembier, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Etingof, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Ostrik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' On Frobenius exact symmetric tensor categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='02372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' [11] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Coulembier, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Ostrik, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Tubbenhauer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Growth rates of the number of indecomposable summands in tensor powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='00885.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' [12] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Etingof, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Golberg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Hensel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Liu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Schwendner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Vaintrob, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Yudovina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Introduction to Representation Theory with historical interludes by Slava Gerovitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Student Mathematical Library, 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' American Mathematical Society, Providence, RI, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' viii+228 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' [13] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Etingof and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Kannan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Lectures on symmetric tensor categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='04878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' [14] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Etingof and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Ostrik, On semisimplification of tensor categories, Representation theory and algebraic geometry—a conference celebrating the birthdays of Sasha Beilinson and Victor Ginzburg, 3–35, Trends Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=', Birkh¨auser/Springer, Cham, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Jantzen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Representations of algebraic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Mathematical Surveys and Monographs, 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' American Mathematical Society, Providence, RI, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Montgomery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Hopf algebras and their actions on rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' CBMS Regional Conference Series in Mathematics, 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Published for the Conference Board of the Mathematical Sciences, Washington, DC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' by the American Mathematical Society, Providence, RI, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' xiv+238 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' [17] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Waterhouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Introduction to Affine Group Schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Springer-Verlag New York, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' [18] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Webb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' A Course in Finite Group Representation Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Cambridge University Press, Cambridge, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Montgomery High School, Skillman, NJ 08558, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content=' Email address: george.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='cao@mtsdstudent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='us Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E3T4oBgHgl3EQfCQlj/content/2301.04274v1.pdf'} +page_content='S.' metadata={'source': 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b/TdFJT4oBgHgl3EQfMSzY/content/tmp_files/2301.11473v1.pdf.txt @@ -0,0 +1,597 @@ +Proof of a conjecture of Krawchuk and Rampersad +Jeffrey Shallit∗ +School of Computer Science +University of Waterloo +Waterloo, ON N2L 3G1 +Canada +shallit@uwaterloo.ca +January 30, 2023 +Abstract +We prove a 2018 conjecture of Krawchuk and Rampersad on the extremal behavior +of c(n), where c(n) counts the number of length-n factors of the Thue-Morse word t, +up to cyclic rotation. +1 +Introduction +Let x be an infinite word. In a recent paper, Krawchuk and Rampersad [4] studied the cyclic +complexity function cx(n), defined to be the number of length-n factors of x, where factors +that are the same, up to cyclic shift, are only counted once. +They observed that for t = 01101001 · · · , the Thue-Morse sequence [1], the function +ct(n) is 2-regular and is specified by a linear representation of rank 50. This means there are +vectors v, w and a matrix-valued morphism γ such that ct(n) = vγ(z)w for all strings z that +are binary representations of n (allowing leading zeros). See [2] for more details. The first +few terms of ct(n) are given in Table 1; it is sequence A360104 in the On-Line Encyclopedia +of Integer Sequences (OEIS) [8]. +n +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +ct(n) +1 +2 +3 +2 +4 +4 +6 +8 +12 +8 +12 +16 +14 +18 +18 +18 +28 +20 +20 +28 +Table 1: First few values of ct(n). +∗Research funded by a grant from NSERC, 2018-04118. +1 +arXiv:2301.11473v1 [math.CO] 27 Jan 2023 + +Krawchuk and Rampersad conjectured that lim sup ct(n)/n = 2 and lim inf ct(n)/n = +4/3. In this paper we prove these two conjectures. The conjectures follow from two inequal- +ities: ct(n) ≤ 2n − 4 for n ≥ 3 and ct(n) ≥ 4 +3n − 4 for n ≥ 0, which we prove in Section 3 +and 4, respectively. +The method of linear representations (as discussed in, for example, [7]) is extremely +powerful for proving statements about automatic sequences, but it has some limitations. +While it can usually be used to prove various equalities, proving inequalities is typically +more problematic. In this paper we use traditional techniques based on induction, together +with linear representations, to prove the desired inequalities. +2 +Preliminary results +Throughout we abbreviate ct(n) by c(n). +Proposition 1. We have +c(2k) = 2k+1 − 4, +(k ≥ 2) +(1) +c(2k + 3) = 5 +3 · 2k − 2 +3(−1)k + 2, +(k ≥ 2) +(2) +c(2k + 1) = 4 +3 · 2k + 2 +3(−1)k − 2, +(k ≥ 2) +(3) +c(2k − 3) = 5 +3 · 2k + 1 +3(−1)k − 5, +(k ≥ 5) +(4) +c(2k − 1) = 4 +3 · 2k − 1 +3(−1)k − 3, +(k ≥ 2) +(5) +c(2k − 5) = 3 +2 · 2k + (−1)k − 7, +(k ≥ 5) +(6) +c(2k − 7) = 3 +2 · 2k − (−1)k − 9, +(k ≥ 3) +(7) +c(12 · 2k − 3) = 56 +3 · 2k − 2 +3(−1)k − 10, +(k ≥ 0). +(8) +Proof. Eq. (1) can be found in [4, Prop. 1]. +The remaining equalities can be proved exactly as in that paper, using the same technique. +The following three identities are crucial to our approach. +Lemma 2. There are 2-automatic sequences a0, a1, a3 such that +c(2i) = 2c(i) + a0(i) +(9) +c(4i + 1) = 2c(i + 1) + c(2i + 1) + a1(i) +(10) +c(4i + 3) = 1 +2c(2i) + c(2i + 3) + 1 +2c(2i + 4) + a3(i) +(11) +2 + +and furthermore +2 ≤ a0(i) ≤ 6, +(i ≥ 3) +(12) +0 ≤ a1(i) ≤ 10, +(i ≥ 2) +(13) +−1 ≤ a3(i) ≤ 3, +(i ≥ 1). +(14) +Proof. For each relation, we compute a linear representation for each term except a0 (resp., +a1, a3) using Walnut [5]; then we compute a linear representation for the difference using +block matrices. Then we minimize the linear representation and use the “semigroup trick” +(see, e.g., [7, §4.11]) to verify that the sequence is automatic and find a deterministic finite +automaton with output (DFAO) for a0 (resp., a1, a3). +We provide more details about the computation of a0 (in part because we will need them +in what follows). We start with the following Walnut code: +def tmfactoreq "At t T[i+t]=T[j+t]": +def tmconj "Et t<=n & $tmfactoreq(j,i+t,n-t) & $tmfactoreq(i,(j+n)-t,t)": +def tmc "Aj j ~$tmconj(i,j,n)": +Here tmc(i,n) asserts that t[i..i + n − 1] is the first occurrence of a factor that is cyclically +equivalent to it. Hence, by counting the number of i for which it evaluates to TRUE, we +determine the number of length-n cyclic factors. +Let us compute the number of cyclic +factors of various lengths using Walnut: +eval tmcc n "$tmc(i,n)": +eval tmcc2 n "$tmc(i,2*n)": +The first two commands produce linear representations for c(n) and c(2n), in Maple +format. They are of rank 50 and 60, respectively. From this we can easily compute a linear +representation for a0(n) := c(2n) − 2c(n), of rank 110. Using Sch¨utzenberger’s algorithm [2, +Chap. 2], this representation can be minimized into a linear representation (v, γ, w) of rank +7, as follows: +vT = [ 1 0 0 0 0 0 0 ] +γ(0) = +� +�� +1 0 0 0 0 0 0 +0 0 1 0 0 0 0 +0 0 0 0 1 0 0 +0 0 0 0 0 1 0 +0 0 0 0 1 0 0 +0 0 0 1 0 0 0 +0 0 0 0 0 0 1 +� +�� +γ(1) = +� +�� +0 1 0 0 +0 +0 0 +0 0 0 1 +0 +0 0 +0 0 0 0 +1 +0 0 +0 0 0 0 1/2 0 0 +0 0 0 0 +1 +0 0 +0 0 0 0 +0 +0 1 +0 0 0 0 1/2 0 0 +� +�� +w = +� +�� +−1 +−1 +−2 +2 +4 +2 +6 +� +�� . +We can now use the “semigroup trick” to show that a0(n) is 2-automatic and find a DFAO +for it. It has 8 states and is displayed in Figure 1. From examining the result, we see that +a0(n) ∈ {−2, −1, 2, 4, 6} and furthermore a0(n) ∈ {2, 4, 6} for n ≥ 3. This proves Eqs. (9) +and (12). +3 + +0/-1 +0 +1/-1 +1 +2/-2 +0 +3/2 +1 +4/4 +0,1 +5/2 +0 +6/2 +1 +0,1 +0 +7/6 +1 +0,1 +1 +0 +Figure 1: DFAO for a0(n). +Using a similar procedure, and the following Walnut commands, we can find the DFAO’s +for a1(n) and a3(n), which are given in Figures 2 and 3. +eval tmcc41 n "$tmc(i,4*n+1)": +eval tmcc1 n "$tmc(i,n+1)": +eval tmcc21 n "$tmc(i,2*n+1)": +eval tmcc43 n "$tmc(i,4*n+3)": +eval tmcc23 n "$tmc(i,2*n+3)": +eval tmcc24 n "$tmc(i,2*n+4)": +0/-4 +0 +1/-4 +1 +2/0 +0 +3/2 +1 +1 +4/4 +0 +5/6 +0 +6/6 +1 +0,1 +7/6 +0 +8/2 +1 +1 +0 +0 +9/10 +1 +0,1 +0 +1 +Figure 2: DFAO for a1(n). +4 + +0/-5/2 +0 +1/-1/2 +1 +2/0 +0 +3/1 +1 +4/-1 +0 +5/3 +1 +6/1 +0 +7/1 +1 +8/0 +0,1 +0 +9/1 +1 +10/0 +0 +11/2 +1 +1 +12/1 +0 +0,1 +0 +1 +13/2 +0 +14/0 +1 +1 +0 +0,1 +0 +1 +1 +15/2 +0 +0,1 +Figure 3: DFAO for a3(n). +Examining the results proves Eqs. (10), (11), (13), (14). +3 +Upper bound +Theorem 3. c(n) ≤ 2n − 4 for all n ≥ 3. +Proof. By induction on n, using Eqs. (9)–(11) and (13)–(14). However, the claim of the +theorem does not seem to be strong enough to carry out the induction, so we actually prove +the following stronger claim by induction: +c(n) ≤ 2n − 7 if n ≥ 12 and n ̸∈ P2, +(15) +where P2 = {2i : i ≥ 0} = {1, 2, 4, 8, 16, . . .}. The base case is n ≤ 44; we can easily check +that Assertion (15) holds for these n. +Now assume n ≥ 45. There are three cases to consider. +Case 1: n ≡ 0 (mod 2), n = 2i. +Case 1a: Assume n/2 ̸∈ P2. Then +c(n) = c(2i) = 2c(i) + a0(i) += 2c(n/2) + a0(n/2) +≤ 2c(n/2) + 6 +≤ 2(n − 7) + 6 +(by induction, since n/2 ≥ 22) += 2n − 8 ≤ 2n − 7. +5 + +Case 1b: If n/2 ∈ P2, then n = 2k for some k ≥ 1. Since n ≥ 45 we have k ≥ 6. Hence +the desired bound c(n) ≤ 2n − 4 follows from Eq. (1). +Case 2: n ≡ 1 (mod 4), n = 4i + 1. +Case 2a: Assume (n + 3)/4 ̸∈ P2 and (n + 1)/2 ̸∈ P2. Then we have +c(n) = c(4i + 1) = 2c(i + 1) + c(2i + 1) + a1(i) +≤ 2c((n + 3)/4) + c((n + 1)/2) + a1((n − 1)/4) +≤ 2((n + 3)/2 − 7) + (n + 1 − 7) + 10 +(by induction, since (n + 1)/2 ≥ (n + 3)/4 ≥ 12) += 2n − 7. +Case 2b: If (n + 3)/4 ∈ P2, then n = 2k − 3 for some k ≥ 2. Since n ≥ 45 we must have +k ≥ 6. Then Eq. (4) implies the desired bound. +Case 2c: If (n + 1)/2 ∈ P2 then n = 2k − 1 for some k ≥ 1. Since n ≥ 45 we have k ≥ 6. +Hence by Eq. (5) we get the desired bound. +Case 3: n ≡ 3 (mod 4), n = 4i + 3. +Case 3a: Assume (n − 3)/2 ̸∈ P2 and (n + 3)/2 ̸∈ P2 and (n + 5)/2 ̸∈ P2. Then we have +c(n) = c(4i + 3) = 1 +2c(2i) + c(2i + 3) + 1 +2c(2i + 4) + a3(i) += 1 +2c((n − 3)/2) + c((n + 3)/2) + 1 +2c((n + 5)/2) + a3((n − 3)/4) +≤ 1 +2(n − 10) + (n + 3 − 7) + 1 +2(n + 5 − 7) + 3 +(by induction, since (n + 5)/2 ≥ (n + 3)/2 ≥ (n − 3)/2 ≥ 21) += 2n − 7. +Case 3b: If (n − 3)/2 ∈ P2 then n = 2k + 3 for some k ≥ 1. Since n ≥ 45 we have k ≥ 6. +So the desired bound follows from Eq. (2). +Case 3c: If (n + 3)/2 ∈ P2 then n = 2k − 3 for some k ≥ 1. Since n ≥ 45 we have k ≥ 6. +So the desired bound follows from Eq. (4). +Case 3d: If (n + 5)/2 ∈ P2 then n = 2k − 5 for some k ≥ 1. Since k ≥ 45 we have k ≥ 6. +So the desired bound follows from Eq. (6). +We have now completed the proof of Assertion (15). To finish the proof of the theorem, +we only need observe that if n ≥ 8 is a power of 2, then c(n) = 2n − 4, and check that +c(n) ≤ 2n − 4 for 3 ≤ n ≤ 11. +We now get the first conjecture of Krawchuk and Rampersad as an immediate corollary: +Corollary 4. lim supn→∞ c(n)/n = 2. +6 + +4 +Lower bound +In this section we prove the the corresponding lower bound on c(n), namely +Theorem 5. c(n) ≥ 4 +3n − 4 for n ≥ 0. +The ideas are similar to those in the proof of the upper bound, but a bit more complicated +because the various exceptional sets are more intricate. +We need a lemma. Define the following exceptional sets. +A = {2k − 1 : k ≥ 1} = {1, 3, 7, 15, 31, . . .} +B = {2k + 1 : k ≥ 2} = {5, 9, 17, 33, . . .} +D = {12 · 2k − 3 : k ≥ 0} = {9, 21, 45, 93, . . .} +J = {(22i+1 + 1)2j : i ≥ 1, j ≥ 0} = {9, 18, 33, 36, 66, 72, 129, 132, 144, 258, . . .}. +Lemma 6. +(i) If n ∈ A, then c(n) ≥ 4 +3n − 2. +(ii) If n ∈ B, then c(n) ≥ 4 +3n − 4. +(iii) If n ∈ D, then c(n) = 2c((n + 3)/4) + c((n + 1)/2). +(iv) If n ∈ J, say n = (22i+1 +1)2j, then c(n) = 8 +322i+j + 4 +32j −4 = 4 +3n−4. for i ≥ 1, j ≥ 0. +(v) If n ∈ 4J + 3, say n = (22i+1 + 1)2j+2 + 3, and (i, j) ̸= (1, 0) (i.e., n ̸= 39), then +c(n) = (104 · 22i+j + 64 · 2j + 4 · 22i(−1)j − 10 · (−1)j + 18)/9 ≥ (4n + 16)/3. +(vi) If n ∈ 2J + 3, say n = (22i+1 + 1)2j+1 + 3, and furthermore j ≥ 1, then c(n) = +(52 · 22i+j + 32 · 2j − 4 · 22i(−1)j + 10 · (−1)j + 18)/9. +(vii) If n ∈ 2J − 5, say n = (22i+1 + 1)2j+1 − 5, then c(n) = 6 · 22i+j + 4 · 2j + 8 +3 · 22i(−1)j − +2 +3(−1)j − 14 for i ≥ 1 and j ≥ 2. +Proof. Items (i) and (ii) follow immediately from Eqs. (3) and (5). +For item (iii), take i = (n − 1)/4 in Eq. (10). Then c(n) = 2c((n + 3)/4) + c((n + 1)/2) +iff a1((n − 1)/4) = 0. But from the DFAO in Figure 2 we see ai(m) = 0 iff m = 3 · 2k − 1 +for k ≥ 0. Hence a1((n − 1)/4) = 0 iff (n − 1)/4 = 3 · 2k − 1 for k ≥ 0, iff n = 12 · 2k − 3 for +k ≥ 0. +For item (iv), we use a variant of the linear representation trick. Let n = (22i+1 + 1)2j. +The base-2 representation of n is 102i10j, so c(n) = vγ(102i10j)w = vγ(1)γ(0)2iγ(1)γ(0)jw. +The minimal polynomial of γ(0) is X2(X − 1)(X − 2)(X + 1), so each entry of γ(0)j for +j ≥ 2 is a linear combination of 2j, (−1)j, and 1. The same is then true for γ(1)γ(0)jw. +Similarly, each entry of γ(0)2i for i ≥ 2 is a linear combination of 22i and 1, and the same +is true for vγ(1)γ(0)2i. Hence the entries of the product vγ(1)γ(0)2iγ(1)γ(0)jw are linear +7 + +combinations of 22i+j, 2j, 22i, 1, 22i(−1)j, and (−1)j. We can deduce the particular constants +by substituting small values of i and j and solving the resulting linear system. The result +now follows. +Parts (v), (vi), and (vii) follow from the same technique. +We are now ready to prove Theorem 5. +Proof. Again, the statement of the theorem does not seem strong enough to make the in- +duction go through. +We will prove the following three claims below by simultaneous induction on n. +(i) For all n we have c(n) ≥ 4 +3n − 4. +(ii) If n is even and n ̸∈ J then c(n) ≥ 4 +3n − 2. +(iii) If n is odd, n ≥ 47, and n ̸∈ A ∪ B, then c(n) ≥ (4n + 16)/3. +The base case is n < 191, which we can check by a short computation. Now assume +n ≥ 191. +Case 1: n ≡ 0 (mod 2), n = 2i. +Case 1a: Suppose n ∈ J. Then from Lemma 6 (iv) we have c(n) = 4 +3n − 4. +Case 1b: Suppose n ̸∈ J. Then +c(n) = c(2i) = c(i) + a0(i) += 2c(n/2) + a0(n/2) +≥ 2c(n/2) + 2 +≥ 2 · (4 +3(n/2) − 2) + 2 +(by induction, since n/2 ≥ 47) += 4 +3n − 2. +Case 2: n ≡ 1 (mod 4), n = 4i + 1. If n ∈ A ∪ B then the inequality c(n) ≥ 4 +3n − 4 follows +from Eqs. (3) and (5). So assume n ̸∈ A ∪ B. +Case 2a: Suppose n ̸∈ D, n+3 +4 +̸∈ J ∪ A ∪ B, n+1 +2 +̸∈ A ∪ B. +Then +c(n) = c(4i + 1) = 2c(i + 1) + c(2i + 1) + a1(i) += 2c((n + 3)/4) + c((n + 1)/2) + a1((n − 1)/4) +≥ 2c((n + 3)/4) + c((n + 1)/2) + 2 +(because n ̸∈ D) +≥ 2 · (4 +3 · ((n + 3)/4) − 2) + (4n + 1 +2 ++ 16)/3 + 2 +(because (n + 3)/4 ̸∈ J ∪ A ∪ B and (n + 1)/2 ̸∈ A ∪ B +and (n + 1)/2 ≥ (n + 3)/4 ≥ 47 by induction) +≥ 4 +3n + 6 ≥ (4n + 16)/3. +8 + +Case 2b: If n ∈ D then Eq. (8) implies that c(n) ≥ (4n/3) − 4. If further n ≥ 47 then it +implies that c(n) ≥ (4n + 16)/3. +Case 2c: Suppose n+3 +4 +∈ J. Then from Lemma 6 (v) we have a closed form for c(n). +Using a routine calculation and the fact that n ̸= 39, we get c(n) ≥ (4n + 16)/3. +Case 2d: Suppose n+3 +4 +∈ A. Then n = 2k − 7 for k ≥ 3, and then by Eq. (7) we have +c(n) ≥ (4n + 16)/3 for k ≥ 5. +Case 2e: Suppose n+3 +4 +∈ B. Then n ∈ B, a contradiction. +Case 2f: Suppose +n+1 +2 +∈ A . +Then n = 2k − 3 for k ≥ 2, and by Eq. (4) we have +c(n) ≥ (4n + 16)/3 for k ≥ 5. +Case 2g: Suppose n+1 +2 +∈ B. Then n ∈ B, a contradiction. +Case 3: n ≡ 3 (mod 4), n = 4i + 3. If n ∈ A ∪ B then c(n) ≥ 4 +3n − 4 follows from Eqs. (3) +and (5). So assume n ̸∈ A ∪ B. +Case 3a: Suppose (n − 3)/2 ̸∈ J ∪ A ∪ B and (n + 3)/2 ̸∈ J ∪ A ∪ B and (n + 5)/2 ̸∈ +J ∪ A ∪ B. Then +c(n) = c(4i + 3) = 1 +2c(2i) + c(2i + 3) + 1 +2c(2i + 4) + a3(i) += 1 +2c((n − 3)/2) + c((n + 3)/2) + c((n + 5)/2) + a3((n − 3)/4) +≥ 1 +2(4 +3(n − 3)/2 − 2) + (4n + 3 +2 ++ 16)/3 + 1 +2(4 +3(n + 5)/2 − 2) − 1 +(by conditions on (n − 3)/2, (n + 3)/2, (n + 5)/2 +and (n + 5)/2 ≥ (n + 3)/2 ≥ (n − 3)/2 ≥ 47) += 4 +3n + 6 ≥ (4n + 16)/3. +Case 3b: Suppose (n − 3)/2 ∈ J. Then n ∈ 2J + 3, so n = (22i+1 + 1)2j+1 + 3. Since +n ≡ 3 (mod 4) we must have j ≥ 1. Then by a routine calculation using Lemma 6 (vi), we +have c(n) ≥ (4n + 16)/3 since n > 39. +Case 3c: Suppose (n − 3)/2 ∈ A ∪ B. But (n − 3)/2 = 2n is even, a contradiction. +Case 3d: Suppose (n + 3)/2 ∈ J. Then n ∈ 2J − 3. But n ≡ 3 (mod 4), so it is easy to +see that this forces n = 22k − 1 ∈ B for k ≥ 2, a contradiction. +Case 3e: Suppose (n + 3)/2 ∈ A. Then n = 2k − 5 and by Eq. (6) we have c(n) ≥ +(4n + 16)/3 for k ≥ 6. +Case 3f: Suppose (n + 3)/2 ∈ B. Then n ∈ A, a contradiction. +Case 3g: Suppose (n + 5)/2 ∈ J. Then n ∈ 2J − 5. Then from Lemma 6 (vii) it follows +by a routine computation that c(n) ≥ (4n + 16)/3. +Case 3h: Suppose (n + 5)/2 ∈ A ∪ B. But (n + 5)/2 = 2n + 4 is even, a contradiction. +This completes the proof by induction of (i), (ii), and (iii). +Corollary 7. lim infn→∞ c(n)/n = 4/3. +9 + +We also have enough to prove +Theorem 8. c(n) = 4 +3n − 4 iff n ∈ J. +Proof. ⇐=: follows from Lemma 6 (v). +=⇒: if n is even and n ̸∈ J then c(n) ≥ 4 +3n − 2 > 4 +3n − 4 by above. +If n ≥ 47 is odd and n ̸∈ A ∪ B, then c(n) ≥ (4n + 16)/3 > 4 +3n − 4 by above. It remains +to check n < 47 and n ∈ A ∪ B. +For n ∈ A, we know from Lemma 6 (i) that c(n) ≥ 4 +3n−2 > 4 +3n−4. For n ∈ B, it follows +from Eq. (3) that c(n) = 4 +3n − 4 iff n = 22k+1 + 1 ∈ J. Finally, n < 47 can be checked with +a computation. +5 +Final remarks +For ordinary subword complexity ρx(n), which counts the number of distinct length-n factors +appearing in x, it is known that if x is an automatic sequence, then there is an automaton +that takes as input the representations of n and y in parallel, and accepts iff y = ρx(n). In +other words, ρx(n) is synchronized for automatic sequences; see [3] for more details. This +means that checking whether ρx(n) ≤ An + B or ρx(n) ≥ An + B for all n is, in general, +decidable for automatic sequences, since we can express these assertions as a first-order +logical formula. +However, for cyclic complexity the function cx(n) is not, in general, synchronized. We +can see this as follows: let p = 11010001 · · · be the characteristic sequence of the powers of +2. Then it is not hard to see that cp(n) = O(log n) and cp(2n) = n + 2 for n ≥ 0. However, +by a theorem about synchronized sequences [6], this kind of growth rate is impossible. +This fundamental difference may help explain why it is so much harder to prove inequal- +ities for cyclic complexity. +References +[1] J.-P. Allouche and J. O. Shallit. +The ubiquitous Prouhet-Thue-Morse sequence. +In +C. Ding, T. Helleseth, and H. Niederreiter, editors, Sequences and Their Applications, +Proceedings of SETA ’98, pp. 1–16. Springer-Verlag, 1999. +[2] J. Berstel and C. Reutenauer. Noncommutative Rational Series With Applications, Vol. +137 of Encyclopedia of Mathematics and Its Applications. Cambridge University Press, +2011. +[3] D. Goˇc, L. Schaeffer, and J. Shallit. Subword complexity and k-synchronization. In M.-P. +B´eal and O. Carton, editors, DLT 2013, Vol. 7907 of Lecture Notes in Computer Science, +pp. 252–263. Springer-Verlag, 2013. +10 + +[4] C. Krawchuk and N. Rampersad. Cyclic complexity of some infinite words and general- +izations. INTEGERS—Elect. J. Comb. Numb. Theory 18A (2018), #A12 (electronic), +math.colgate.edu/~integers/vol18a.html. +[5] H. Mousavi. Automatic theorem proving in Walnut. Arxiv preprint arXiv:1603.06017 +[cs.FL], available at http://arxiv.org/abs/1603.06017, 2016. +[6] J. Shallit. Synchronized sequences. In T. Lecroq and S. Puzynina, editors, WORDS +2021, Vol. 12847 of Lecture Notes in Computer Science, pp. 1–19. Springer-Verlag, 2021. +[7] J. Shallit. The Logical Approach To Automatic Sequences: Exploring Combinatorics on +Words with Walnut, Vol. 482 of London Math. Society Lecture Note Series. Cambridge +University Press, 2022. +[8] N. J. A. Sloane et al. The on-line encyclopedia of integer sequences, 2023. Available at +https://oeis.org. +11 + diff --git a/TdFJT4oBgHgl3EQfMSzY/content/tmp_files/load_file.txt b/TdFJT4oBgHgl3EQfMSzY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..93d79d25e70ec3e1ed983e827f0f52630387a3ca --- /dev/null +++ b/TdFJT4oBgHgl3EQfMSzY/content/tmp_files/load_file.txt @@ -0,0 +1,314 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf,len=313 +page_content='Proof of a conjecture of Krawchuk and Rampersad Jeffrey Shallit∗ School of Computer Science University of Waterloo Waterloo, ON N2L 3G1 Canada shallit@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content='ca January 30, 2023 Abstract We prove a 2018 conjecture of Krawchuk and Rampersad on the extremal behavior of c(n), where c(n) counts the number of length-n factors of the Thue-Morse word t, up to cyclic rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' 1 Introduction Let x be an infinite word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' In a recent paper, Krawchuk and Rampersad [4] studied the cyclic complexity function cx(n), defined to be the number of length-n factors of x, where factors that are the same, up to cyclic shift, are only counted once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' They observed that for t = 01101001 · · · , the Thue-Morse sequence [1], the function ct(n) is 2-regular and is specified by a linear representation of rank 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' This means there are vectors v, w and a matrix-valued morphism γ such that ct(n) = vγ(z)w for all strings z that are binary representations of n (allowing leading zeros).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' See [2] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' The first few terms of ct(n) are given in Table 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' it is sequence A360104 in the On-Line Encyclopedia of Integer Sequences (OEIS) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ct(n) 1 2 3 2 4 4 6 8 12 8 12 16 14 18 18 18 28 20 20 28 Table 1: First few values of ct(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' ∗Research funded by a grant from NSERC, 2018-04118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content='11473v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content='CO] 27 Jan 2023 Krawchuk and Rampersad conjectured that lim sup ct(n)/n = 2 and lim inf ct(n)/n = 4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' In this paper we prove these two conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' The conjectures follow from two inequal- ities: ct(n) ≤ 2n − 4 for n ≥ 3 and ct(n) ≥ 4 3n − 4 for n ≥ 0, which we prove in Section 3 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' The method of linear representations (as discussed in, for example, [7]) is extremely powerful for proving statements about automatic sequences, but it has some limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' While it can usually be used to prove various equalities, proving inequalities is typically more problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' In this paper we use traditional techniques based on induction, together with linear representations, to prove the desired inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' 2 Preliminary results Throughout we abbreviate ct(n) by c(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' We have c(2k) = 2k+1 − 4, (k ≥ 2) (1) c(2k + 3) = 5 3 · 2k − 2 3(−1)k + 2, (k ≥ 2) (2) c(2k + 1) = 4 3 · 2k + 2 3(−1)k − 2, (k ≥ 2) (3) c(2k − 3) = 5 3 · 2k + 1 3(−1)k − 5, (k ≥ 5) (4) c(2k − 1) = 4 3 · 2k − 1 3(−1)k − 3, (k ≥ 2) (5) c(2k − 5) = 3 2 · 2k + (−1)k − 7, (k ≥ 5) (6) c(2k − 7) = 3 2 · 2k − (−1)k − 9, (k ≥ 3) (7) c(12 · 2k − 3) = 56 3 · 2k − 2 3(−1)k − 10, (k ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (1) can be found in [4, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' The remaining equalities can be proved exactly as in that paper, using the same technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' The following three identities are crucial to our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' There are 2-automatic sequences a0, a1, a3 such that c(2i) = 2c(i) + a0(i) (9) c(4i + 1) = 2c(i + 1) + c(2i + 1) + a1(i) (10) c(4i + 3) = 1 2c(2i) + c(2i + 3) + 1 2c(2i + 4) + a3(i) (11) 2 and furthermore 2 ≤ a0(i) ≤ 6, (i ≥ 3) (12) 0 ≤ a1(i) ≤ 10, (i ≥ 2) (13) −1 ≤ a3(i) ≤ 3, (i ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (14) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' For each relation, we compute a linear representation for each term except a0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=', a1, a3) using Walnut [5];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' then we compute a linear representation for the difference using block matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then we minimize the linear representation and use the “semigroup trick” (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=', [7, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content='11]) to verify that the sequence is automatic and find a deterministic finite automaton with output (DFAO) for a0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=', a1, a3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' We provide more details about the computation of a0 (in part because we will need them in what follows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' We start with the following Walnut code: def tmfactoreq "At t T[i+t]=T[j+t]": def tmconj "Et t<=n & $tmfactoreq(j,i+t,n-t) & $tmfactoreq(i,(j+n)-t,t)": def tmc "Aj j ~$tmconj(i,j,n)": Here tmc(i,n) asserts that t[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content='.i + n − 1] is the first occurrence of a factor that is cyclically equivalent to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Hence, by counting the number of i for which it evaluates to TRUE, we determine the number of length-n cyclic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Let us compute the number of cyclic factors of various lengths using Walnut: eval tmcc n "$tmc(i,n)": eval tmcc2 n "$tmc(i,2*n)": The first two commands produce linear representations for c(n) and c(2n), in Maple format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' They are of rank 50 and 60, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' From this we can easily compute a linear representation for a0(n) := c(2n) − 2c(n), of rank 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Using Sch¨utzenberger’s algorithm [2, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' 2], this representation can be minimized into a linear representation (v, γ, w) of rank 7, as follows: vT = [ 1 0 0 0 0 0 0 ] γ(0) = � �� 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 � �� γ(1) = � �� 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1/2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1/2 0 0 � �� w = � �� −1 −1 −2 2 4 2 6 � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' We can now use the “semigroup trick” to show that a0(n) is 2-automatic and find a DFAO for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' It has 8 states and is displayed in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' From examining the result, we see that a0(n) ∈ {−2, −1, 2, 4, 6} and furthermore a0(n) ∈ {2, 4, 6} for n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' This proves Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (9) and (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' 3 0/-1 0 1/-1 1 2/-2 0 3/2 1 4/4 0,1 5/2 0 6/2 1 0,1 0 7/6 1 0,1 1 0 Figure 1: DFAO for a0(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Using a similar procedure, and the following Walnut commands, we can find the DFAO’s for a1(n) and a3(n), which are given in Figures 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' eval tmcc41 n "$tmc(i,4*n+1)": eval tmcc1 n "$tmc(i,n+1)": eval tmcc21 n "$tmc(i,2*n+1)": eval tmcc43 n "$tmc(i,4*n+3)": eval tmcc23 n "$tmc(i,2*n+3)": eval tmcc24 n "$tmc(i,2*n+4)": 0/-4 0 1/-4 1 2/0 0 3/2 1 1 4/4 0 5/6 0 6/6 1 0,1 7/6 0 8/2 1 1 0 0 9/10 1 0,1 0 1 Figure 2: DFAO for a1(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' 4 0/-5/2 0 1/-1/2 1 2/0 0 3/1 1 4/-1 0 5/3 1 6/1 0 7/1 1 8/0 0,1 0 9/1 1 10/0 0 11/2 1 1 12/1 0 0,1 0 1 13/2 0 14/0 1 1 0 0,1 0 1 1 15/2 0 0,1 Figure 3: DFAO for a3(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Examining the results proves Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (10), (11), (13), (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' 3 Upper bound Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' c(n) ≤ 2n − 4 for all n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' By induction on n, using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (9)–(11) and (13)–(14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' However, the claim of the theorem does not seem to be strong enough to carry out the induction, so we actually prove the following stronger claim by induction: c(n) ≤ 2n − 7 if n ≥ 12 and n ̸∈ P2, (15) where P2 = {2i : i ≥ 0} = {1, 2, 4, 8, 16, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' The base case is n ≤ 44;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' we can easily check that Assertion (15) holds for these n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Now assume n ≥ 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' There are three cases to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 1: n ≡ 0 (mod 2), n = 2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 1a: Assume n/2 ̸∈ P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then c(n) = c(2i) = 2c(i) + a0(i) = 2c(n/2) + a0(n/2) ≤ 2c(n/2) + 6 ≤ 2(n − 7) + 6 (by induction, since n/2 ≥ 22) = 2n − 8 ≤ 2n − 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' 5 Case 1b: If n/2 ∈ P2, then n = 2k for some k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Since n ≥ 45 we have k ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Hence the desired bound c(n) ≤ 2n − 4 follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 2: n ≡ 1 (mod 4), n = 4i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 2a: Assume (n + 3)/4 ̸∈ P2 and (n + 1)/2 ̸∈ P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then we have c(n) = c(4i + 1) = 2c(i + 1) + c(2i + 1) + a1(i) ≤ 2c((n + 3)/4) + c((n + 1)/2) + a1((n − 1)/4) ≤ 2((n + 3)/2 − 7) + (n + 1 − 7) + 10 (by induction, since (n + 1)/2 ≥ (n + 3)/4 ≥ 12) = 2n − 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 2b: If (n + 3)/4 ∈ P2, then n = 2k − 3 for some k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Since n ≥ 45 we must have k ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (4) implies the desired bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 2c: If (n + 1)/2 ∈ P2 then n = 2k − 1 for some k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Since n ≥ 45 we have k ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Hence by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (5) we get the desired bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 3: n ≡ 3 (mod 4), n = 4i + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 3a: Assume (n − 3)/2 ̸∈ P2 and (n + 3)/2 ̸∈ P2 and (n + 5)/2 ̸∈ P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then we have c(n) = c(4i + 3) = 1 2c(2i) + c(2i + 3) + 1 2c(2i + 4) + a3(i) = 1 2c((n − 3)/2) + c((n + 3)/2) + 1 2c((n + 5)/2) + a3((n − 3)/4) ≤ 1 2(n − 10) + (n + 3 − 7) + 1 2(n + 5 − 7) + 3 (by induction, since (n + 5)/2 ≥ (n + 3)/2 ≥ (n − 3)/2 ≥ 21) = 2n − 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 3b: If (n − 3)/2 ∈ P2 then n = 2k + 3 for some k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Since n ≥ 45 we have k ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' So the desired bound follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 3c: If (n + 3)/2 ∈ P2 then n = 2k − 3 for some k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Since n ≥ 45 we have k ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' So the desired bound follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 3d: If (n + 5)/2 ∈ P2 then n = 2k − 5 for some k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Since k ≥ 45 we have k ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' So the desired bound follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' We have now completed the proof of Assertion (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' To finish the proof of the theorem, we only need observe that if n ≥ 8 is a power of 2, then c(n) = 2n − 4, and check that c(n) ≤ 2n − 4 for 3 ≤ n ≤ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' We now get the first conjecture of Krawchuk and Rampersad as an immediate corollary: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' lim supn→∞ c(n)/n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' 6 4 Lower bound In this section we prove the the corresponding lower bound on c(n), namely Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' c(n) ≥ 4 3n − 4 for n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' The ideas are similar to those in the proof of the upper bound, but a bit more complicated because the various exceptional sets are more intricate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' We need a lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Define the following exceptional sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' A = {2k − 1 : k ≥ 1} = {1, 3, 7, 15, 31, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content='} B = {2k + 1 : k ≥ 2} = {5, 9, 17, 33, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content='} D = {12 · 2k − 3 : k ≥ 0} = {9, 21, 45, 93, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content='} J = {(22i+1 + 1)2j : i ≥ 1, j ≥ 0} = {9, 18, 33, 36, 66, 72, 129, 132, 144, 258, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (i) If n ∈ A, then c(n) ≥ 4 3n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (ii) If n ∈ B, then c(n) ≥ 4 3n − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (iii) If n ∈ D, then c(n) = 2c((n + 3)/4) + c((n + 1)/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (iv) If n ∈ J, say n = (22i+1 +1)2j, then c(n) = 8 322i+j + 4 32j −4 = 4 3n−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' for i ≥ 1, j ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (v) If n ∈ 4J + 3, say n = (22i+1 + 1)2j+2 + 3, and (i, j) ̸= (1, 0) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=', n ̸= 39), then c(n) = (104 · 22i+j + 64 · 2j + 4 · 22i(−1)j − 10 · (−1)j + 18)/9 ≥ (4n + 16)/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (vi) If n ∈ 2J + 3, say n = (22i+1 + 1)2j+1 + 3, and furthermore j ≥ 1, then c(n) = (52 · 22i+j + 32 · 2j − 4 · 22i(−1)j + 10 · (−1)j + 18)/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (vii) If n ∈ 2J − 5, say n = (22i+1 + 1)2j+1 − 5, then c(n) = 6 · 22i+j + 4 · 2j + 8 3 · 22i(−1)j − 2 3(−1)j − 14 for i ≥ 1 and j ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Items (i) and (ii) follow immediately from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (3) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' For item (iii), take i = (n − 1)/4 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then c(n) = 2c((n + 3)/4) + c((n + 1)/2) iff a1((n − 1)/4) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' But from the DFAO in Figure 2 we see ai(m) = 0 iff m = 3 · 2k − 1 for k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Hence a1((n − 1)/4) = 0 iff (n − 1)/4 = 3 · 2k − 1 for k ≥ 0, iff n = 12 · 2k − 3 for k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' For item (iv), we use a variant of the linear representation trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Let n = (22i+1 + 1)2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' The base-2 representation of n is 102i10j, so c(n) = vγ(102i10j)w = vγ(1)γ(0)2iγ(1)γ(0)jw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' The minimal polynomial of γ(0) is X2(X − 1)(X − 2)(X + 1), so each entry of γ(0)j for j ≥ 2 is a linear combination of 2j, (−1)j, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' The same is then true for γ(1)γ(0)jw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Similarly, each entry of γ(0)2i for i ≥ 2 is a linear combination of 22i and 1, and the same is true for vγ(1)γ(0)2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Hence the entries of the product vγ(1)γ(0)2iγ(1)γ(0)jw are linear 7 combinations of 22i+j, 2j, 22i, 1, 22i(−1)j, and (−1)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' We can deduce the particular constants by substituting small values of i and j and solving the resulting linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' The result now follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Parts (v), (vi), and (vii) follow from the same technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' We are now ready to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Again, the statement of the theorem does not seem strong enough to make the in- duction go through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' We will prove the following three claims below by simultaneous induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (i) For all n we have c(n) ≥ 4 3n − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (ii) If n is even and n ̸∈ J then c(n) ≥ 4 3n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (iii) If n is odd, n ≥ 47, and n ̸∈ A ∪ B, then c(n) ≥ (4n + 16)/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' The base case is n < 191, which we can check by a short computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Now assume n ≥ 191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 1: n ≡ 0 (mod 2), n = 2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 1a: Suppose n ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then from Lemma 6 (iv) we have c(n) = 4 3n − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 1b: Suppose n ̸∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then c(n) = c(2i) = c(i) + a0(i) = 2c(n/2) + a0(n/2) ≥ 2c(n/2) + 2 ≥ 2 · (4 3(n/2) − 2) + 2 (by induction, since n/2 ≥ 47) = 4 3n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 2: n ≡ 1 (mod 4), n = 4i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' If n ∈ A ∪ B then the inequality c(n) ≥ 4 3n − 4 follows from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (3) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' So assume n ̸∈ A ∪ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 2a: Suppose n ̸∈ D, n+3 4 ̸∈ J ∪ A ∪ B, n+1 2 ̸∈ A ∪ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then c(n) = c(4i + 1) = 2c(i + 1) + c(2i + 1) + a1(i) = 2c((n + 3)/4) + c((n + 1)/2) + a1((n − 1)/4) ≥ 2c((n + 3)/4) + c((n + 1)/2) + 2 (because n ̸∈ D) ≥ 2 · (4 3 · ((n + 3)/4) − 2) + (4n + 1 2 + 16)/3 + 2 (because (n + 3)/4 ̸∈ J ∪ A ∪ B and (n + 1)/2 ̸∈ A ∪ B and (n + 1)/2 ≥ (n + 3)/4 ≥ 47 by induction) ≥ 4 3n + 6 ≥ (4n + 16)/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' 8 Case 2b: If n ∈ D then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (8) implies that c(n) ≥ (4n/3) − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' If further n ≥ 47 then it implies that c(n) ≥ (4n + 16)/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 2c: Suppose n+3 4 ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then from Lemma 6 (v) we have a closed form for c(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Using a routine calculation and the fact that n ̸= 39, we get c(n) ≥ (4n + 16)/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 2d: Suppose n+3 4 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then n = 2k − 7 for k ≥ 3, and then by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (7) we have c(n) ≥ (4n + 16)/3 for k ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 2e: Suppose n+3 4 ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then n ∈ B, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 2f: Suppose n+1 2 ∈ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then n = 2k − 3 for k ≥ 2, and by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (4) we have c(n) ≥ (4n + 16)/3 for k ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 2g: Suppose n+1 2 ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then n ∈ B, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 3: n ≡ 3 (mod 4), n = 4i + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' If n ∈ A ∪ B then c(n) ≥ 4 3n − 4 follows from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (3) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' So assume n ̸∈ A ∪ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 3a: Suppose (n − 3)/2 ̸∈ J ∪ A ∪ B and (n + 3)/2 ̸∈ J ∪ A ∪ B and (n + 5)/2 ̸∈ J ∪ A ∪ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then c(n) = c(4i + 3) = 1 2c(2i) + c(2i + 3) + 1 2c(2i + 4) + a3(i) = 1 2c((n − 3)/2) + c((n + 3)/2) + c((n + 5)/2) + a3((n − 3)/4) ≥ 1 2(4 3(n − 3)/2 − 2) + (4n + 3 2 + 16)/3 + 1 2(4 3(n + 5)/2 − 2) − 1 (by conditions on (n − 3)/2, (n + 3)/2, (n + 5)/2 and (n + 5)/2 ≥ (n + 3)/2 ≥ (n − 3)/2 ≥ 47) = 4 3n + 6 ≥ (4n + 16)/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 3b: Suppose (n − 3)/2 ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then n ∈ 2J + 3, so n = (22i+1 + 1)2j+1 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Since n ≡ 3 (mod 4) we must have j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then by a routine calculation using Lemma 6 (vi), we have c(n) ≥ (4n + 16)/3 since n > 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 3c: Suppose (n − 3)/2 ∈ A ∪ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' But (n − 3)/2 = 2n is even, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 3d: Suppose (n + 3)/2 ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then n ∈ 2J − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' But n ≡ 3 (mod 4), so it is easy to see that this forces n = 22k − 1 ∈ B for k ≥ 2, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 3e: Suppose (n + 3)/2 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then n = 2k − 5 and by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (6) we have c(n) ≥ (4n + 16)/3 for k ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 3f: Suppose (n + 3)/2 ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then n ∈ A, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 3g: Suppose (n + 5)/2 ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then n ∈ 2J − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then from Lemma 6 (vii) it follows by a routine computation that c(n) ≥ (4n + 16)/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Case 3h: Suppose (n + 5)/2 ∈ A ∪ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' But (n + 5)/2 = 2n + 4 is even, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' This completes the proof by induction of (i), (ii), and (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' lim infn→∞ c(n)/n = 4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' 9 We also have enough to prove Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' c(n) = 4 3n − 4 iff n ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' ⇐=: follows from Lemma 6 (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' =⇒: if n is even and n ̸∈ J then c(n) ≥ 4 3n − 2 > 4 3n − 4 by above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' If n ≥ 47 is odd and n ̸∈ A ∪ B, then c(n) ≥ (4n + 16)/3 > 4 3n − 4 by above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' It remains to check n < 47 and n ∈ A ∪ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' For n ∈ A, we know from Lemma 6 (i) that c(n) ≥ 4 3n−2 > 4 3n−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' For n ∈ B, it follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' (3) that c(n) = 4 3n − 4 iff n = 22k+1 + 1 ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Finally, n < 47 can be checked with a computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' 5 Final remarks For ordinary subword complexity ρx(n), which counts the number of distinct length-n factors appearing in x, it is known that if x is an automatic sequence, then there is an automaton that takes as input the representations of n and y in parallel, and accepts iff y = ρx(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' In other words, ρx(n) is synchronized for automatic sequences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' see [3] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' This means that checking whether ρx(n) ≤ An + B or ρx(n) ≥ An + B for all n is, in general, decidable for automatic sequences, since we can express these assertions as a first-order logical formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' However, for cyclic complexity the function cx(n) is not, in general, synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' We can see this as follows: let p = 11010001 · · · be the characteristic sequence of the powers of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Then it is not hard to see that cp(n) = O(log n) and cp(2n) = n + 2 for n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' However, by a theorem about synchronized sequences [6], this kind of growth rate is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' This fundamental difference may help explain why it is so much harder to prove inequal- ities for cyclic complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Allouche and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Shallit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' The ubiquitous Prouhet-Thue-Morse sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' In C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Ding, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Helleseth, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Niederreiter, editors, Sequences and Their Applications, Proceedings of SETA ’98, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' 1–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Springer-Verlag, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Berstel and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Reutenauer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Noncommutative Rational Series With Applications, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' 137 of Encyclopedia of Mathematics and Its Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Cambridge University Press, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Goˇc, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Schaeffer, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Shallit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Subword complexity and k-synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' In M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' B´eal and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Carton, editors, DLT 2013, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' 7907 of Lecture Notes in Computer Science, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' 252–263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Springer-Verlag, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' 10 [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Krawchuk and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Rampersad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Cyclic complexity of some infinite words and general- izations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' INTEGERS—Elect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Numb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Theory 18A (2018), #A12 (electronic), math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content='colgate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content='edu/~integers/vol18a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' [5] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Mousavi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Automatic theorem proving in Walnut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Arxiv preprint arXiv:1603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content='06017 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content='FL], available at http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content='org/abs/1603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content='06017, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Shallit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Synchronized sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' In T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Lecroq and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Puzynina, editors, WORDS 2021, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' 12847 of Lecture Notes in Computer Science, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' 1–19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Springer-Verlag, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Shallit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' The Logical Approach To Automatic Sequences: Exploring Combinatorics on Words with Walnut, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' 482 of London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Society Lecture Note Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Cambridge University Press, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' [8] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Sloane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' The on-line encyclopedia of integer sequences, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' Available at https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} +page_content=' 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFJT4oBgHgl3EQfMSzY/content/2301.11473v1.pdf'} diff --git a/TtE2T4oBgHgl3EQfCgZi/content/tmp_files/2301.03614v1.pdf.txt b/TtE2T4oBgHgl3EQfCgZi/content/tmp_files/2301.03614v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4cefe7275ac10604aeefa2b48b306a90075a4103 --- /dev/null +++ b/TtE2T4oBgHgl3EQfCgZi/content/tmp_files/2301.03614v1.pdf.txt @@ -0,0 +1,1799 @@ +MNRAS 000, 1–13 (2023) +Preprint 11 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Fountain-driven gas accretion feeding star formation over the disc of +NGC 2403 +Anqi Li1★, Filippo Fraternali1, Antonino Marasco2,3, Scott C. Trager1, Gabriele Pezzulli1, +Pavel E. Mancera Piña1,4,5 and Marc A. W. Verheijen1 +1Kapteyn Astronomical Institute, University of Groningen, Landleven 12, 9747 AD Groningen, The Netherlands +2INAF–Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122, Padova, Italy +3INAF–Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50157, Firenze, Italy +4 ASTRON, Netherlands Institute for Radio Astronomy, Postbus 2, NL-7900 AA Dwingeloo, The Netherlands +5 Leiden Observatory, Leiden University, P.O.Box 9513, NL-2300 AA Leiden, The Netherlands +Accepted January 4, 2023; Received September 14, 2022; in original form February 11, 2022 +ABSTRACT +We use a dynamical model of galactic fountain to study the neutral extraplanar gas (EPG) in the nearby spiral galaxy NGC 2403. +We have modelled the EPG as a combination of material ejected from the disc by stellar feedback (i.e. galactic fountain) and gas +accreting from the inner circumgalactic medium (CGM). This accretion is expected to occur because of cooling/condensation of +the hot CGM (corona) triggered by the fountain. Our dynamical model reproduces the distribution and kinematics of the EPG H i +emission in NGC 2403 remarkably well and suggests a total EPG mass of 4.7+1.2 +−0.9 ×108 M⊙, with a typical scale height of around +1 kpc and a vertical gradient of the rotation velocity of −10.0 ± 2.7 km s−1 kpc−1. The best-fitting model requires a characteristic +outflow velocity of 50 ± 10 km s−1. The outflowing gas starts out mostly ionised and only becomes neutral later in the trajectory. +The accretion rate from the condensation of the inner hot CGM inferred by the model is 0.8 M⊙ yr−1, approximately equal to +the star formation rate in this galaxy (0.6 M⊙ yr−1). We show that the accretion profile, which peaks at a radius of about 4.5 kpc, +predicts a disc growth rate compatible with the observed value. Our results indicate that fountain-driven corona condensation is +a likely mechanism to sustain star formation as well as the disc inside-out growth in local disc galaxies. +Key words: galaxies: haloes – galaxies: ISM – galaxies: evolution – galaxies: intergalactic medium – ISM: structure – ISM: +kinematics and dynamics +1 INTRODUCTION +Nearby spiral galaxies have been forming stars, across their lifetimes, +at an approximately constant or gently declining rate, despite the fact +that the gas in their interstellar medium (ISM) would, without re- +plenishment, be consumed in a few Gyr (Aumer & Binney 2009; +Tacconi et al. 2018). An external gas reservoir is therefore needed +from which galaxies accrete gas at a rate compatible with their SFR +(e.g. Fraternali & Tomassetti 2012). Gas-rich mergers are not pro- +viding a sufficient contribution, at least in the local Universe (Sancisi +et al. 2008; Di Teodoro & Fraternali 2014). Therefore the majority +of the accretion must come from the diffuse gas that resides outside +galaxies. +The multi-phase circumgalactic medium (CGM) is expected to +host a significant fraction of the baryons associated with dark matter +halos in normal spiral galaxies (e.g. Crain et al. 2007; Tumlinson +et al. 2011; Li et al. 2018), which makes it the most probable gas +reservoir eligible for accretion. A prominent component of the CGM +is hot gas (𝑇 ∼ 106−7 K) in the form of a diffuse ‘corona’ at nearly +the virial temperature and in nearly hydrostatic equilibrium with +★ E-mail: li@astro.rug.nl +the dark matter potential (e.g. White & Frenk 1991; Pezzulli et al. +2017). Galactic coronae are thought to surround galaxies and to be +extended to their virial radii (Fukugita & Peebles 2006; Faerman +et al. 2020). Direct detection of the hot coronae in X-rays is limited +to the innermost few tens of kpc in massive galaxies with stellar +mass beyond 1011 M⊙ (e.g. Anderson & Bregman 2011; Walker et al. +2015; Anderson et al. 2016), while indirect evidence of their presence +extends further (e.g. Gatto et al. 2013; Putman et al. 2021). Cool CGM +(𝑇 ∼ 104 K) gas has also been detected, mostly in absorption along +quasar sightlines, in several studies (e.g. Heckman et al. 2017; Rubin +et al. 2018; Zahedy et al. 2019). Like the hot corona, also these cool +absorbers extend to large distances (up to and sometimes beyond the +virial radius) and their origin and fate remain debated (Rubin et al. +2010; Schroetter et al. 2019; Pointon et al. 2019; Afruni et al. 2021). +Although gas accretion from the CGM is crucial to feed star for- +mation (Hopkins et al. 2008; Sancisi et al. 2008; Kereš et al. 2009), +how precisely it takes place is still unknown. One possible accretion +scenario is that cold filaments reach the outer disc (Lagos et al. 2017; +El-Badry et al. 2018; Trapp et al. 2022) and are transported into the +inner star-forming regions via radial motions, although Di Teodoro & +Peek (2021) found that radial inflows in nearby galaxies alone could +not sustain the star formation rates. Other possible mechanisms in- +© 2023 The Authors +arXiv:2301.03614v1 [astro-ph.GA] 9 Jan 2023 + +2 +A. Li et al. +clude cold gas filaments directly feeding the inner regions of a galaxy +or the cooling of the hot corona (Kereš et al. 2005; Nelson et al. 2013; +Voit et al. 2015). The spontaneous cooling of the corona via thermal +instability is still under debate as a number of works suggest that +the combination of buoyancy and thermal conduction can suppress +the growth of thermal perturbations (e.g. Binney et al. 2009; Nipoti +2010; Joung et al. 2012). Some authors have proposed that coronal +condensation could be triggered by the ejection of gas from the disc +due to stellar feedback, such as in supernova-powered superbubbles +(Fraternali 2017, and references therein). In this scenario, the cooling +of the hot gas is due to the mixing with the cool gas ejected from +the disc and occurs within the fountain cycle. This process can be +detected in high-quality data as it leaves a mark in the kinematics of +the ejected disc gas (Fraternali & Binney 2008; Marasco et al. 2012). +To gain insight into the gas exchange processes between the disc +and the inner hot CGM, one must focus on the disc-halo interface +region. Deep H i observations have shown that disc galaxies, includ- +ing the Milky Way, are surrounded by a neutral gas layer extending +up to a few kpcs from their disc planes (e.g. Wakker 2001; San- +cisi et al. 2008; Hess et al. 2009; Marasco & Fraternali 2011). This +gas layer, known as extraplanar gas (EPG), is nearly ubiquitous in +late-type galaxies and has a mass of 10–30 per cent of the mass +of the H i in the disc (Marasco et al. 2019). The kinematics of the +EPG is primarily characterised by differential rotation, similar to the +disc, but with a negative rotational gradient (lag) ranging from −10 +to −20 km s−1 kpc−1 in the vertical direction (e.g. Oosterloo et al. +2007; Zschaechner et al. 2011). Non-circular motions, especially +large-scale inflows are also often found (e.g. Fraternali et al. 2002; +Barbieri et al. 2005; Marasco et al. 2019). Ionised EPG has also been +detected, both in the Milky Way (Dettmar 1990; Lehner et al. 2012, +2022) and in several other galaxies (Heald et al. 2005; Levy et al. +2019), with similar kinematics as the neutral EPG (Kamphuis et al. +2007; Li et al. 2021; Marasco et al. 2022). +The similarity between EPG and disc kinematics strongly suggests +that EPG originates mostly from the disc, very likely pushed out of +the plane due to stellar feedback and pulled back by gravity. This +phenomenon is also known as ‘galactic fountain’ (Shapiro & Field +1976; Bregman 1980). Fraternali & Binney (2006, hereafter FB06) +built ballistic models of galactic fountain flows, which successfully +reproduced many of the observed properties of the EPG in the two +nearby galaxies NGC 891 and NGC 2403. It is worth noticing that +ballistic models also describe very well the properties of the warm +gas (neutral and ionised) in the hydrodynamical TIGRESS simula- +tions (Vijayan et al. 2020). However, a pure fountain model failed +to reproduce the net inward flow (instead, an outward flow was pre- +dicted) and underestimated the rotation lag compared to the observed +EPG in NGC 891 and NGC 2403. Fraternali & Binney (2008, here- +after FB08) mitigated these issues by introducing an external factor +that could lower the angular momentum of fountain gas: accretion +from the ambient gas. Although initially introduced to reproduce the +kinematics of the EPG, the net inflow rate derived from this model +turned out to be consistent with the SFR of the two galaxies, sug- +gesting that the accretion triggered by the fountain cycle could be a +viable mechanism to maintain the star formation activity. +An unsolved issue of the above fountain-driven accretion scenario +was the source of the accretion. This has been explored by Marinacci +et al. (2010) with hydrodynamical simulations. Their simulations of +fountain gas clouds interacting with the hot corona indicated that +the corona was a possible accretion source. During the interaction +process, part of the fountain gas is stripped and mixed with the hot +gas. The mixture has a typical temperature of 𝑇 ∼ 105 K, where the +cooling function peaks, and also higher metallicity and density than +the hot corona. As a consequence, the cooling time is reduced to a +value shorter than the travel time of fountain gas. This result has been +confirmed by other simulations with increasing levels of complexity +(Armillotta et al. 2016; Gronke & Oh 2018; Kooij et al. 2021). Some +studies have upgraded the approach of FB08, taking into account +the results of hydrodynamical simulations, using physical properties +of the EPG and the hot corona as adjustable parameters, and man- +aged to reproduce the phase-space distribution of both neutral and +ionised EPG in the disc–halo interface of Milky Way remarkably +well (Marasco et al. 2013; Fraternali et al. 2013; Marasco et al. 2012, +hereafter M12). The best-fitting model predicted a net inflow rate +which is consistent with the SFR of the Milky Way. +The aforementioned studies strongly suggest that fountain-driven +accretion takes place in the Milky Way and provides a promising +explanation for how galaxies like our own can sustain their star for- +mation with time. However, so far the Milky Way remains the only +galaxy for which a state-of-the-art model of the galactic fountain has +been applied to the observations using a parametric fitting method- +ology, which is required to robustly characterise the fountain flow +and to quantify the properties of the accreting gas. The earlier mod- +els in FB08 did not statistically explore the parameter space, and +furthermore, did not include the condensation of the corona, since +hydrodynamical simulations were not available by then. In this pa- +per, we revisit this by applying our state-of-the-art fountain model to +NGC 2403, using high-quality H i data (with a beam size of 30′′× +29′′and an rms-noise of 0.19 mJy beam−1) from Fraternali et al. +(2002), which were later included in the HALOGAS survey (Heald +et al. 2011). Table 1 summarises the main physical properties of +NGC 2403. +In Section 2 we provide a description of our dynamical model of +the galactic fountain. In Section 3 we discuss the customisation we +have made to implement the model for the case of NGC 2403. In +Section 4 we present the modelling results. In Section 5 we discuss +the reliability of our results and possible implications. We summarise +our analysis in Section 6. +2 THE MODEL +In this Section, we describe the main ingredients of our model and +discuss its main free parameters. Further details can be found in +FB06, FB08 and M12. We consider two different types of models: +a ‘pure fountain’ ballistic model and a ‘fountain + corona accretion’ +model which takes the interaction of fountain clouds with the hot +coronal gas into consideration. In both scenarios, the models have a +quasi-stationary state and are axisymmetric. The neutral EPG in the +disc–halo interface region is modelled as a collection of clouds that +are ejected from the disc at different radii with a given distribution +of initial velocities and angles, and whose orbits are then integrated +in time and followed across the halo region until they return to the +disc. +Since galactic fountains are powered by stellar feedback, we as- +sume that the amount of gas ejected from each location in the disc +is proportional to the SFR surface density at that radius. In practice, +we incorporate this assumption by assigning, to each of our mod- +elled clouds, a weight proportional to the SFR surface density at the +ejection radius. This weight is then factored in when creating the +mock datacube to be compared with observations (see also further +explanations below). +In our pure fountain ballistic models, the trajectories of the fountain +clouds are integrated using a numerical approximation of the galaxy +gravitational potential, derived as described in Section 3.1. For foun- +MNRAS 000, 1–13 (2023) + +Fountain-driven corona accretion in NGC 2403 +3 +Galaxy Name +RA +DEC +PA +INCL +Distance +Hubble Type +MB +M∗ +MHI,EPG +SFR +[◦] +[◦] +[Mpc] +[108 M⊙] +[108 M⊙] +[M⊙ yr−1] +(1) +(2) +(3) +(4) +(5) +(6) +(7) +(8) +(9) +(10) +(11) +NGC 2403 +07h36m51.s4 ++65◦36′09.′′2 +124.6 +62.5 +3.2 +SAcd +−19.68 +71.9 +5.9 +0.6 +Table 1. Galaxy properties. Columns: (1) Galaxy name. (2)–(3): Coordinates (J2000). (4)–(5): Position-angle and inclination. (6) Distance. (7) Hubble type. (8) +Absolute magnitude in the 𝐵-band. (9) Stellar mass (see Pezzulli et al. 2015). (10) Total mass of H i extraplanar gas. (11) Total star formation rate of the galaxy. +Values in this table are taken from Marasco et al. (2019) unless otherwise mentioned. +tain + corona accretion models, hydrodynamical forces due to the +interaction between the clouds and the hot corona are parameterised +in simple forms described in Section 2.3. +The positions and velocities of the clouds along their orbits are +recorded at each time-step (0.3 Myr), projected along the line-of- +sight of the observer, weighted by the local SFR surface density at +the ejection radius and transferred into a synthetic datacube, which +is then adapted to a specific galaxy (NGC 2403 in our case) by +assuming a distance, inclination (INCL), and position angle (PA), and +using the same observational setup (beam shape, spectral resolution, +pixel size, etc.) of the data under consideration. The outcome of +the dynamical model is therefore a synthetic datacube which can +be directly compared with the observational H i data of our target +galaxy. +Construction of the model involves several parameters but we +will focus preferentially on three (for pure fountain models) or four +(only for fountain + corona accretion models) that regulate the initial +outflow speed of the clouds, their neutral gas fraction, the EPG +total mass and, for models that include interaction with the corona, +an additional parameter that regulates the condensation efficiency. +Below we discuss these parameters in detail. Other ingredients are +fixed by the observations, in particular the galaxy potential (which +affects the trajectory of the cloud) and the SFR surface density profile +(which regulates the ejection rate), as described in Section 3. +2.1 Outflow velocity +Fountain clouds are initially located within the galaxy disc and rotate +at the circular speed set by our gravitational potential1. Each cloud +receives a ‘kick’ with a velocity 𝑣k at certain angles 𝜃, which is +defined as the angle between the velocity vector and the direction +normal to the disc plane. The probability distribution of the ejection +as a function of 𝑣k and 𝜃 (assuming a uniform probability in the +azimuthal direction) follows FB06 and is given by +P(𝑣k, 𝜃) ∝ exp +� +− +𝑣2 +𝑘 +2ℎ2𝑣 cos2Γ 𝜃 +� +, +(1) +where ℎ𝑣 is the characteristic velocity, and Γ determines the level +of collimation of the ejected clouds. Larger values of ℎ𝑣 increase +the probability that a cloud is kicked at high speed. The larger Γ, the +more collimated the ejection. FB06 have tested models with different +values for Γ and found that more collimated ejections agree better +with the data. We have therefore fixed Γ = 10 (highly collimated). +The outflow velocity of a cloud affects the maximum height and the +1 They also feature an additional velocity component, with an amplitude +randomly extracted from a Gaussian distribution with rms of 8 km s−1 and +a random (isotropic) direction, to simulate the typical velocity dispersion of +the neutral ISM (Iorio et al. 2017; Bacchini et al. 2019; Mancera Piña et al. +2021). +trajectory of the orbit and therefore influences the final model. We, +therefore, let the characteristic velocity ℎ𝑣 be a free parameter with +a flat prior in the range 40–100 km s−1. This range covers the typical +characteristic ejection speeds of the warm gas in high-resolution +hydrodynamical simulations of galactic fountains (Kim & Ostriker +2018). It also agrees with theoretical estimates of the typical blow-out +speed of individual superbubbles (e.g. Mac Low & McCray 1988; +Keller et al. 2014). +2.2 Phase change +Previous studies have found that the neutral EPG in some spiral +galaxies (including the Milky Way) shows a tentative preference for +vertical inflow (Marasco et al. 2019; French et al. 2021, for example), +which can be interpreted as due to a change of phase during the +fountain cloud orbit: gas is largely ionised when ejected from the +star-forming region of the disc but later recombines and becomes +visible in H i at some point during its trajectory. To account for this +effect in our model, we assume that a cloud is only visible in the H i +phase when +𝑣𝑧(𝑡) < 𝑣𝑧,0(1 − 𝑓ion), +(2) +where 𝑣𝑧 is the vertical velocity (that is, in the direction perpendicular +to the disc) of the cloud, 𝑣𝑧,0 is the vertical component of the initial +outflow velocity and 𝑓ion is the ionisation fraction parameter, which +we set as a free parameter with a flat prior and varies from zero to +one. When 𝑓ion equals zero, the cloud is visible in the whole orbit, +while when 𝑓ion equals one, the cloud is only visible when 𝑣𝑧 < 0 +(i.e., the descending stage). +2.3 Interaction with the corona +In our model, the hot corona is modelled as a smooth, volume-filling +gas layer that rotates at a lower speed than the disc, which is justified +on both observational (Hodges-Kluck et al. 2016) and theoretical +(Pezzulli et al. 2017) grounds. We assume that the corona maintains +a temperature of ∼ 106 K, which implicitly implies some heating by +either supernova feedback (e.g. Stinson et al. 2013) or active galac- +tic nucleus feedback (for galaxies with ongoing AGN activities; e.g. +Ciotti & Ostriker 2012). The condensation and accretion of the hot +corona is triggered by the cool (𝑇 ∼ 104 K) fountain clouds ejected +from the disc, which mix efficiently with the former and produce a +mixture at 𝑇 ∼ 105 K, dramatically reducing the cooling time of the +hot corona. The above processes have been investigated in the hydro- +dynamical simulations of cloud–corona interactions (Marinacci et al. +2010). A follow-up analysis (Marinacci et al. 2011) indicate that there +is a net transfer of momentum from the fountain to the corona until the +relative velocity between these two, 𝑣rel, reaches a certain threshold +𝑣thres. Marinacci et al. (2011) suggested 𝑣thres ≈ 75 km s−1 for initial +conditions valid for the Milky Way but pointed out that 𝑣thres can +MNRAS 000, 1–13 (2023) + +4 +A. Li et al. +vary in the range 45–105 km s−1 (see also Fraternali 2017). As soon +as 𝑣rel becomes smaller than this threshold 𝑣thres, the net momentum +transfer ceases as the condensation of corona recaptures angular mo- +mentum lost by fountain gas. For this reason, we set the azimuthal +speed of the corona to be always lower than the local circular speed +𝑣c by 𝑣thres, and in this case, 𝑣c − 75 km s−1. In Section 5.1 we ex- +plore models with different value of 𝑣thres, corresponding to different +rotational speeds for the coronal gas. +In the above scenario, the cloud acceleration due to interaction +with the corona is defined as +�𝒗 = +� +− C𝜌hot 𝜎cloud(𝑣rel−𝑣thres) +𝑀cloud +𝒗rel − 𝛼𝒗rel, +𝑣rel ≥ 𝑣thres +−𝛼𝒗rel, +𝑣rel < 𝑣thres, +(3) +where 𝒗rel is the cloud-corona relative velocity vector, 𝑣rel is the +modulus of 𝒗rel, 𝑀cloud and 𝜎cloud are the mass and the cross-section +of the cloud (defined as 𝜋𝑅2 +cloud, with 𝑅cloud the radius of the cloud), +𝜌hot is the density of the corona, C is a dimensionless constant of +order unity (in our model C=1) to account for the geometry of the +cloud, and 𝛼 is the condensation rate of the coronal gas onto the +cloud, such that the mass of the cloud 𝑀cloud grows with time as +�𝑀cloud = 𝛼𝑀cloud. We assume a corona density of 10−3 cm−3, a +cloud radius of 100 pc and an initial mass of 2× 104 M⊙, consistent +with typical values of fountain clouds suggested by observations (Hsu +et al. 2011). +The first term on the right-hand side of equation 3 represents the +drag experienced by the fountain cloud as it moves through the coro- +nal gas: the cloud speed decreases as long as its velocity stays above +𝑣thres. The second term is due to the condensation of coronal gas onto +the cloud: as the total mass of the cloud increases, conservation of +the total momentum implies lower velocity (see Fraternali & Binney +2008). We have also derived the drag timescale 𝑡drag = 724 Myr us- +ing equation(6) in Fraternali (2017), which is larger than the fountain +orbit time (∼100 Myr), we therefore expect that drag only has a minor +effect. +In fountain + corona accretion models, we let 𝛼 be a free parameter +with a flat prior in the range 𝛼 = 0–6 Gyr−1. +2.4 EPG mass +The normalisation of the H i flux presented in the final galactic foun- +tain model sets the total H i EPG mass, which is another free param- +eter. We use a fiducial EPG mass of 5.9 × 108 M⊙ from Marasco +et al. (2019) as an initial guess, but allow the EPG mass to vary, +multiplying the fiducial value by a normalisation scaling factor in the +range 0.1–10. +3 IMPLEMENTATION OF THE MODEL +In this section, we describe the gravitational potential and the SFR +surface density radial profile for NGC 2403, as they are necessary +ingredients to construct our dynamical models. We then describe +how we fit the model parameters to the data. +3.1 The gravitational potential +We use the gravitational potential grid derived by FB06 for NGC 2403 +without modification. Below we briefly describe how the potential +model is built. +The gravitational potential was derived from an axisymmetric +mass model, which consists of three components: a stellar disc, a +(𝑀/𝐿)∗ +𝑅∗ +ℎ∗ +𝑅gas +ℎgas +𝜌0,DM +r𝑠 +[kpc] +[kpc] +[kpc] +[kpc] +[M⊙ kpc−3] +[kpc] +(1) +(2) +(3) +(4) +(5) +(6) +(7) +1.70 +2.0 +0.4 +5.7 +0.1 +3.1 × 107 +4.5 +Table 2. Mass models for NGC 2403. Columns: (1) Mass-to-light ratio in +the 𝐵-band of the stellar disc. (2)–(3): Scale length and scale height of the +stellar disc. (4)–(5): Scale length and scale height of the gaseous disc. (6)–(7) +Central density and scale radius of the NFW dark matter halo. +gaseous disc, and an NFW dark matter halo (Navarro et al. 1997). +FB06 performed a mass decomposition of the H i rotation curve of +NGC 2403 (Fraternali et al. 2002) using the three components men- +tioned above. The stellar and the gaseous discs’ density distributions +were given by exponential profiles, along both the radial (𝑅) and the +vertical (𝑧) direction. The scale length of the stellar (gaseous) disc +𝑅∗ (𝑅gas) was derived by fitting an exponential profile to the stellar +(gaseous) surface brightness radial profile. The scale height of the +stellar disc was set to one-fifth of its scale length (see van der Kruit +& Freeman 2011 and references therein), and the scale height of the +gaseous disc was set to 100 pc (typical of the inner gaseous disc, see +Marasco et al. 2017; Bacchini et al. 2019; Mancera Piña et al. 2022). +The mass-to-light ratio of the stellar disc was derived via the rota- +tion curve decomposition. The above parameters of the mass model +are listed in Table 2. Once the parameters of all components are +decided, the galactic potential and forces are calculated numerically +in the (𝑅, 𝑧) cylindrical coordinate system, using a grid with a cell +size of 0.1 kpc within 𝑅 < 25 kpc and 𝑧 < 5 kpc, and of 0.5 kpc for +25 < 𝑅 < 100 kpc and 5 < 𝑧 < 100 kpc. Potential and forces are +determined at any (𝑅,𝑧) via a bilinear interpolation of these grids +(see FB06 for details). +3.2 Star-formation-rate surface-density profiles +In this paper, we directly use the SFR surface density radial profiles +from previous observations, as opposed to FB06, which used the +Schmidt–Kennicutt law (Kennicutt 1989), and M12, which used an- +other empirical star formation law (directly derived from 17 galaxies +with known gas and SFR surface densities) to estimate the SFR. +The SFR surface-density profile of NGC 2403 is mainly taken from +Leroy et al. (2008), which derived the SFR using a combination of +far ultraviolet (FUV) and 24 𝜇m data, and is then complemented +with the SFR surface density profile from Bigiel et al. (2010), which +is derived from FUV data with a lower resolution but larger radial +extent compared to Leroy et al. (2008). We refer the readers to Bac- +chini et al. (2019, 2020) for more details about collecting SFR data +of NGC 2403. Fig. 1 shows the SFR surface-density data and the +interpolated profile (in steps of 0.5 kpc) which we used as an input +for our fountain models. +3.3 Separation of the EPG emission +Before modelling the EPG in the NGC 2403 datacube, we first need +to isolate its emission from the underlying disc and from external +regions (foreground and background emission) that are clearly not +associated with the galaxy. For this purpose, we follow the procedure +described in Marasco et al. (2019). +The emission from regions external to the galaxy is filtered out by +spatially smoothing the datacube by a 2D Gaussian kernel with a full +width half maximum (FWHM) of 64.′′5 × 54.′′6, which is five times +MNRAS 000, 1–13 (2023) + +Fountain-driven corona accretion in NGC 2403 +5 +0 +2 +4 +6 +8 +10 +12 +14 +R [kpc] +10 +−4 +10 +−3 +10 +−2 +Σ +SFR +[Msun +kpc +−2 +yr +−1 +] +Leroy08 +Bigiel10 +interpolated SFR +Figure 1. Star formation rate surface density versus galactocentric distance +in NGC 2403. Blue dots represent data from Leroy et al. (2008) while orange +points are from Bigiel et al. (2010). The green curve shows the interpolated +profiles with steps of 0.5 kpc and is used as an input for our fountain model. +larger than the spatial resolution of the data, calculating a smoothed +rms noise level, and then sigma-clipping at S/N = 4. This produces +a mask that is applied to the original (not smoothed) data to exclude +the regions external to the main galaxy. +In intermediate-inclination galaxies like NGC 2403, the emission +from the EPG overlaps spatially with that from the regularly rotating +disc but can be (at least in part) separated from the latter in the +velocity space, provided that the velocity resolution is sufficient. +Here, we employ the disc–EPG separation method introduced by +Fraternali et al. (2002), which works as follows. For any given H i +velocity profile at a certain location in the sky, the disc component +is assumed to be described by a Gaussian profile. The EPG adds a +wing to the profile, which is typically due to the lagging of EPG +and located toward the systemic-velocity side; although wings on +both sides can be seen at some spatial locations across the disc due +to other non-circular (mostly vertical) motions (see also Boomsma +et al. 2008). Despite the disc and EPG profiles are blended together, it +is reasonable to neglect the contribution of the EPG around the peak +of each velocity profile since EPG mass is only a small percentage +(∼ 20 per cent for NGC 2403, Marasco et al. 2019) of the total H i +mass. We therefore use the ‘peak’ region to fit the disc emission +by performing a Gaussian fit using only the upper 40 per cent (in +intensity) of the line profile. This Gaussian profile is considered to +be the contribution of emission from the disc component alone. Pixels +with disc emission (estimated from the Gaussian profile) larger than +𝑁 times the rms noise are clipped (see Marasco et al. 2019 and Li +et al. 2021 for a more detailed explanation of this methodology). The +scaling factor 𝑁 is decided empirically as a compromise between +keeping enough EPG emission for the modelling and alleviating the +disc contamination. We set 𝑁 = 2 for NGC 2403. +Some peculiar features in NGC 2403, in particular, a long filament +of unknown origin (see also de Blok et al. 2014) have also been +manually filtered out (see blank regions in Figs. 2 and 3). We discuss +this further in Section 5.1. +After passing through the above mask, only EPG emission and +noise remain in the datacube. We then implement sigma-clipping at +S/N = 2 to mask the random noise. For consistency, the same mask +has also been applied to the model datacube that we describe below. +3.4 Model construction and evaluation +Our EPG models have three or four free parameters: the characteristic +outflow velocity ℎ𝑣, the ionisation fraction 𝑓ion, the condensation rate +𝛼 (for fountain + corona accretion models), and the EPG mass MEPG. +We build three(four)-dimensional grids for pure fountain (fountain ++ corona accretion) models with ℎ𝑣 varying from 40 to 100 km s−1 +in steps of 10 km s−1, 𝑓ion varying from 0.0 to 1.0 in steps of 0.2, 𝛼 +varying from 0 to 6 Gyr−1 in steps of 0.6 Gyr−1, and scaling factor +of the initial EPG mass varying from 0.1 to 10 in steps of factor of +100.2. The ranges and steps of the free parameters are summarised +in Table 3. +The best-fitting parameters are estimated by a Bayesian approach. +For each cell in our 3D (4D) parameter grid, we compute the posterior +probability of our model. For a chosen parameter vector x and given +our data D, the posterior probability P is given by +P(x|D) ∝ P(D|x)P(x), +(4) +where P(D|x) is the likelihood function and P(x) is the prior. +The prior for each parameter is uniform within the parameter space +(uniform in the logarithmic scale for the normalisation parameter). +The likelihood function is given by +P(D|x) +∝ +� +𝑛.𝑣𝑜𝑥𝑒𝑙𝑠 +exp +� +− |M(x) − D| +𝜀 +� += +exp +� +− +∑︁ +𝑛.𝑣𝑜𝑥𝑒𝑙𝑠 +|M(x) − D| +𝜀 +� += +exp[−R(x)/𝜀], +(5) +where M represents the model datacube built from parameter vector +x, 𝜀 is the uncertainty of the data, and R is the sum of the absolute +residuals between the data and the model, which is defined as the +sum of absolute difference in each pixel: Res = � |data − model|. +Note that both the model and the data have been masked using the +method described in Section 3.3, i.e, the voxels where EPG emission +is detected at more than 2𝜎 are considered in the determination of +the residuals. In equation 5, 𝜀 regulates how rapidly the likelihood +drops when our model deviates from the data. Assuming 𝜀 equal to +the rms-noise of the data is a poor choice, which leads to very nar- +row posterior probability distributions and severely underestimates +the uncertainties in our model parameters. This occurs because our +model is smooth and axisymmetric, and cannot possibly capture the +complexity of the data down to the noise level. Numerical solu- +tions to this problem can be worked out (see Section 2.5 in Marasco +et al. 2019), but in this work, we prefer to set 𝜀 a posteriori, in a +way that the 2-𝜎 uncertainty on the derived parameters corresponds +to models that look very different from the data by visual inspec- +tion. In the end, we assume 𝜀 = 0.38 Jy beam−1. We marginalise the +multi-dimensional posterior distribution to determine the probability +distribution of individual parameters. Best-fitting values are defined +as the median of these marginalised posterior distributions, and the +uncertainties are taken as half the difference between the 84th and +16th percentiles of the distribution. +4 RESULTS +4.1 Residuals and position-velocity diagrams +In this Section, we show the best-fitting results of the pure fountain +and the fountain + corona accretion models. The 2D marginalised +posterior probability distributions are shown in Appendix A. The +MNRAS 000, 1–13 (2023) + +6 +A. Li et al. +Parameter +description +range +step +units +ℎ𝑣 +Characteristic outflow velocity (equation 1) +[40,100] +10 +km s−1 +𝑓ion +Ionisation fraction during the ascending part of the orbits(equation 2) +[0,1.0] +0.2 +𝛼 +condensation rate of coronal gas (equation 3) +[0,6.0] +0.6 +Gyr−1 +Norm +EPG mass scaling factor 𝑎 +[0.1,10] +100.2 +Table 3. Free parameters of our galactic fountain model. The third column lists the range explored in our residual calculations, using a grid size given by the +forth column. 𝑎 a value of 1 corresponds to the EPG mass determined by Marasco et al. (2019) (5.9 × 108 M⊙). +best-fitting values and uncertainties, obtained with the method de- +scribed in Section 3.4, are listed in Table 4. . +The position–velocity (pv) slices of the best-fitting models are +compared with the data in Figs. 2 and 3. In general, both the pure +fountain and fountain + corona accretion models recover the EPG +emission, but we find that the former reproduces the data poorly for +pv slices parallel to the minor axis. Instead, the fountain + corona +accretion model performs better in the same locations. This is better +shown in Fig. 4 where we compare the two models for a pv slice +parallel to the minor axis with an offset 4′ from the centre. The best- +fitting pure fountain model fails to reproduce the emission marked +out by the red arrow and predicts extra emission in the blank region +marked out by the black arrow. Instead, the best-fitting fountain + +corona accretion model generates the same asymmetry shown by the +data. Previous studies (Fraternali et al. 2002; Marasco et al. 2019) +have shown that this asymmetric feature can be produced by radial +inflows. In a fountain model, EPG emission shows outward radial +flows, but accretion from low-angular momentum material can invert +this trend and produce an inward flow (especially evident for clouds +ejected from the outer regions of the disc; Fraternali 2017), which is +required to best reproduce the data. +The above visual comparison prefers the fountain + corona ac- +cretion model. This result has been already inferred by FB08, but +we now have its statistical confirmation using the likelihood values +derived by equation 5. We find − ln [P(D|x)] = 232.6 for the best- +fitting pure fountain model, while − ln [P(D|x)] = 224.5 for the +best-fitting fountain + corona accretion model, as shown in Table 4. +We use the Bayesian information criterion (BIC; Schwarz 1978) to +infer which of the two different scenarios (pure fountain or fountain ++ corona accretion) is statistically preferred by the data, given that +they make use of a different number of free parameters. The BIC is +derived as +BIC = −2 ln L + 𝑘 ln N, +(6) +where L is the likelihood of the model (equation 5), 𝑘 is the number +of parameters estimated by the model, and N is the number of inde- +pendent data points used in the fit. When comparing similar models +with different numbers of free parameters, a model with a lower +BIC is to be preferred, as the BIC penalises extra parameters that do +not significantly lower the likelihood. The BIC for the pure fountain +model is 490.6 while for the accretion model is 482.9, indicating that +the fountain + corona accretion model is statistically preferred by +BIC. +The above results show that the H i EPG of NGC 2403 is consti- +tuted by a combination of material ejected from the disc by stellar +feedback and gas cooling from the inner hot CGM and accreting +onto the disc. This is also consistent with previous indication from +kinematic modelling of the EPG which shows radial and vertical +inflow (Marasco et al. 2019). The best-fitting fountain + corona ac- +cretion model requires an outflow with a characteristic velocity of +50 ± 10 km s−1, starting out mostly ionised and becoming neutral +when the vertical velocity has been reduced by around 40%. The +inferred H i total mass of the EPG (4.7+1.2 +−0.9 × 108 M⊙) is similar to +that derived in Marasco et al. (2019) (5.9 × 108 M⊙). The accretion +rate given by our best-fitting model (0.8+0.4 +−0.2 M⊙ yr−1) is compati- +ble with the star formation rate of NGC 2403 (0.6 M⊙ yr−1; Heald +et al. 2012)2, indicating that the mechanism of fountain-driven gas +accretion can sustain the ongoing star formation in NGC 2403. It is +noteworthy that the values of both outflow speed and accretion rate +found with our statistical analysis are in agreement with those found +by FB08 by trial and error. The present analysis, however, allows us to +further our understanding of fountain-driven accretion in NGC 2403. +4.2 Properties of the extraplanar gas layer in NGC 2403 +This is the first time that a dynamical fountain model including corona +condensation has been applied to an external galaxy with a statistical +fitting method. The best-fitting fountain + corona accretion model +reproduces most of the EPG features in NGC 2403. Assuming our +model is reliable and correct (see discussion in Section 5.1), we can +therefore extract physical properties of the EPG layer, as well as a +predicted gas accretion profile, from the model. +4.2.1 Thickness of the neutral extraplanar gas layer +We determine the thickness of the EPG layer in our best-fitting model +by fitting the vertical density profiles at different radii with exponen- +tial functions. Fig. 5 shows the scale height of the EPG in our best- +fitting fountain + corona accretion model as a function of radius. The +scale height is calculated only out to 𝑅 = 12.5 kpc, as fountain clouds +beyond this radius are too rare to provide a reliable vertical profile. +Overall, the thickness of the gas layer increases slightly with radius, +which is what we would expect given that the gravitational potential +is shallower in the outer parts of the galaxy (we have assumed that +ℎ𝑣 is constant with radius for simplicity, see also Section 5.1). This +makes the orbits more extended in the outer region than in the inner +region. The flux-weighted average scale height of our EPG model +is 0.93±0.003 kpc, compatible with the scale height derived in the +kinematic model in Marasco et al. (2019). Thus, the EPG layer of +NGC 2403 is significantly thicker than its H i disc, which has scale +height comprised between 100 and 600 pc (Mancera Piña et al. 2022). +4.2.2 EPG rotational lag +Fig. 6 shows the rotation curves of the EPG layer at different heights +above the disc. These curves are derived from our best-fitting fountain ++ corona accretion model by taking the flux-weighted mean value +2 This estimate has an uncertainty of around ±0.3 dex or better, based on the +algorithm Heald et al. (2012) used to derive the SFR (Kennicutt et al. 2009). +MNRAS 000, 1–13 (2023) + +Fountain-driven corona accretion in NGC 2403 +7 +Model +𝑣thres +ℎ𝑣 +𝑓ion +𝛼 +�𝑚 +MEPG +− ln L +BIC +[km s−1] +[km s−1] +[Gyr−1] +[M⊙ yr−1] +[108 M⊙] +(1) +(2) +(3) +(4) +(5) +(6) +(7) +(8) +(9) +pure fountain +𝑁 /𝐴 +50 ± 10 +0.6±0.2 +𝑁 /𝐴 +𝑁 /𝐴 +5.9+1.5 +−1.2 +232.6 +490.6 +fountain + corona accretion +75 +50 ± 10 +0.4±0.2 +2.4+1.8 +−0.6 +0.8+0.4 +−0.2 +4.7+1.2 +−0.9 +224.5 +482.9 +fountain + corona accretion +45 +50 ± 10 +0.4+0.2 +−0.4 +4.2±1.2 +1.1+0.3 +−0.2 +4.7+1.2 +−0.9 +223.5 +480.9 +Table 4. The best-fitting values and uncertainties (obtained with the method described in Section 3.4) for our fountain (+ corona accretion) models of the EPG +of NGC 2403. We focus on the first two models in this Section and further discuss the third model in Section 5.1. (1) Model type. (2) The velocity threshold +for fountain + corona accretion models. The net transfer of momentum from the fountain to the corona ceases when the relative velocity between these two +decreases below this threshold (see Section 2.3). (3) Characteristic outflow velocity. (4) Ionisation fraction of the fountain gas. (5) Condensation rate of the hot +gas. (6) Global accretion rate of the condensed hot gas onto the disc. Note that this is not a free parameter but a value derived from the best-fitting model. (7) H i +EPG mass. (8) Logarithm of the likelihood values P(D |x) of the best-fitting models, calculated in equation 5. (9) The BIC values of the best-fitting models, +calculated from equation 6. +10 +0 +10 +100 +0 +100 +VHEL-VSYS [km/s] +-4′ +10 +0 +10 +-2′ +10 +0 +10 +P.A.=124 +pure fountain +0′ +10 +0 +10 +2′ +10 +0 +10 +4′ +10 +0 +10 +100 +0 +100 +VHEL-VSYS [km/s] +-4′ +10 +0 +10 +-2′ +10 +0 +10 +offset [′] +0′ +10 +0 +10 +2′ +10 +0 +10 +4′ +Figure 2. Position–velocity (pv) slices from the data (shown in black contours and blue colour scale) and from the best-fitting pure fountain model (red contours); +from outer to inner regions, contour levels are (2, 4, 8, 16)-𝜎, respectively, and a negative contour -2𝜎 is shown as the dashed grey contour.. The (irregular) +blank region represents the disc mask and the square blank region represents the manual mask that filters out the irregular filament in NGC 2403. Top panels are +pv slices parallel to the major axis with offsets −4′, −2′, 0′, 2′, 4′. Bottom panels are pv slices parallel to the minor axis with offsets −4′, −2′, 0′, 2′, 4′. +10 +0 +10 +100 +0 +100 +VHEL-VSYS [km/s] +-4′ +10 +0 +10 +-2′ +10 +0 +10 +P.A.=124 +fountain + corona accretion +0′ +10 +0 +10 +2′ +10 +0 +10 +4′ +10 +0 +10 +100 +0 +100 +VHEL-VSYS [km/s] +-4′ +10 +0 +10 +-2′ +10 +0 +10 +offset [′] +0′ +10 +0 +10 +2′ +10 +0 +10 +4′ +Figure 3. As in Fig. 2, but for the best-fitting fountain + corona accretion model of NGC 2403. +MNRAS 000, 1–13 (2023) + +8 +A. Li et al. +Figure 4. As in Figs. 2 and 3, but focusing on the pv slice parallel to the minor axis with offset 4′. Left: best-fitting pure fountain model. Right: best-fitting +fountain + corona accretion model. The red arrows mark regions where EPG emission is present in the data and in the fountain + corona accretion model, but +not in the pure fountain model. The black arrows mark out the region where the pure fountain model predicts extra emission with respect to the data, while the +fountain + corona accretion model correctly predicts a lack of emission. +0 +2 +4 +6 +8 +10 +12 +R [kpc] +0.8 +0.9 +1.0 +1.1 +1.2 +1.3 +scale height [kpc] +Figure 5. The scale height of the EPG layer predicted by our best-fitting +fountain + corona accretion model for NGC 2403. +of the azimuthal velocities of the particles in a given bin of radius +and height. We find that the rotation velocity of the EPG decreases +with height. At 𝑅 = 5.5 kpc (the half-mass radius of the EPG in +NGC 2403), the velocity gradient is around −10.0±2.7 km s−1 kpc−1. +This gradient is consistent with the velocity gradient of −11.7 ± +0.5 km s−1 kpc−1 inferred by Marasco et al. (2019), who modelled +the EPG of NGC 2403 with simplified geometric and kinematic +assumptions, and therefore intrinsically differs from our dynamical +model. Our results are also comparable with the velocity gradient +−15 ± 0.5 km s−1 kpc−1 directly measured in the edge-on galaxy +NGC 891. +4.3 Gas flows and accretion in NGC 2403 +Fig. 7 shows the inflow and outflow rates as a function of radius +predicted by our best-fitting fountain + corona accretion model. The +Figure 6. Rotational velocities for the EPG layer at different heights from +the plane (solid/dashed/dotted lines), compared to the disc rotation curve +(black squares with error bars) given by Fraternali et al. (2002). Velocities +are derived from our best-fitting fountain + corona accretion model by taking +the flux-weighted average of azimuthal velocity 𝑣𝜙 at given (𝑅, 𝑧) locations. +shape of the outflow rate profile strictly follows that of the SFR profile +shown in Fig. 1. This is true by construction, as explained in Section 2. +The mass loading factor (defined as the ratio of the mass outflow rate +to the SFR and therefore is proportional to the normalisation factor +free parameter in our model) is however a prediction of our model, +and we find a value of around 9.5. The inflow rate at a given radius +is given by the combination of fountain clouds and accreted coronal +particles that fall onto the disc per unit time and area. Since fountain +clouds do not fall back onto the disc at the same radius as they are +ejected and collect additional gas condensed from the corona as they +fall, the inflow rates do not precisely follow the outflow-rate trend +but show a somewhat smoother distribution. +We also present the net flow rate (where inflow is defined as +positive value) as a function of radius in Fig. 7 top panel. The first +MNRAS 000, 1–13 (2023) + +150 +150 +pure fountain +fountain + corona accretion +-100 +-100 +VHEL-Vsys [km/s] +50 +-50 +0 +0 +50 +50 +100 +100 +0 +150 +150 +-10 +-5 +0 +5 +10 +-10 +-5 +0 +5 +10 +offset' +offsetr!140 +120 +100 +Vμ[kms-1] +80 +60 +disc +40 +1 kpc +2 kpc +3 kpc +20 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +R [kpc]Fountain-driven corona accretion in NGC 2403 +9 +evident feature is that the net flow is much lower than both outflow +and inflow across the disc, except for the very outer parts. Also, except +for some fluctuation in the innermost region (within 𝑅 = 4 kpc), the +overall tendency is net inflow in the inner region (𝑅 < 10.5 kpc, +the vertical dashed line in Fig. 7 top panel) and net outflow in the +outer region. The net inflow is mostly due to condensation of the hot +corona, while the net outflow in the outer region can be explained +by the fact that the interaction between fountain gas and the corona +results in inward orbits for the former: cloud particles are more likely +to fall back to the plane at a radius smaller than their ejected radius +(see Fig. 8 in Fraternali 2017). +As we discussed in Section 1, accretion of the CGM onto the disc +is crucial for feeding star formation and is also a key process in the +evolution of a galaxy. The details of this process are however not well +understood. Now with our best-fitting fountain + corona accretion +model, we can predict the accretion rate as a function of radius, +shown in the bottom panel of Fig. 7. Despite star formation being the +origin of the fountain cycle, the fountain-driven accretion rate does +not follow the profile of the SFR surface density (shown in Fig. 1) and +in particular, it is more skewed towards larger radii compared with +the SFR surface density profile. This is due to a number of effects, the +most important of which is a radially increasing orbital time, which +is in turn a consequence of a varying gravitational potential with +radius, as also discussed in Section 4.2.1. A longer orbital time causes +an increase in the total condensation along a given orbit, even with +a fixed accretion efficiency per unit time (i.e. 𝛼), as assumed in our +model. The accretion profile has a well-defined peak at intermediate +radii and its exact position is determined by an interplay between +a radially declining SFR surface density and a radially increasing +duration of the orbits (see also M12 for the Milky Way). +The gas accretion rate that comes from corona condensation is +at every radius a minor fraction of the overall gas inflow (∼ 10%; +see Fig. 7). Compared to the total accretion rate of 0.8 M⊙ yr−1, the +total inflow and outflow rates are 6.48 M⊙ yr−1 and 5.69 M⊙ yr−1, +respectively. Most of the gas inflow occurs as a consequence of the +return to the disc of the gas ejected by the fountain. However, the +fountain cycle by itself does not add any new gas to the disc and +would not help to sustain the star formation. Instead, our model +predicts that the fountain flow "captures" new gas from the corona +that is then added everywhere across the disc to sustain the local star +formation. Remarkably, the accretion rate that is needed to reproduce +the seemingly independent kinematics of the EPG in NGC 2403 turns +out to be very similar to the one needed to sustain its star formation. +Overall, the accretion rate peaks at around 4.5 kpc and the cumu- +lative accretion rate reaches 50 per cent of the total accretion rate +at 6.25 kpc. As we mentioned, this distribution is shifted outwards +with respect to the SFR surface density distribution, which peaks in +the centre of NGC 2403 and reaches 50 per cent of the total SFR +at 3.3 kpc. The relevance of this difference is further discussed in +Section 5.2. +5 DISCUSSION +5.1 Reliability of the fountain + corona accretion model +In this paper, we have investigated gas accretion as the potential mech- +anism to maintain star formation in NGC 2403 and found a remark- +able consistency between the accretion rate predicted by our model +and the SFR. However, accretion is not the only fuelling mechanism. +Several studies have pointed out the importance of stellar mass loss +in extending gas consumption timescales (e.g. Sandage 1986; Ken- +nicutt et al. 1994) and sustaining star formation (e.g. Schaye et al. +0 +2 +4 +6 +8 +10 +12 +14 +R[kpc] +−0.02 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +Mass +flow +rate +[M +⊙ +yr +−1 +kpc +−2 +] +inflow +outflow +net flow +0 +2 +4 +6 +8 +10 +12 +14 +R[kpc] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Coronal +gas +inflow +rate +[10 +−3 +M +⊙ +yr +−1 +kpc +−2 +] +Figure 7. Inflow and outflow rate surface density as a function of radius +predicted by our best-fitting fountain + corona accretion model of NGC 2403. +Top panel: inflow rates (blue bars), outflow rates (black bars), and net flow +rates (red bars: inflow−outflow; positive values indicate net inflow). The +vertical dashed line at 10.5 kpc marks the boundary where the net flow changes +from inflow to outflow. Bottom panel: inflow rate surface density contributed +by corona accretion, the integration of which gives us the global accretion +rate of 0.8 M⊙ yr−1. +2010; Leitner & Kravtsov 2011). In particular, Leitner & Kravtsov +(2011, hereafter LK11) has estimated the current stellar mass loss +rate of NGC 2403 to be 0.5 − 0.79 M⊙ yr−1 (depending on the un- +derlying initial mass function), which seems to eliminate the need of +gas accretion. However, this mass loss rate was calculated in LK11 +assuming a SFR of 1.3 M⊙ yr−1, implying that the stellar mass loss +can sustain at most 60% of the SFR of NGC2403, while at least 40% +must be due to gas accretion. Note that the estimation of the mass loss +rate is dependent on the SFR: a lower SFR would result in a lower +mass loss rate (although not necessarily in proportion). Overall, we +conclude that gas accretion is still necessary to sustain the SFR in +NGC 2403 within the circumstances explored by the LK11 model. +In Section 4 we explored four free parameters that are crucial +for our EPG dynamical model. However, construction of the model +also involves other parameters and ingredients for which we make +specific choices. Below we discuss the limitations and reliability of +our model. +MNRAS 000, 1–13 (2023) + +10 +A. Li et al. +The gravitational potential of NGC 2403 used in this paper is gen- +erated from a mass model consisting of three components: a stellar +disc, a gaseous disc, and a dark matter halo. The parameters of the +mass model are inferred via rotation curve decomposition (FB06). +Given that the circular velocity generated from the mass model is +consistent with the rotation curve of NGC 2403 (see FB06), we con- +clude that the gravitational potential is robust. The only uncertainty is +related to the fraction of the stellar disc contribution to the potential, +parametrised by the mass-to-light ratio. The gravitational potential +used in the above analysis was based on the maximum-disc model +shown in Table 2. It is however noteworthy that the minimum disc +potential in FB06 is in fair agreement with those derived more re- +cently with more sophisticated methods (Mancera Piña et al. 2022). +FB06 have experimented with both maximum disc and minimum +disc potentials and showed that the dynamics of the EPG does not +change significantly. +An assumption of our model is the existence of a uniform charac- +teristic outflow velocity at all radii, whereas the varying stellar feed- +back activities might lead to outflow velocities changing with radius. +Allowing spatial variations in the characteristic outflow velocity is a +potential improvement for this kind of study. This has been briefly +explored in FB06 to generate specific features in N2403 (e.g. the +filament shown in channel 104.1 km s−1 and channel 135.0 km s−1 +of Fig. 14 in FB06) that are otherwise not reproduced. However, +exploring the variation of ℎ𝑣 with radius would introduce at least +one extra free parameter, which would significantly complicate our +exploration of the parameter space. Overall, the global kinematics of +the EPG in NGC 2403 appears to be well reproduced by a constant +characteristic outflow speed across the disc. +In the fountain + corona accretion scenario, the acceleration of +fountain gas is directly dictated, besides by gravity, by the veloc- +ity difference between the fountain and the corona. In our model, +we assume a relative azimuthal velocity of 75 km s−1 between the +fountain gas and the corona, based on hydrodynamical simulations +(Marinacci et al. 2011). Such a high relative velocity would imply +a rather slowly rotating corona in NGC 2403, given the disc rota- +tion of around 130 km s−1 (FB06). We have therefore tested models +with a lower relative velocity of 45 km s−1 that result in nearly iden- +tical best-fitting parameters as in Section 4.1 except for a higher +condensation rate (4.2±1.2 Gyr−1), which corresponds to a global +accretion rate of 1.1+0.3 +−0.2 M⊙ yr−1 (the best-fitting results are listed +in Table 4). This higher rate is not surprising. In our model, as a +consequence of condensation, the coronal gas joins the cold/warm +phase of the fountain gas such that the velocity of a single cloud +evolves as a combination (mass-weighted average) of the kinemat- +ics of the two components (cloud and condensed material). If the +velocity difference between these two components is reduced, one +needs a larger accretion rate (more condensed material) to produce +the same effect in the combined kinematics. It is noteworthy that +EPG models built with a lower relative velocity have lower veloc- +ity gradients than what we show in Fig. 6. However, the difference +(1.0 km s−1 kpc−1) is negligible, given that the uncertainty for our +measurement is 2.7 km s−1 kpc−1. +The separation of EPG emission from the datacube is an important +ingredient of our method. The reliability of our strategy for masking +the disc emission has been verified in several previous studies (e.g. +Fraternali et al. 2002; Marasco et al. 2019; Li et al. 2021). We +have tested the robustness of our results by fitting the data without +masking the peculiar H i filament of NGC 2403, finding the same +normalisation factor as shown in Table 4, but an ℎ𝑣 of 60 km s−1, an +𝑓ion of 0, a condensation rate of 4.8 Gyr−1, leading to an accretion +rate of 1.28 M⊙ yr−1 (all parameters are compatible with those of our +fiducial model within the errors.). Thus models with slightly higher +outflow velocities and condensation rates are preferred to account for +the filament in NGC 2403, but the overall validity of our results is +not particularly affected by our masking. +In conclusion, the construction of our dynamical model is robust. +The variation of certain ingredients leads to small changes in the +model best-fitting parameters but does not alter our main conclusion: +the EPG of NGC 2403 is produced by a combination of galactic +fountain clouds and gas accretion from the condensation of the hot +CGM at a rate compatible with the SFR of the galaxy. +5.2 Can the fountain + corona accretion sustain the inside-out +growth of the disc? +Since accretion is a key source to fuel further star formation, the +outward shift of the accretion (compared to the SFR) shown in Sec- +tion 4.3 suggests a potential inside-out redistribution of gas and star +formation activities in the future, which has been predicted by cosmo- +logical simulations (e.g. Grand et al. 2017) and supported by many +observations (e.g. Wang et al. 2011; van der Wel et al. 2014; Pez- +zulli et al. 2015). Pezzulli et al. (2015) also provided measurements +of the specific radial growth rate, 𝜈𝑅 ≡ (1/𝑅∗) × d𝑅∗/d𝑡, where +𝑅∗ is the scale length of the stellar disc, for a sample of galaxies +including NGC 2403. Furthermore, a cosmological/zoom-in simula- +tion (Grand et al. 2019) also found that fountain clouds can acquire +angular momentum via interaction with the CGM. +To verify whether the gas accretion due to a galactic fountain can +be deemed responsible for this growth, we calculated the variation +in time of the specific angular momentum d𝑗/d𝑡 of the stellar disc (a +direct tracer of disc growth; Mo et al. 1998; Posti et al. 2019) due to +accretion, under the simplifying assumption that the next generation +of stars will be formed out of the newly accreted gas. This gives +d𝑗 +d𝑡 += +d(𝐽/𝑀) +d𝑡 += +1 +𝑀 +d𝐽 +d𝑡 − +𝐽 +𝑀2 +d𝑀 +d𝑡 , +(7) +where 𝐽 and 𝑀 (7.2 × 109 M⊙) are the angular momentum and mass +of the stellar disc. We estimate 𝐽 as 𝐽 = 2𝑀𝑉flat𝑅∗ (Romanowsky & +Fall 2012), where 𝑉flat is the rotational velocity of the flat part of the +rotation curve (130 km s−1) and 𝑅∗ = 2.0 kpc (values from Fraternali +et al. 2002). The time derivative of the angular momentum d𝐽/d𝑡 is +given by +d𝐽 +d𝑡 += +d𝐽in +d𝑡 +− d𝐽out +d𝑡 += +2𝜋 +∫ R +0 +𝑅′2Fin(𝑅′)𝑉in(𝑅′) d𝑅′ +−2𝜋 +∫ R +0 +𝑅′2Fout(𝑅′)𝑉out(𝑅′) d𝑅′, +(8) +where Fin (Fout) is the inflow (outflow) surface density rate given in +Section 4.3, 𝑉in(𝑅′) (𝑉out(𝑅′)) is the average rotational velocity of +all cloud particles falling onto (ejected from) the disc at radius 𝑅′, +obtained from our model by tracking the outflow and inflow radius +and velocity of all fountain clouds. The time derivative of the mass, +d𝑀/d𝑡, is by definition the accretion rate of new gas given by the +model. +Implementing the above equation to our best-fitting model, we +have d𝑗/d𝑡 = −2.6 × 10−8 km s−1 kpc yr−1. This would indicate that +the gas accreted through the fountain cannot be solely responsible for +the observed inside-out growth of the disc. Part of this growth should +MNRAS 000, 1–13 (2023) + +Fountain-driven corona accretion in NGC 2403 +11 +then be ascribed to gas that is already present in the disc. This is a +viable option, as the gas in the disc is known to be located, on average, +at larger radii compared to the stellar component (e.g. Fraternali +et al. 2002). This solution is, however, only partly satisfactory, as +the gas reservoir at these large radii would, without replacement, +be consumed on a relatively short timescale (a few Gyr; see e.g. +Fraternali & Tomassetti 2012), implying that the growth of the disc +would not be sustainable in the long term. +With these considerations in mind, we stress that our calculation +of d𝑗/d𝑡, presented above, very much depends on the value that we +are assuming for the rotational speed of the corona, which is, as we +discussed above, very uncertain. Interestingly, when assuming the +rotational lag between the fountain and the hot gas is 45 km s−1 (the +third model in Table 4), we have d𝑗/d𝑡 = 1.5×10−8 km s−1 kpc yr−1, +which indicates an inside-out growth. Combining the current value +of the specific angular momentum 𝑗 and its derivative d𝑗/d𝑡, we can +easily derive the specific angular momentum growth rate, which we +define (following Pezzulli et al. 2015) as 𝜈 𝑗 ≡ (1/𝑗) × d𝑗/d𝑡. We +find a value of 𝜈 𝑗 = 2.88 × 10−2 Gyr−1, in excellent agreement with +the specific radial growth rate 𝜈𝑅 = (2.93 ± 0.16) × 10−2 Gyr−1 +measured by Pezzulli et al. (2015) for NGC 2403. The two quantities +𝜈 𝑗 and 𝜈R are comparable and are in fact expected to be equal, as long +as the rotation curve of the galaxy can be considered approximately +stationary with time3. We have therefore found that our model with a +reduced rotational lag is in remarkable quantitative agreement with +the galactic fountain being the main source of the observed inside-out +growth in NGC 2403. +It is important to note that in the absence of triggered condensa- +tion, a galactic corona will be expected to cool in the very inner parts, +where its density tends to be higher, thus producing the accretion of +low angular momentum gas that then would need to be expelled via +strong feedback (e.g. Brook et al. 2012). Instead, when the cooling +is triggered by the fountain, the location of the bulk of the gas ac- +cretion is naturally shifted to outer radii for the reasons described +in Section 4.3. This phenomenon had been indicated as plausibly +compatible with the inside-out growth of discs (Pezzulli & Fraternali +2016), but this is the first time that quantitative evidence is provided. +6 CONCLUSION +In this work, we have modelled the distribution and kinematics of +the neutral extra-planar gas (EPG) in the late-type nearby galaxy +NGC 2403 using a dynamical model of galactic fountain. In this +model, stellar feedback activities continuously eject gas from the +galaxy disc, which travels through the halo and falls back to the +disc. This gas cycle brings metal-rich and cold/warm gas to mix +and interact with the hot corona, significantly reducing its cooling +time, and leading to condensation and accretion of some coronal +gas onto the disc. Due to angular momentum exchange between the +fountain clouds and the corona, this interaction is expected to leave a +signature in the kinematics of the H i gas at the disc–halo interface. +The application of our models to the data leverage this signature to +infer, along with other parameters, the efficiency of the condensation +process and the accretion rate of coronal gas onto the disc. +While these models have been applied extensively to the EPG of +the Milky Way (Marasco et al. 2012, 2013; Fraternali et al. 2013, +3 This is immediately seen by taking the time derivative of the equation +𝑗 = 2𝑉flat𝑅∗. +2015), so far applications to external galaxies were limited to the pre- +liminary studies of FB06 and FB08, which did not include a rotating +corona nor a statistically meaningful exploration of the parameter +space. This study presents the first detailed application of the current +fountain accretion framework to an external galaxy. Our results are +summarised as follows: +(i) The galactic fountain framework can reproduce most of the +neutral EPG features in NGC 2403. A model where the fountain +clouds interact with the hot corona is statistically preferred compared +to a pure fountain model without interaction with the hot CGM. +(ii) The best-fitting model requires a fountain with a characteristic +outflow velocity of 50±10 km s−1, with the gas being ionised for some +time after ejection and then recombining. Recombination appears to +occur on average when its vertical velocity has been reduced by about +40 per cent. +(iii) The H i EPG in NGC 2403 inferred from the best-fitting +model has a total EPG mass of 4.7+1.2 +−0.9 × 108 M⊙, with an average +scale height of 0.93 ± 0.003 kpc and a vertical gradient in rotational +velocity of −10.0±2.7 km s−1 kpc−1. Our values are compatible with +a previous estimate of Marasco et al. (2019), which was derived with +simpler phenomenological approaches. +(iv) Our model predicts a condensation rate of 2.4 Gyr−1 +(4.2 Gyr−1 ) for the hot CGM, leading to a total accretion rate of +0.8 M⊙ yr−1 (1.1 M⊙ yr−1) when assuming the rotational lag be- +tween the fountain and the hot gas is 75 km s−1 (45 km s−1), similar to +the star formation rate 0.6 M⊙ yr−1 of NGC 2403, suggesting corona +accretion as a viable mechanism to maintain the star-formation rate +in this galaxy. +(v) The accretion rate surface density profile predicted by our +model is radially more extended than the star-formation-rate surface +density. We have also shown that, if the rotation velocity of the corona +is larger than a certain threshold, the specific angular momentum +growth rate predicted by our model is in excellent agreement with +the observed inside-out growth rate in NGC 2403. The fountain- +driven accretion process can therefore be responsible for the inside- +out growth of its stellar disc. +ACKNOWLEDGEMENTS +The authors would like to thank an anonymous referee for helpful +comments and Cecilia Bacchini for collecting and providing the H i, +H2, and star-formation-rate data of NGC 2403. AL was supported +by the Netherlands Research School for Astronomy (Nederlandse +Onderzoekschool voor Astronomie, NOVA), Network 1, Project +10.1.5.9 WEAVE. GP acknowledges support from the Netherlands +Research School for Astronomy (Nederlandse Onderzoekschool voor +Astronomie, NOVA) through project 10.1.5.18. +DATA AVAILABILITY +The data underlying this article were obtained by Fraternali +et al. (2002) with the CS configuration of the VLA and were +later included in the HALOGAS survey, which is available at +https://www.astron.nl/halogas. +REFERENCES +Afruni A., Fraternali F., Pezzulli G., 2021, MNRAS, 501, 5575 +Anderson M. E., Bregman J. N., 2011, ApJ, 737, 22 +MNRAS 000, 1–13 (2023) + +12 +A. Li et al. +Anderson M. E., Churazov E., Bregman J. N., 2016, MNRAS, 455, 227 +Armillotta L., Fraternali F., Marinacci F., 2016, MNRAS, 462, 4157 +Aumer M., Binney J. J., 2009, MNRAS, 397, 1286 +Bacchini C., Fraternali F., Iorio G., Pezzulli G., 2019, A&A, 622, A64 +Bacchini C., Fraternali F., Pezzulli G., Marasco A., 2020, A&A, 644, A125 +Barbieri C. V., F., Oosterloo T., Bertin G., Boomsma R., Sancisi R., 2005, +A&A, 439, 947 +Bigiel F., Leroy A., Walter F., Blitz L., Brinks E., de Blok W. J. G., Madore +B., 2010, AJ, 140, 1194 +Binney J., Nipoti C., Fraternali F., 2009, MNRAS, 397, 1804 +Boomsma R., Oosterloo T. A., Fraternali F., van der Hulst J. M., Sancisi R., +2008, A&A, 490, 555 +Bregman J. N., 1980, ApJ, 236, 577 +Brook C. B., Stinson G., Gibson B. K., Roškar R., Wadsley J., Quinn T., 2012, +MNRAS, 419, 771 +Ciotti L., Ostriker J. P., 2012, in Kim D.-W., Pellegrini S., eds, Astrophysics +and Space Science Library Vol. 378, Astrophysics and Space Science +Library. p. 83 (arXiv:1104.2238), doi:10.1007/978-1-4614-0580-1_4 +Crain R. A., Eke V. R., Frenk C. S., Jenkins A., McCarthy I. G., Navarro J. F., +Pearce F. R., 2007, MNRAS, 377, 41 +Dettmar R. J., 1990, A&A, 232, L15 +Di Teodoro E. M., Fraternali F., 2014, A&A, 567, A68 +Di Teodoro E. M., Peek J. E. G., 2021, ApJ, 923, 220 +El-Badry K., et al., 2018, MNRAS, 473, 1930 +Faerman Y., Sternberg A., McKee C. F., 2020, ApJ, 893, 82 +Fraternali F., 2017, in Fox A., Davé R., eds, ASSL, Vol. 430, Gas Ac- +cretion onto Galaxies. Springer International Publishing AG, p. 323, +doi:10.1007/978-3-319-52512-9_14 +Fraternali F., Binney J. J., 2006, MNRAS, 366, 449 +Fraternali F., Binney J. J., 2008, MNRAS, 386, 935 +Fraternali F., Tomassetti M., 2012, MNRAS, 426, 2166 +Fraternali F., van Moorsel G., Sancisi R., Oosterloo T., 2002, AJ, 123, 3124 +Fraternali F., Marasco A., Marinacci F., Binney J., 2013, ApJ, 764, L21 +Fraternali F., Marasco A., Armillotta L., Marinacci F., 2015, MNRAS, 447, +L70 +French D. M., et al., 2021, ApJ, 923, 50 +Fukugita M., Peebles P. J. E., 2006, ApJ, 639, 590 +Gatto A., Fraternali F., Read J. I., Marinacci F., Lux H., Walch S., 2013, +MNRAS, 433, 2749 +Grand R. J. J., et al., 2017, MNRAS, 467, 179 +Grand R. J. J., et al., 2019, MNRAS, 490, 4786 +Gronke M., Oh S. P., 2018, MNRAS, 480, L111 +Heald G. H., Rand R. J., Benjamin R. A., Bershady M. A., Collins J. A., +Bland-Hawthorn J., 2005, in AAS Meeting Abstracts. p. 164.05 +Heald G., et al., 2011, A&A, 526, A118 +Heald G., et al., 2012, The Westerbork Hydrogen Accretion in LOcal GAlax- +ieS (HALOGAS) survey (Corrigendum). I. Survey description and pilot +observations, Astronomy & Astrophysics, Volume 544, id.C1, 1 pp., +doi:10.1051/0004-6361/201015938e +Heckman T., Borthakur S., Wild V., Schiminovich D., Bordoloi R., 2017, +ApJ, 846, 151 +Hess K. M., Pisano D. J., Wilcots E. M., Chengalur J. N., 2009, ApJ, 699, 76 +Hodges-Kluck E. J., Miller M. J., Bregman J. N., 2016, ApJ, 822, 21 +Hopkins A. M., McClure-Griffiths N. M., Gaensler B. M., 2008, ApJ, 682, +L13 +Hsu W. H., Putman M. E., Heitsch F., Stanimirović S., Peek J. E. G., Clark +S. E., 2011, AJ, 141, 57 +Iorio G., Fraternali F., Nipoti C., Di Teodoro E., Read J. I., Battaglia G., 2017, +MNRAS, 466, 4159 +Joung M. R., Bryan G. L., Putman M. E., 2012, ApJ, 745, 148 +Kamphuis P., Peletier R. F., Dettmar R. J., van der Hulst J. M., van der Kruit +P. C., Allen R. J., 2007, A&A, 468, 951 +Keller B. W., Wadsley J., Benincasa S. M., Couchman H. M. P., 2014, MN- +RAS, 442, 3013 +Kennicutt Robert C. J., 1989, ApJ, 344, 685 +Kennicutt Robert C. J., Tamblyn P., Congdon C. E., 1994, ApJ, 435, 22 +Kennicutt Robert C. J., et al., 2009, ApJ, 703, 1672 +Kereš D., Katz N., Weinberg D. H., Davé R., 2005, MNRAS, 363, 2 +Kereš D., Katz N., Fardal M., Davé R., Weinberg D. H., 2009, MNRAS, 395, +160 +Kim C.-G., Ostriker E. C., 2018, ApJ, 853, 173 +Kooij R., Grønnow A., Fraternali F., 2021, MNRAS, 502, 1263 +Lagos C. d. P., Theuns T., Stevens A. R. H., Cortese L., Padilla N. D., Davis +T. A., Contreras S., Croton D., 2017, MNRAS, 464, 3850 +Lehner N., Howk J. C., Thom C., Fox A. J., Tumlinson J., Tripp T. M., Meiring +J. D., 2012, MNRAS, 424, 2896 +Lehner N., Howk J. C., Marasco A., Fraternali F., 2022, MNRAS, +Leitner S. N., Kravtsov A. V., 2011, ApJ, 734, 48 +Leroy A. K., Walter F., Brinks E., Bigiel F., de Blok W. J. G., Madore B., +Thornley M. D., 2008, AJ, 136, 2782 +Levy R. C., et al., 2019, ApJ, 882, 84 +Li J.-T., Bregman J. N., Wang Q. D., Crain R. A., Anderson M. E., 2018, ApJ, +855, L24 +Li A., Marasco A., Fraternali F., Trager S., Verheijen M. A. W., 2021, MN- +RAS, 504, 3013 +Mac Low M.-M., McCray R., 1988, ApJ, 324, 776 +Mancera Piña P. E., Posti L., Fraternali F., Adams E. A. K., Oosterloo T., +2021, A&A, 647, A76 +Mancera Piña P. E., Fraternali F., Oosterloo T., Adams E. A. K., di Teodoro +E., Bacchini C., Iorio G., 2022, MNRAS, 514, 3329 +Marasco A., Fraternali F., 2011, A&A, 525, A134 +Marasco A., Fraternali F., Binney J. J., 2012, MNRAS, 419, 1107 +Marasco A., Marinacci F., Fraternali F., 2013, MNRAS, 433, 1634 +Marasco A., Fraternali F., van der Hulst J. M., Oosterloo T., 2017, A&A, 607, +A106 +Marasco A., et al., 2019, A&A, 631, A50 +Marasco A., Fraternali F., Lehner N., Howk J. C., 2022, MNRAS, 515, 4176 +Marinacci F., Binney J., Fraternali F., Nipoti C., Ciotti L., Londrillo P., 2010, +MNRAS, 404, 1464 +Marinacci F., Fraternali F., Nipoti C., Binney J., Ciotti L., Londrillo P., 2011, +MNRAS, 415, 1534 +Mo H. J., Mao S., White S. D. M., 1998, MNRAS, 295, 319 +Navarro J. F., Frenk C. S., White S. D. M., 1997, ApJ, 490, 493 +Nelson D., Vogelsberger M., Genel S., Sijacki D., Kereš D., Springel V., +Hernquist L., 2013, MNRAS, 429, 3353 +Nipoti C., 2010, MNRAS, 406, 247 +Oosterloo T., Fraternali F., Sancisi R., 2007, AJ, 134, 1019 +Pezzulli G., Fraternali F., 2016, MNRAS, 455, 2308 +Pezzulli G., Fraternali F., Boissier S., Muñoz-Mateos J. C., 2015, MNRAS, +451, 2324 +Pezzulli G., Fraternali F., Binney J., 2017, MNRAS, 467, 311 +Pointon S. K., Kacprzak G. G., Nielsen N. M., Muzahid S., Murphy M. T., +Churchill C. W., Charlton J. C., 2019, ApJ, 883, 78 +Posti L., Marasco A., Fraternali F., Famaey B., 2019, A&A, 629, A59 +Putman M. E., Zheng Y., Price-Whelan A. M., Grcevich J., Johnson A. C., +Tollerud E., Peek J. E. G., 2021, ApJ, 913, 53 +Romanowsky A. J., Fall S. M., 2012, ApJS, 203, 17 +Rubin K. H. R., Prochaska J. X., Koo D. C., Phillips A. C., Weiner B. J., +2010, ApJ, 712, 574 +Rubin K. H. R., Diamond-Stanic A. M., Coil A. L., Crighton N. H. M., +Moustakas J., 2018, ApJ, 853, 95 +Sancisi R., Fraternali F., Oosterloo T., van der Hulst T., 2008, A&ARv, 15, +189 +Sandage A., 1986, A&A, 161, 89 +Schaye J., et al., 2010, MNRAS, 402, 1536 +Schroetter I., et al., 2019, MNRAS, 490, 4368 +Schwarz G., 1978, Annals of Statistics, 6, 461 +Shapiro P. R., Field G. B., 1976, ApJ, 205, 762 +Stinson G. S., Brook C., Macciò A. V., Wadsley J., Quinn T. R., Couchman +H. M. P., 2013, MNRAS, 428, 129 +Tacconi L. J., et al., 2018, ApJ, 853, 179 +Trapp C. W., et al., 2022, MNRAS, 509, 4149 +Tumlinson J., et al., 2011, Science, 334, 948 +Vijayan A., Kim C.-G., Armillotta L., Ostriker E. C., Li M., 2020, ApJ, 894, +12 +Voit G. M., Bryan G. L., O’Shea B. W., Donahue M., 2015, ApJ, 808, L30 +MNRAS 000, 1–13 (2023) + +Fountain-driven corona accretion in NGC 2403 +13 +Wakker B. P., 2001, ApJS, 136, 463 +Walker S. A., Bagchi J., Fabian A. C., 2015, MNRAS, 449, 3527 +Wang J., et al., 2011, MNRAS, 412, 1081 +White S. D. M., Frenk C. S., 1991, ApJ, 379, 52 +Zahedy F. S., Chen H.-W., Johnson S. D., Pierce R. M., Rauch M., Huang +Y.-H., Weiner B. J., Gauthier J.-R., 2019, MNRAS, 484, 2257 +Zschaechner L. K., Rand R. J., Heald G. H., Gentile G., Kamphuis P., 2011, +ApJ, 740, 35 +de Blok W. J. G., et al., 2014, A&A, 569, A68 +van der Kruit P. C., Freeman K. C., 2011, ARA&A, 49, 301 +van der Wel A., et al., 2014, ApJ, 788, 28 +APPENDIX A: 2D MARGINALISED POSTERIOR +PROBABILITY DISTRIBUTION +2D marginalised posterior probability distribution maps and contours +of the three(four)-dimensional grids of free parameters (summarised +in Table 3) for pure fountain (fountain + corona accretion) models, +shown in Figs. A1 and A2 respectively. +MNRAS 000, 1–13 (2023) + +14 +A. Li et al. +Figure A1. 2D marginalised posterior probability distribution for our pure fountain models onto different 2D spaces: upper left – (ℎ𝑣, 𝑓ion), upper right – +(ℎ𝑣,Norm), lower-left – ( 𝑓ion,Norm). Iso-probability contours (in yellow) correspond to 2.51e-07, 1.06e-06, 4.47e-06, 1.88e-05, 7.94e-05, 3.34e-04, 1.41e-03, +5.96e-03,2.51e-02, 1.05e-01. +MNRAS 000, 1–13 (2023) + +90 +1.8e-01 +4.0e-02 +80 +8.9e-03 +2.0e-03 +70 +h +4.5e-04 +60 +1.0e-04 +2.2e-05 +50 +5.0e-06 +40 +1.1e-06 +2.5e-07 +10-1 +100 +Norm0.8 +1.8e-01 +4.0e-02 +0.6 +8.9e-03 +2.0e-03 +Tion +0.4 +4.5e-04 +1.0e-04 +0.2 +2.2e-05 +5.0e-06 +0.0 +1.1e-06 +2.5e-07 +10-1 +100 +Norm90 +1.8e-01 +4.0e-02 +80 +8.9e-03 +2.0e-03 +70 +4.5e-04 +60 +1.0e-04 +2.2e-05 +50 +5.0e-06 +40 +1.1e-06 +2.5e-07 +0.0 +0.2 +0.4 +0.6 +0.8 +fionFountain-driven corona accretion in NGC 2403 +15 +Figure A2. 2D marginalised posterior probability distribution for our fountain + corona accretion models onto different 2D spaces: upper-left – (ℎ𝑣, 𝑓ion), +upper-right – (ℎ𝑣,𝛼), middle-left – (ℎ𝑣,Norm), middle-right – ( 𝑓ion,𝛼), lower-left – ( 𝑓ion,Norm), lower-right – (𝛼,Norm). Iso-probability contours (in yellow) +correspond to 2.51e-07, 1.06e-06, 4.47e-06, 1.88e-05, 7.94e-05, 3.34e-04, 1.41e-03, 5.96e-03,2.51e-02, 1.05e-01. +MNRAS 000, 1–13 (2023) + +90 +1.8e-01 +4.0e-02 +80 +8.9e-03 +70 +2.0e-03 +4.5e-04 +60 +1.0e-04 +2.2e-05 +50 +5.0e-06 +40 +1.1e-06 +2.5e-07 +0.0 +0.2 +0.4 +0.6 +0.8 +fion90 +1.8e-01 +4.0e-02 +80 +8.9e-03 +hv [km s-1] +2.0e-03 +70 +4.5e-04 +60 +1.0e-04 +2.2e-05 +50 +5.0e-06 +40 +1.1e-06 +2.5e-07 +0 +i +2 +3 +4 +5 +α[Gyr-1]90 +1.8e-01 +4.0e-02 +80 +8.9e-03 +hv [km s-1] +2.0e-03 +70 +4.5e-04 +60 +1.0e-04 +2.2e-05 +50 +5.0e-06 +40 +1.1e-06 +2.5e-07 +10-1 +100 +Norm0.8 +1.8e-01 +4.0e-02 +0.6 +8.9e-03 +2.0e-03 +4.5e-04 +1.0e-04 +0.2 +2.2e-05 +5.0e-06 +0.0 +1.1e-06 +2.5e-07 +0 +1 +2 +3 +4 +5 +α[Gyr-1]0.8 +1.8e-01 +4.0e-02 +0.6 +8.9e-03 +2.0e-03 +Tion +0.4 +4.5e-04 +1.0e-04 +0.2 +2.2e-05 +5.0e-06 +0.0 +1.1e-06 +2.5e-07 +10-1 +100 +Norm1.8e-01 +5 +4.0e-02 +4 +8.9e-03 +α[Gyr-1] +2.0e-03 +3 +4.5e-04 +2 +1.0e-04 +2.2e-05 +1 +5.0e-06 +1.1e-06 +0 +2.5e-07 +10-1 +100 +Norm \ No newline at end of file diff --git a/TtE2T4oBgHgl3EQfCgZi/content/tmp_files/load_file.txt b/TtE2T4oBgHgl3EQfCgZi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..de410c23bc5ca218742f62c6a5d81ee2601e878c --- /dev/null +++ b/TtE2T4oBgHgl3EQfCgZi/content/tmp_files/load_file.txt @@ -0,0 +1,1617 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf,len=1616 +page_content='MNRAS 000, 1–13 (2023) Preprint 11 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0 Fountain-driven gas accretion feeding star formation over the disc of NGC 2403 Anqi Li1★, Filippo Fraternali1, Antonino Marasco2,3, Scott C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Trager1, Gabriele Pezzulli1, Pavel E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Mancera Piña1,4,5 and Marc A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Verheijen1 1Kapteyn Astronomical Institute, University of Groningen, Landleven 12, 9747 AD Groningen, The Netherlands 2INAF–Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122, Padova, Italy 3INAF–Osservatorio Astrofisico di Arcetri, Largo E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Fermi 5, I-50157, Firenze, Italy 4 ASTRON, Netherlands Institute for Radio Astronomy, Postbus 2, NL-7900 AA Dwingeloo, The Netherlands 5 Leiden Observatory, Leiden University, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='Box 9513, NL-2300 AA Leiden, The Netherlands Accepted January 4, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Received September 14, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' in original form February 11, 2022 ABSTRACT We use a dynamical model of galactic fountain to study the neutral extraplanar gas (EPG) in the nearby spiral galaxy NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We have modelled the EPG as a combination of material ejected from the disc by stellar feedback (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' galactic fountain) and gas accreting from the inner circumgalactic medium (CGM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This accretion is expected to occur because of cooling/condensation of the hot CGM (corona) triggered by the fountain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Our dynamical model reproduces the distribution and kinematics of the EPG H i emission in NGC 2403 remarkably well and suggests a total EPG mass of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='7+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='9 ×108 M⊙, with a typical scale height of around 1 kpc and a vertical gradient of the rotation velocity of −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='7 km s−1 kpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The best-fitting model requires a characteristic outflow velocity of 50 ± 10 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The outflowing gas starts out mostly ionised and only becomes neutral later in the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The accretion rate from the condensation of the inner hot CGM inferred by the model is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='8 M⊙ yr−1, approximately equal to the star formation rate in this galaxy (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='6 M⊙ yr−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We show that the accretion profile, which peaks at a radius of about 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 kpc, predicts a disc growth rate compatible with the observed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Our results indicate that fountain-driven corona condensation is a likely mechanism to sustain star formation as well as the disc inside-out growth in local disc galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Key words: galaxies: haloes – galaxies: ISM – galaxies: evolution – galaxies: intergalactic medium – ISM: structure – ISM: kinematics and dynamics 1 INTRODUCTION Nearby spiral galaxies have been forming stars, across their lifetimes, at an approximately constant or gently declining rate, despite the fact that the gas in their interstellar medium (ISM) would, without re- plenishment, be consumed in a few Gyr (Aumer & Binney 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Tacconi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' An external gas reservoir is therefore needed from which galaxies accrete gas at a rate compatible with their SFR (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Fraternali & Tomassetti 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Gas-rich mergers are not pro- viding a sufficient contribution, at least in the local Universe (Sancisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Di Teodoro & Fraternali 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Therefore the majority of the accretion must come from the diffuse gas that resides outside galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The multi-phase circumgalactic medium (CGM) is expected to host a significant fraction of the baryons associated with dark matter halos in normal spiral galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Crain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Tumlinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2018), which makes it the most probable gas reservoir eligible for accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' A prominent component of the CGM is hot gas (𝑇 ∼ 106−7 K) in the form of a diffuse ‘corona’ at nearly the virial temperature and in nearly hydrostatic equilibrium with ★ E-mail: li@astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='rug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='nl the dark matter potential (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' White & Frenk 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Pezzulli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Galactic coronae are thought to surround galaxies and to be extended to their virial radii (Fukugita & Peebles 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Faerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Direct detection of the hot coronae in X-rays is limited to the innermost few tens of kpc in massive galaxies with stellar mass beyond 1011 M⊙ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Anderson & Bregman 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Anderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2016), while indirect evidence of their presence extends further (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Gatto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Putman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Cool CGM (𝑇 ∼ 104 K) gas has also been detected, mostly in absorption along quasar sightlines, in several studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Heckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Rubin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Zahedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Like the hot corona, also these cool absorbers extend to large distances (up to and sometimes beyond the virial radius) and their origin and fate remain debated (Rubin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Schroetter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Pointon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Afruni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Although gas accretion from the CGM is crucial to feed star for- mation (Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Sancisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Kereš et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2009), how precisely it takes place is still unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' One possible accretion scenario is that cold filaments reach the outer disc (Lagos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' El-Badry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Trapp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2022) and are transported into the inner star-forming regions via radial motions, although Di Teodoro & Peek (2021) found that radial inflows in nearby galaxies alone could not sustain the star formation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Other possible mechanisms in- © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='03614v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='GA] 9 Jan 2023 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' clude cold gas filaments directly feeding the inner regions of a galaxy or the cooling of the hot corona (Kereš et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Voit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The spontaneous cooling of the corona via thermal instability is still under debate as a number of works suggest that the combination of buoyancy and thermal conduction can suppress the growth of thermal perturbations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Binney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Nipoti 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Joung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Some authors have proposed that coronal condensation could be triggered by the ejection of gas from the disc due to stellar feedback, such as in supernova-powered superbubbles (Fraternali 2017, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In this scenario, the cooling of the hot gas is due to the mixing with the cool gas ejected from the disc and occurs within the fountain cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This process can be detected in high-quality data as it leaves a mark in the kinematics of the ejected disc gas (Fraternali & Binney 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' To gain insight into the gas exchange processes between the disc and the inner hot CGM, one must focus on the disc-halo interface region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Deep H i observations have shown that disc galaxies, includ- ing the Milky Way, are surrounded by a neutral gas layer extending up to a few kpcs from their disc planes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Wakker 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' San- cisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Hess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Marasco & Fraternali 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This gas layer, known as extraplanar gas (EPG), is nearly ubiquitous in late-type galaxies and has a mass of 10–30 per cent of the mass of the H i in the disc (Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The kinematics of the EPG is primarily characterised by differential rotation, similar to the disc, but with a negative rotational gradient (lag) ranging from −10 to −20 km s−1 kpc−1 in the vertical direction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Oosterloo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Zschaechner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Non-circular motions, especially large-scale inflows are also often found (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Fraternali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Barbieri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Ionised EPG has also been detected, both in the Milky Way (Dettmar 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Lehner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2012, 2022) and in several other galaxies (Heald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Levy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2019), with similar kinematics as the neutral EPG (Kamphuis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The similarity between EPG and disc kinematics strongly suggests that EPG originates mostly from the disc, very likely pushed out of the plane due to stellar feedback and pulled back by gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This phenomenon is also known as ‘galactic fountain’ (Shapiro & Field 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Bregman 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Fraternali & Binney (2006, hereafter FB06) built ballistic models of galactic fountain flows, which successfully reproduced many of the observed properties of the EPG in the two nearby galaxies NGC 891 and NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' It is worth noticing that ballistic models also describe very well the properties of the warm gas (neutral and ionised) in the hydrodynamical TIGRESS simula- tions (Vijayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' However, a pure fountain model failed to reproduce the net inward flow (instead, an outward flow was pre- dicted) and underestimated the rotation lag compared to the observed EPG in NGC 891 and NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Fraternali & Binney (2008, here- after FB08) mitigated these issues by introducing an external factor that could lower the angular momentum of fountain gas: accretion from the ambient gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Although initially introduced to reproduce the kinematics of the EPG, the net inflow rate derived from this model turned out to be consistent with the SFR of the two galaxies, sug- gesting that the accretion triggered by the fountain cycle could be a viable mechanism to maintain the star formation activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' An unsolved issue of the above fountain-driven accretion scenario was the source of the accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This has been explored by Marinacci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2010) with hydrodynamical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Their simulations of fountain gas clouds interacting with the hot corona indicated that the corona was a possible accretion source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' During the interaction process, part of the fountain gas is stripped and mixed with the hot gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The mixture has a typical temperature of 𝑇 ∼ 105 K, where the cooling function peaks, and also higher metallicity and density than the hot corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' As a consequence, the cooling time is reduced to a value shorter than the travel time of fountain gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This result has been confirmed by other simulations with increasing levels of complexity (Armillotta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Gronke & Oh 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Kooij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Some studies have upgraded the approach of FB08, taking into account the results of hydrodynamical simulations, using physical properties of the EPG and the hot corona as adjustable parameters, and man- aged to reproduce the phase-space distribution of both neutral and ionised EPG in the disc–halo interface of Milky Way remarkably well (Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Fraternali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2012, hereafter M12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The best-fitting model predicted a net inflow rate which is consistent with the SFR of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The aforementioned studies strongly suggest that fountain-driven accretion takes place in the Milky Way and provides a promising explanation for how galaxies like our own can sustain their star for- mation with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' However, so far the Milky Way remains the only galaxy for which a state-of-the-art model of the galactic fountain has been applied to the observations using a parametric fitting method- ology, which is required to robustly characterise the fountain flow and to quantify the properties of the accreting gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The earlier mod- els in FB08 did not statistically explore the parameter space, and furthermore, did not include the condensation of the corona, since hydrodynamical simulations were not available by then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In this pa- per, we revisit this by applying our state-of-the-art fountain model to NGC 2403, using high-quality H i data (with a beam size of 30′′× 29′′and an rms-noise of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='19 mJy beam−1) from Fraternali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2002), which were later included in the HALOGAS survey (Heald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Table 1 summarises the main physical properties of NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In Section 2 we provide a description of our dynamical model of the galactic fountain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In Section 3 we discuss the customisation we have made to implement the model for the case of NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In Section 4 we present the modelling results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In Section 5 we discuss the reliability of our results and possible implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We summarise our analysis in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2 THE MODEL In this Section, we describe the main ingredients of our model and discuss its main free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Further details can be found in FB06, FB08 and M12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We consider two different types of models: a ‘pure fountain’ ballistic model and a ‘fountain + corona accretion’ model which takes the interaction of fountain clouds with the hot coronal gas into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In both scenarios, the models have a quasi-stationary state and are axisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The neutral EPG in the disc–halo interface region is modelled as a collection of clouds that are ejected from the disc at different radii with a given distribution of initial velocities and angles, and whose orbits are then integrated in time and followed across the halo region until they return to the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Since galactic fountains are powered by stellar feedback, we as- sume that the amount of gas ejected from each location in the disc is proportional to the SFR surface density at that radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In practice, we incorporate this assumption by assigning, to each of our mod- elled clouds, a weight proportional to the SFR surface density at the ejection radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This weight is then factored in when creating the mock datacube to be compared with observations (see also further explanations below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In our pure fountain ballistic models, the trajectories of the fountain clouds are integrated using a numerical approximation of the galaxy gravitational potential, derived as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' For foun- MNRAS 000, 1–13 (2023) Fountain-driven corona accretion in NGC 2403 3 Galaxy Name RA DEC PA INCL Distance Hubble Type MB M∗ MHI,EPG SFR [◦] [◦] [Mpc] [108 M⊙] [108 M⊙] [M⊙ yr−1] (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) NGC 2403 07h36m51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='s4 +65◦36′09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='′′2 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='6 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 SAcd −19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='68 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='6 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Galaxy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Columns: (1) Galaxy name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2)–(3): Coordinates (J2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (4)–(5): Position-angle and inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (6) Distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (7) Hubble type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (8) Absolute magnitude in the 𝐵-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (9) Stellar mass (see Pezzulli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (10) Total mass of H i extraplanar gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (11) Total star formation rate of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Values in this table are taken from Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2019) unless otherwise mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' tain + corona accretion models, hydrodynamical forces due to the interaction between the clouds and the hot corona are parameterised in simple forms described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The positions and velocities of the clouds along their orbits are recorded at each time-step (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='3 Myr), projected along the line-of- sight of the observer, weighted by the local SFR surface density at the ejection radius and transferred into a synthetic datacube, which is then adapted to a specific galaxy (NGC 2403 in our case) by assuming a distance, inclination (INCL), and position angle (PA), and using the same observational setup (beam shape, spectral resolution, pixel size, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=') of the data under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The outcome of the dynamical model is therefore a synthetic datacube which can be directly compared with the observational H i data of our target galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Construction of the model involves several parameters but we will focus preferentially on three (for pure fountain models) or four (only for fountain + corona accretion models) that regulate the initial outflow speed of the clouds, their neutral gas fraction, the EPG total mass and, for models that include interaction with the corona, an additional parameter that regulates the condensation efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Below we discuss these parameters in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Other ingredients are fixed by the observations, in particular the galaxy potential (which affects the trajectory of the cloud) and the SFR surface density profile (which regulates the ejection rate), as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1 Outflow velocity Fountain clouds are initially located within the galaxy disc and rotate at the circular speed set by our gravitational potential1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Each cloud receives a ‘kick’ with a velocity 𝑣k at certain angles 𝜃, which is defined as the angle between the velocity vector and the direction normal to the disc plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The probability distribution of the ejection as a function of 𝑣k and 𝜃 (assuming a uniform probability in the azimuthal direction) follows FB06 and is given by P(𝑣k, 𝜃) ∝ exp � − 𝑣2 𝑘 2ℎ2𝑣 cos2Γ 𝜃 � , (1) where ℎ𝑣 is the characteristic velocity, and Γ determines the level of collimation of the ejected clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Larger values of ℎ𝑣 increase the probability that a cloud is kicked at high speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The larger Γ, the more collimated the ejection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' FB06 have tested models with different values for Γ and found that more collimated ejections agree better with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We have therefore fixed Γ = 10 (highly collimated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The outflow velocity of a cloud affects the maximum height and the 1 They also feature an additional velocity component, with an amplitude randomly extracted from a Gaussian distribution with rms of 8 km s−1 and a random (isotropic) direction, to simulate the typical velocity dispersion of the neutral ISM (Iorio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Bacchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Mancera Piña et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' trajectory of the orbit and therefore influences the final model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We, therefore, let the characteristic velocity ℎ𝑣 be a free parameter with a flat prior in the range 40–100 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This range covers the typical characteristic ejection speeds of the warm gas in high-resolution hydrodynamical simulations of galactic fountains (Kim & Ostriker 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' It also agrees with theoretical estimates of the typical blow-out speed of individual superbubbles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Mac Low & McCray 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Keller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 Phase change Previous studies have found that the neutral EPG in some spiral galaxies (including the Milky Way) shows a tentative preference for vertical inflow (Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' French et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2021, for example), which can be interpreted as due to a change of phase during the fountain cloud orbit: gas is largely ionised when ejected from the star-forming region of the disc but later recombines and becomes visible in H i at some point during its trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' To account for this effect in our model, we assume that a cloud is only visible in the H i phase when 𝑣𝑧(𝑡) < 𝑣𝑧,0(1 − 𝑓ion), (2) where 𝑣𝑧 is the vertical velocity (that is, in the direction perpendicular to the disc) of the cloud, 𝑣𝑧,0 is the vertical component of the initial outflow velocity and 𝑓ion is the ionisation fraction parameter, which we set as a free parameter with a flat prior and varies from zero to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' When 𝑓ion equals zero, the cloud is visible in the whole orbit, while when 𝑓ion equals one, the cloud is only visible when 𝑣𝑧 < 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', the descending stage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='3 Interaction with the corona In our model, the hot corona is modelled as a smooth, volume-filling gas layer that rotates at a lower speed than the disc, which is justified on both observational (Hodges-Kluck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2016) and theoretical (Pezzulli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2017) grounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We assume that the corona maintains a temperature of ∼ 106 K, which implicitly implies some heating by either supernova feedback (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Stinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2013) or active galac- tic nucleus feedback (for galaxies with ongoing AGN activities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Ciotti & Ostriker 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The condensation and accretion of the hot corona is triggered by the cool (𝑇 ∼ 104 K) fountain clouds ejected from the disc, which mix efficiently with the former and produce a mixture at 𝑇 ∼ 105 K, dramatically reducing the cooling time of the hot corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The above processes have been investigated in the hydro- dynamical simulations of cloud–corona interactions (Marinacci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' A follow-up analysis (Marinacci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2011) indicate that there is a net transfer of momentum from the fountain to the corona until the relative velocity between these two, 𝑣rel, reaches a certain threshold 𝑣thres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Marinacci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2011) suggested 𝑣thres ≈ 75 km s−1 for initial conditions valid for the Milky Way but pointed out that 𝑣thres can MNRAS 000, 1–13 (2023) 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' vary in the range 45–105 km s−1 (see also Fraternali 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' As soon as 𝑣rel becomes smaller than this threshold 𝑣thres, the net momentum transfer ceases as the condensation of corona recaptures angular mo- mentum lost by fountain gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' For this reason, we set the azimuthal speed of the corona to be always lower than the local circular speed 𝑣c by 𝑣thres, and in this case, 𝑣c − 75 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1 we ex- plore models with different value of 𝑣thres, corresponding to different rotational speeds for the coronal gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In the above scenario,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' the cloud acceleration due to interaction with the corona is defined as �𝒗 = � − C𝜌hot 𝜎cloud(𝑣rel−𝑣thres) 𝑀cloud 𝒗rel − 𝛼𝒗rel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 𝑣rel ≥ 𝑣thres −𝛼𝒗rel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 𝑣rel < 𝑣thres,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (3) where 𝒗rel is the cloud-corona relative velocity vector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 𝑣rel is the modulus of 𝒗rel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 𝑀cloud and 𝜎cloud are the mass and the cross-section of the cloud (defined as 𝜋𝑅2 cloud,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' with 𝑅cloud the radius of the cloud),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 𝜌hot is the density of the corona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' C is a dimensionless constant of order unity (in our model C=1) to account for the geometry of the cloud,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' and 𝛼 is the condensation rate of the coronal gas onto the cloud,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' such that the mass of the cloud 𝑀cloud grows with time as �𝑀cloud = 𝛼𝑀cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We assume a corona density of 10−3 cm−3, a cloud radius of 100 pc and an initial mass of 2× 104 M⊙, consistent with typical values of fountain clouds suggested by observations (Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The first term on the right-hand side of equation 3 represents the drag experienced by the fountain cloud as it moves through the coro- nal gas: the cloud speed decreases as long as its velocity stays above 𝑣thres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The second term is due to the condensation of coronal gas onto the cloud: as the total mass of the cloud increases, conservation of the total momentum implies lower velocity (see Fraternali & Binney 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We have also derived the drag timescale 𝑡drag = 724 Myr us- ing equation(6) in Fraternali (2017), which is larger than the fountain orbit time (∼100 Myr), we therefore expect that drag only has a minor effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In fountain + corona accretion models, we let 𝛼 be a free parameter with a flat prior in the range 𝛼 = 0–6 Gyr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='4 EPG mass The normalisation of the H i flux presented in the final galactic foun- tain model sets the total H i EPG mass, which is another free param- eter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We use a fiducial EPG mass of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='9 × 108 M⊙ from Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2019) as an initial guess, but allow the EPG mass to vary, multiplying the fiducial value by a normalisation scaling factor in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 3 IMPLEMENTATION OF THE MODEL In this section, we describe the gravitational potential and the SFR surface density radial profile for NGC 2403, as they are necessary ingredients to construct our dynamical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We then describe how we fit the model parameters to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1 The gravitational potential We use the gravitational potential grid derived by FB06 for NGC 2403 without modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Below we briefly describe how the potential model is built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The gravitational potential was derived from an axisymmetric mass model, which consists of three components: a stellar disc, a (𝑀/𝐿)∗ 𝑅∗ ℎ∗ 𝑅gas ℎgas 𝜌0,DM r𝑠 [kpc] [kpc] [kpc] [kpc] [M⊙ kpc−3] [kpc] (1) (2) (3) (4) (5) (6) (7) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1 × 107 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Mass models for NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Columns: (1) Mass-to-light ratio in the 𝐵-band of the stellar disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2)–(3): Scale length and scale height of the stellar disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (4)–(5): Scale length and scale height of the gaseous disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (6)–(7) Central density and scale radius of the NFW dark matter halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' gaseous disc, and an NFW dark matter halo (Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' FB06 performed a mass decomposition of the H i rotation curve of NGC 2403 (Fraternali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2002) using the three components men- tioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The stellar and the gaseous discs’ density distributions were given by exponential profiles, along both the radial (𝑅) and the vertical (𝑧) direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The scale length of the stellar (gaseous) disc 𝑅∗ (𝑅gas) was derived by fitting an exponential profile to the stellar (gaseous) surface brightness radial profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The scale height of the stellar disc was set to one-fifth of its scale length (see van der Kruit & Freeman 2011 and references therein), and the scale height of the gaseous disc was set to 100 pc (typical of the inner gaseous disc, see Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Bacchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Mancera Piña et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The mass-to-light ratio of the stellar disc was derived via the rota- tion curve decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The above parameters of the mass model are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Once the parameters of all components are decided, the galactic potential and forces are calculated numerically in the (𝑅, 𝑧) cylindrical coordinate system, using a grid with a cell size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1 kpc within 𝑅 < 25 kpc and 𝑧 < 5 kpc, and of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 kpc for 25 < 𝑅 < 100 kpc and 5 < 𝑧 < 100 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Potential and forces are determined at any (𝑅,𝑧) via a bilinear interpolation of these grids (see FB06 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 Star-formation-rate surface-density profiles In this paper, we directly use the SFR surface density radial profiles from previous observations, as opposed to FB06, which used the Schmidt–Kennicutt law (Kennicutt 1989), and M12, which used an- other empirical star formation law (directly derived from 17 galaxies with known gas and SFR surface densities) to estimate the SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The SFR surface-density profile of NGC 2403 is mainly taken from Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2008), which derived the SFR using a combination of far ultraviolet (FUV) and 24 𝜇m data, and is then complemented with the SFR surface density profile from Bigiel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2010), which is derived from FUV data with a lower resolution but larger radial extent compared to Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We refer the readers to Bac- chini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2019, 2020) for more details about collecting SFR data of NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 1 shows the SFR surface-density data and the interpolated profile (in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 kpc) which we used as an input for our fountain models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='3 Separation of the EPG emission Before modelling the EPG in the NGC 2403 datacube, we first need to isolate its emission from the underlying disc and from external regions (foreground and background emission) that are clearly not associated with the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' For this purpose, we follow the procedure described in Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The emission from regions external to the galaxy is filtered out by spatially smoothing the datacube by a 2D Gaussian kernel with a full width half maximum (FWHM) of 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='′′5 × 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='′′6, which is five times MNRAS 000, 1–13 (2023) Fountain-driven corona accretion in NGC 2403 5 0 2 4 6 8 10 12 14 R [kpc] 10 −4 10 −3 10 −2 Σ SFR [Msun kpc −2 yr −1 ] Leroy08 Bigiel10 interpolated SFR Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Star formation rate surface density versus galactocentric distance in NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Blue dots represent data from Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2008) while orange points are from Bigiel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The green curve shows the interpolated profiles with steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 kpc and is used as an input for our fountain model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' larger than the spatial resolution of the data, calculating a smoothed rms noise level, and then sigma-clipping at S/N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This produces a mask that is applied to the original (not smoothed) data to exclude the regions external to the main galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In intermediate-inclination galaxies like NGC 2403, the emission from the EPG overlaps spatially with that from the regularly rotating disc but can be (at least in part) separated from the latter in the velocity space, provided that the velocity resolution is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Here, we employ the disc–EPG separation method introduced by Fraternali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2002), which works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' For any given H i velocity profile at a certain location in the sky, the disc component is assumed to be described by a Gaussian profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The EPG adds a wing to the profile, which is typically due to the lagging of EPG and located toward the systemic-velocity side;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' although wings on both sides can be seen at some spatial locations across the disc due to other non-circular (mostly vertical) motions (see also Boomsma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Despite the disc and EPG profiles are blended together, it is reasonable to neglect the contribution of the EPG around the peak of each velocity profile since EPG mass is only a small percentage (∼ 20 per cent for NGC 2403, Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2019) of the total H i mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We therefore use the ‘peak’ region to fit the disc emission by performing a Gaussian fit using only the upper 40 per cent (in intensity) of the line profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This Gaussian profile is considered to be the contribution of emission from the disc component alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Pixels with disc emission (estimated from the Gaussian profile) larger than 𝑁 times the rms noise are clipped (see Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2019 and Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2021 for a more detailed explanation of this methodology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The scaling factor 𝑁 is decided empirically as a compromise between keeping enough EPG emission for the modelling and alleviating the disc contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We set 𝑁 = 2 for NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Some peculiar features in NGC 2403, in particular, a long filament of unknown origin (see also de Blok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2014) have also been manually filtered out (see blank regions in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We discuss this further in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' After passing through the above mask, only EPG emission and noise remain in the datacube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We then implement sigma-clipping at S/N = 2 to mask the random noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' For consistency, the same mask has also been applied to the model datacube that we describe below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='4 Model construction and evaluation Our EPG models have three or four free parameters: the characteristic outflow velocity ℎ𝑣, the ionisation fraction 𝑓ion, the condensation rate 𝛼 (for fountain + corona accretion models), and the EPG mass MEPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We build three(four)-dimensional grids for pure fountain (fountain + corona accretion) models with ℎ𝑣 varying from 40 to 100 km s−1 in steps of 10 km s−1, 𝑓ion varying from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0 in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2, 𝛼 varying from 0 to 6 Gyr−1 in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='6 Gyr−1, and scaling factor of the initial EPG mass varying from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1 to 10 in steps of factor of 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The ranges and steps of the free parameters are summarised in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The best-fitting parameters are estimated by a Bayesian approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' For each cell in our 3D (4D) parameter grid, we compute the posterior probability of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' For a chosen parameter vector x and given our data D, the posterior probability P is given by P(x|D) ∝ P(D|x)P(x), (4) where P(D|x) is the likelihood function and P(x) is the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The prior for each parameter is uniform within the parameter space (uniform in the logarithmic scale for the normalisation parameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The likelihood function is given by P(D|x) ∝ � 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='𝑣𝑜𝑥𝑒𝑙𝑠 exp � − |M(x) − D| 𝜀 � = exp � − ∑︁ 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='𝑣𝑜𝑥𝑒𝑙𝑠 |M(x) − D| 𝜀 � = exp[−R(x)/𝜀], (5) where M represents the model datacube built from parameter vector x, 𝜀 is the uncertainty of the data, and R is the sum of the absolute residuals between the data and the model, which is defined as the sum of absolute difference in each pixel: Res = � |data − model|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Note that both the model and the data have been masked using the method described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='e, the voxels where EPG emission is detected at more than 2𝜎 are considered in the determination of the residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In equation 5, 𝜀 regulates how rapidly the likelihood drops when our model deviates from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Assuming 𝜀 equal to the rms-noise of the data is a poor choice, which leads to very nar- row posterior probability distributions and severely underestimates the uncertainties in our model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This occurs because our model is smooth and axisymmetric, and cannot possibly capture the complexity of the data down to the noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Numerical solu- tions to this problem can be worked out (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 in Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2019), but in this work, we prefer to set 𝜀 a posteriori, in a way that the 2-𝜎 uncertainty on the derived parameters corresponds to models that look very different from the data by visual inspec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In the end, we assume 𝜀 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='38 Jy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We marginalise the multi-dimensional posterior distribution to determine the probability distribution of individual parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Best-fitting values are defined as the median of these marginalised posterior distributions, and the uncertainties are taken as half the difference between the 84th and 16th percentiles of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 4 RESULTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1 Residuals and position-velocity diagrams In this Section, we show the best-fitting results of the pure fountain and the fountain + corona accretion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The 2D marginalised posterior probability distributions are shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The MNRAS 000, 1–13 (2023) 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Parameter description range step units ℎ𝑣 Characteristic outflow velocity (equation 1) [40,100] 10 km s−1 𝑓ion Ionisation fraction during the ascending part of the orbits(equation 2) [0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 𝛼 condensation rate of coronal gas (equation 3) [0,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='6 Gyr−1 Norm EPG mass scaling factor 𝑎 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1,10] 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Free parameters of our galactic fountain model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The third column lists the range explored in our residual calculations, using a grid size given by the forth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 𝑎 a value of 1 corresponds to the EPG mass determined by Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2019) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='9 × 108 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' best-fitting values and uncertainties, obtained with the method de- scribed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='4, are listed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The position–velocity (pv) slices of the best-fitting models are compared with the data in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In general, both the pure fountain and fountain + corona accretion models recover the EPG emission, but we find that the former reproduces the data poorly for pv slices parallel to the minor axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Instead, the fountain + corona accretion model performs better in the same locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This is better shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 4 where we compare the two models for a pv slice parallel to the minor axis with an offset 4′ from the centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The best- fitting pure fountain model fails to reproduce the emission marked out by the red arrow and predicts extra emission in the blank region marked out by the black arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Instead, the best-fitting fountain + corona accretion model generates the same asymmetry shown by the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Previous studies (Fraternali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2019) have shown that this asymmetric feature can be produced by radial inflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In a fountain model, EPG emission shows outward radial flows, but accretion from low-angular momentum material can invert this trend and produce an inward flow (especially evident for clouds ejected from the outer regions of the disc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Fraternali 2017), which is required to best reproduce the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The above visual comparison prefers the fountain + corona ac- cretion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This result has been already inferred by FB08, but we now have its statistical confirmation using the likelihood values derived by equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We find − ln [P(D|x)] = 232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='6 for the best- fitting pure fountain model, while − ln [P(D|x)] = 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 for the best-fitting fountain + corona accretion model, as shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We use the Bayesian information criterion (BIC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Schwarz 1978) to infer which of the two different scenarios (pure fountain or fountain + corona accretion) is statistically preferred by the data, given that they make use of a different number of free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The BIC is derived as BIC = −2 ln L + 𝑘 ln N, (6) where L is the likelihood of the model (equation 5), 𝑘 is the number of parameters estimated by the model, and N is the number of inde- pendent data points used in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' When comparing similar models with different numbers of free parameters, a model with a lower BIC is to be preferred, as the BIC penalises extra parameters that do not significantly lower the likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The BIC for the pure fountain model is 490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='6 while for the accretion model is 482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='9, indicating that the fountain + corona accretion model is statistically preferred by BIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The above results show that the H i EPG of NGC 2403 is consti- tuted by a combination of material ejected from the disc by stellar feedback and gas cooling from the inner hot CGM and accreting onto the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This is also consistent with previous indication from kinematic modelling of the EPG which shows radial and vertical inflow (Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The best-fitting fountain + corona ac- cretion model requires an outflow with a characteristic velocity of 50 ± 10 km s−1, starting out mostly ionised and becoming neutral when the vertical velocity has been reduced by around 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The inferred H i total mass of the EPG (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='7+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='9 × 108 M⊙) is similar to that derived in Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2019) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='9 × 108 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The accretion rate given by our best-fitting model (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='8+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 M⊙ yr−1) is compati- ble with the star formation rate of NGC 2403 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='6 M⊙ yr−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Heald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2012)2, indicating that the mechanism of fountain-driven gas accretion can sustain the ongoing star formation in NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' It is noteworthy that the values of both outflow speed and accretion rate found with our statistical analysis are in agreement with those found by FB08 by trial and error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The present analysis, however, allows us to further our understanding of fountain-driven accretion in NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 Properties of the extraplanar gas layer in NGC 2403 This is the first time that a dynamical fountain model including corona condensation has been applied to an external galaxy with a statistical fitting method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The best-fitting fountain + corona accretion model reproduces most of the EPG features in NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Assuming our model is reliable and correct (see discussion in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1), we can therefore extract physical properties of the EPG layer, as well as a predicted gas accretion profile, from the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1 Thickness of the neutral extraplanar gas layer We determine the thickness of the EPG layer in our best-fitting model by fitting the vertical density profiles at different radii with exponen- tial functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 5 shows the scale height of the EPG in our best- fitting fountain + corona accretion model as a function of radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The scale height is calculated only out to 𝑅 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 kpc, as fountain clouds beyond this radius are too rare to provide a reliable vertical profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Overall, the thickness of the gas layer increases slightly with radius, which is what we would expect given that the gravitational potential is shallower in the outer parts of the galaxy (we have assumed that ℎ𝑣 is constant with radius for simplicity, see also Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This makes the orbits more extended in the outer region than in the inner region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The flux-weighted average scale height of our EPG model is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='93±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='003 kpc, compatible with the scale height derived in the kinematic model in Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Thus, the EPG layer of NGC 2403 is significantly thicker than its H i disc, which has scale height comprised between 100 and 600 pc (Mancera Piña et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 EPG rotational lag Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 6 shows the rotation curves of the EPG layer at different heights above the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' These curves are derived from our best-fitting fountain + corona accretion model by taking the flux-weighted mean value 2 This estimate has an uncertainty of around ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='3 dex or better, based on the algorithm Heald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2012) used to derive the SFR (Kennicutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023) Fountain-driven corona accretion in NGC 2403 7 Model 𝑣thres ℎ𝑣 𝑓ion 𝛼 �𝑚 MEPG − ln L BIC [km s−1] [km s−1] [Gyr−1] [M⊙ yr−1] [108 M⊙] (1) (2) (3) (4) (5) (6) (7) (8) (9) pure fountain 𝑁 /𝐴 50 ± 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 𝑁 /𝐴 𝑁 /𝐴 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='9+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='6 490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='6 fountain + corona accretion 75 50 ± 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='4+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='8+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='7+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='9 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='9 fountain + corona accretion 45 50 ± 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='7+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='9 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='9 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The best-fitting values and uncertainties (obtained with the method described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='4) for our fountain (+ corona accretion) models of the EPG of NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We focus on the first two models in this Section and further discuss the third model in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (1) Model type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2) The velocity threshold for fountain + corona accretion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The net transfer of momentum from the fountain to the corona ceases when the relative velocity between these two decreases below this threshold (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (3) Characteristic outflow velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (4) Ionisation fraction of the fountain gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (5) Condensation rate of the hot gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (6) Global accretion rate of the condensed hot gas onto the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Note that this is not a free parameter but a value derived from the best-fitting model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (7) H i EPG mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (8) Logarithm of the likelihood values P(D |x) of the best-fitting models, calculated in equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (9) The BIC values of the best-fitting models, calculated from equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 10 0 10 100 0 100 VHEL-VSYS [km/s] 4′ 10 0 10 2′ 10 0 10 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='=124 pure fountain 0′ 10 0 10 2′ 10 0 10 4′ 10 0 10 100 0 100 VHEL-VSYS [km/s] 4′ 10 0 10 2′ 10 0 10 offset [′] 0′ 10 0 10 2′ 10 0 10 4′ Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Position–velocity (pv) slices from the data (shown in black contours and blue colour scale) and from the best-fitting pure fountain model (red contours);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' from outer to inner regions, contour levels are (2, 4, 8, 16)-𝜎, respectively, and a negative contour -2𝜎 is shown as the dashed grey contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='. The (irregular) blank region represents the disc mask and the square blank region represents the manual mask that filters out the irregular filament in NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Top panels are pv slices parallel to the major axis with offsets −4′, −2′, 0′, 2′, 4′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Bottom panels are pv slices parallel to the minor axis with offsets −4′, −2′, 0′, 2′, 4′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 10 0 10 100 0 100 VHEL-VSYS [km/s] 4′ 10 0 10 2′ 10 0 10 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='=124 fountain + corona accretion 0′ 10 0 10 2′ 10 0 10 4′ 10 0 10 100 0 100 VHEL-VSYS [km/s] 4′ 10 0 10 2′ 10 0 10 offset [′] 0′ 10 0 10 2′ 10 0 10 4′ Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2, but for the best-fitting fountain + corona accretion model of NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023) 8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' As in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2 and 3, but focusing on the pv slice parallel to the minor axis with offset 4′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Left: best-fitting pure fountain model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Right: best-fitting fountain + corona accretion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The red arrows mark regions where EPG emission is present in the data and in the fountain + corona accretion model, but not in the pure fountain model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The black arrows mark out the region where the pure fountain model predicts extra emission with respect to the data, while the fountain + corona accretion model correctly predicts a lack of emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 0 2 4 6 8 10 12 R [kpc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='3 scale height [kpc] Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The scale height of the EPG layer predicted by our best-fitting fountain + corona accretion model for NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' of the azimuthal velocities of the particles in a given bin of radius and height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We find that the rotation velocity of the EPG decreases with height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' At 𝑅 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 kpc (the half-mass radius of the EPG in NGC 2403), the velocity gradient is around −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='7 km s−1 kpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This gradient is consistent with the velocity gradient of −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 km s−1 kpc−1 inferred by Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2019), who modelled the EPG of NGC 2403 with simplified geometric and kinematic assumptions, and therefore intrinsically differs from our dynamical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Our results are also comparable with the velocity gradient −15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 km s−1 kpc−1 directly measured in the edge-on galaxy NGC 891.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='3 Gas flows and accretion in NGC 2403 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 7 shows the inflow and outflow rates as a function of radius predicted by our best-fitting fountain + corona accretion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Rotational velocities for the EPG layer at different heights from the plane (solid/dashed/dotted lines), compared to the disc rotation curve (black squares with error bars) given by Fraternali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Velocities are derived from our best-fitting fountain + corona accretion model by taking the flux-weighted average of azimuthal velocity 𝑣𝜙 at given (𝑅, 𝑧) locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' shape of the outflow rate profile strictly follows that of the SFR profile shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This is true by construction, as explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The mass loading factor (defined as the ratio of the mass outflow rate to the SFR and therefore is proportional to the normalisation factor free parameter in our model) is however a prediction of our model, and we find a value of around 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The inflow rate at a given radius is given by the combination of fountain clouds and accreted coronal particles that fall onto the disc per unit time and area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Since fountain clouds do not fall back onto the disc at the same radius as they are ejected and collect additional gas condensed from the corona as they fall, the inflow rates do not precisely follow the outflow-rate trend but show a somewhat smoother distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We also present the net flow rate (where inflow is defined as positive value) as a function of radius in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 7 top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=" The first MNRAS 000, 1–13 (2023) 150 150 pure fountain fountain + corona accretion 100 100 VHEL-Vsys [km/s] 50 50 0 0 50 50 100 100 0 150 150 10 5 0 5 10 10 5 0 5 10 offset' offsetr!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='140 120 100 Vμ[kms-1] 80 60 disc 40 1 kpc 2 kpc 3 kpc 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0 R [kpc]Fountain-driven corona accretion in NGC 2403 9 evident feature is that the net flow is much lower than both outflow and inflow across the disc, except for the very outer parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Also, except for some fluctuation in the innermost region (within 𝑅 = 4 kpc), the overall tendency is net inflow in the inner region (𝑅 < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 kpc, the vertical dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 7 top panel) and net outflow in the outer region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The net inflow is mostly due to condensation of the hot corona, while the net outflow in the outer region can be explained by the fact that the interaction between fountain gas and the corona results in inward orbits for the former: cloud particles are more likely to fall back to the plane at a radius smaller than their ejected radius (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 8 in Fraternali 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' As we discussed in Section 1, accretion of the CGM onto the disc is crucial for feeding star formation and is also a key process in the evolution of a galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The details of this process are however not well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Now with our best-fitting fountain + corona accretion model, we can predict the accretion rate as a function of radius, shown in the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Despite star formation being the origin of the fountain cycle, the fountain-driven accretion rate does not follow the profile of the SFR surface density (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 1) and in particular, it is more skewed towards larger radii compared with the SFR surface density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This is due to a number of effects, the most important of which is a radially increasing orbital time, which is in turn a consequence of a varying gravitational potential with radius, as also discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' A longer orbital time causes an increase in the total condensation along a given orbit, even with a fixed accretion efficiency per unit time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 𝛼), as assumed in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The accretion profile has a well-defined peak at intermediate radii and its exact position is determined by an interplay between a radially declining SFR surface density and a radially increasing duration of the orbits (see also M12 for the Milky Way).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The gas accretion rate that comes from corona condensation is at every radius a minor fraction of the overall gas inflow (∼ 10%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Compared to the total accretion rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='8 M⊙ yr−1, the total inflow and outflow rates are 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='48 M⊙ yr−1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='69 M⊙ yr−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Most of the gas inflow occurs as a consequence of the return to the disc of the gas ejected by the fountain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' However, the fountain cycle by itself does not add any new gas to the disc and would not help to sustain the star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Instead, our model predicts that the fountain flow "captures" new gas from the corona that is then added everywhere across the disc to sustain the local star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Remarkably, the accretion rate that is needed to reproduce the seemingly independent kinematics of the EPG in NGC 2403 turns out to be very similar to the one needed to sustain its star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Overall, the accretion rate peaks at around 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 kpc and the cumu- lative accretion rate reaches 50 per cent of the total accretion rate at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='25 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' As we mentioned, this distribution is shifted outwards with respect to the SFR surface density distribution, which peaks in the centre of NGC 2403 and reaches 50 per cent of the total SFR at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='3 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The relevance of this difference is further discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 5 DISCUSSION 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1 Reliability of the fountain + corona accretion model In this paper, we have investigated gas accretion as the potential mech- anism to maintain star formation in NGC 2403 and found a remark- able consistency between the accretion rate predicted by our model and the SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' However, accretion is not the only fuelling mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Several studies have pointed out the importance of stellar mass loss in extending gas consumption timescales (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Sandage 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Ken- nicutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 1994) and sustaining star formation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Schaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 0 2 4 6 8 10 12 14 R[kpc] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='14 Mass flow rate [M ⊙ yr −1 kpc −2 ] inflow outflow net flow 0 2 4 6 8 10 12 14 R[kpc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0 Coronal gas inflow rate [10 −3 M ⊙ yr −1 kpc −2 ] Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Inflow and outflow rate surface density as a function of radius predicted by our best-fitting fountain + corona accretion model of NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Top panel: inflow rates (blue bars), outflow rates (black bars), and net flow rates (red bars: inflow−outflow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' positive values indicate net inflow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The vertical dashed line at 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 kpc marks the boundary where the net flow changes from inflow to outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Bottom panel: inflow rate surface density contributed by corona accretion, the integration of which gives us the global accretion rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='8 M⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Leitner & Kravtsov 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In particular, Leitner & Kravtsov (2011, hereafter LK11) has estimated the current stellar mass loss rate of NGC 2403 to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='79 M⊙ yr−1 (depending on the un- derlying initial mass function), which seems to eliminate the need of gas accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' However, this mass loss rate was calculated in LK11 assuming a SFR of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='3 M⊙ yr−1, implying that the stellar mass loss can sustain at most 60% of the SFR of NGC2403, while at least 40% must be due to gas accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Note that the estimation of the mass loss rate is dependent on the SFR: a lower SFR would result in a lower mass loss rate (although not necessarily in proportion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Overall, we conclude that gas accretion is still necessary to sustain the SFR in NGC 2403 within the circumstances explored by the LK11 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In Section 4 we explored four free parameters that are crucial for our EPG dynamical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' However, construction of the model also involves other parameters and ingredients for which we make specific choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Below we discuss the limitations and reliability of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023) 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The gravitational potential of NGC 2403 used in this paper is gen- erated from a mass model consisting of three components: a stellar disc, a gaseous disc, and a dark matter halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The parameters of the mass model are inferred via rotation curve decomposition (FB06).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Given that the circular velocity generated from the mass model is consistent with the rotation curve of NGC 2403 (see FB06), we con- clude that the gravitational potential is robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The only uncertainty is related to the fraction of the stellar disc contribution to the potential, parametrised by the mass-to-light ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The gravitational potential used in the above analysis was based on the maximum-disc model shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' It is however noteworthy that the minimum disc potential in FB06 is in fair agreement with those derived more re- cently with more sophisticated methods (Mancera Piña et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' FB06 have experimented with both maximum disc and minimum disc potentials and showed that the dynamics of the EPG does not change significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' An assumption of our model is the existence of a uniform charac- teristic outflow velocity at all radii, whereas the varying stellar feed- back activities might lead to outflow velocities changing with radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Allowing spatial variations in the characteristic outflow velocity is a potential improvement for this kind of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This has been briefly explored in FB06 to generate specific features in N2403 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' the filament shown in channel 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1 km s−1 and channel 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0 km s−1 of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 14 in FB06) that are otherwise not reproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' However, exploring the variation of ℎ𝑣 with radius would introduce at least one extra free parameter, which would significantly complicate our exploration of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Overall, the global kinematics of the EPG in NGC 2403 appears to be well reproduced by a constant characteristic outflow speed across the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In the fountain + corona accretion scenario, the acceleration of fountain gas is directly dictated, besides by gravity, by the veloc- ity difference between the fountain and the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In our model, we assume a relative azimuthal velocity of 75 km s−1 between the fountain gas and the corona, based on hydrodynamical simulations (Marinacci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Such a high relative velocity would imply a rather slowly rotating corona in NGC 2403, given the disc rota- tion of around 130 km s−1 (FB06).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We have therefore tested models with a lower relative velocity of 45 km s−1 that result in nearly iden- tical best-fitting parameters as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1 except for a higher condensation rate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 Gyr−1), which corresponds to a global accretion rate of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 M⊙ yr−1 (the best-fitting results are listed in Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This higher rate is not surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In our model, as a consequence of condensation, the coronal gas joins the cold/warm phase of the fountain gas such that the velocity of a single cloud evolves as a combination (mass-weighted average) of the kinemat- ics of the two components (cloud and condensed material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' If the velocity difference between these two components is reduced, one needs a larger accretion rate (more condensed material) to produce the same effect in the combined kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' It is noteworthy that EPG models built with a lower relative velocity have lower veloc- ity gradients than what we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' However, the difference (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0 km s−1 kpc−1) is negligible, given that the uncertainty for our measurement is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='7 km s−1 kpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The separation of EPG emission from the datacube is an important ingredient of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The reliability of our strategy for masking the disc emission has been verified in several previous studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Fraternali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We have tested the robustness of our results by fitting the data without masking the peculiar H i filament of NGC 2403, finding the same normalisation factor as shown in Table 4, but an ℎ𝑣 of 60 km s−1, an 𝑓ion of 0, a condensation rate of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='8 Gyr−1, leading to an accretion rate of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='28 M⊙ yr−1 (all parameters are compatible with those of our fiducial model within the errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Thus models with slightly higher outflow velocities and condensation rates are preferred to account for the filament in NGC 2403, but the overall validity of our results is not particularly affected by our masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In conclusion, the construction of our dynamical model is robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The variation of certain ingredients leads to small changes in the model best-fitting parameters but does not alter our main conclusion: the EPG of NGC 2403 is produced by a combination of galactic fountain clouds and gas accretion from the condensation of the hot CGM at a rate compatible with the SFR of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 Can the fountain + corona accretion sustain the inside-out growth of the disc?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Since accretion is a key source to fuel further star formation, the outward shift of the accretion (compared to the SFR) shown in Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='3 suggests a potential inside-out redistribution of gas and star formation activities in the future, which has been predicted by cosmo- logical simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Grand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2017) and supported by many observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' van der Wel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Pez- zulli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Pezzulli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2015) also provided measurements of the specific radial growth rate, 𝜈𝑅 ≡ (1/𝑅∗) × d𝑅∗/d𝑡, where 𝑅∗ is the scale length of the stellar disc, for a sample of galaxies including NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Furthermore, a cosmological/zoom-in simula- tion (Grand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2019) also found that fountain clouds can acquire angular momentum via interaction with the CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' To verify whether the gas accretion due to a galactic fountain can be deemed responsible for this growth, we calculated the variation in time of the specific angular momentum d𝑗/d𝑡 of the stellar disc (a direct tracer of disc growth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Posti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2019) due to accretion, under the simplifying assumption that the next generation of stars will be formed out of the newly accreted gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This gives d𝑗 d𝑡 = d(𝐽/𝑀) d𝑡 = 1 𝑀 d𝐽 d𝑡 − 𝐽 𝑀2 d𝑀 d𝑡 , (7) where 𝐽 and 𝑀 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 × 109 M⊙) are the angular momentum and mass of the stellar disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We estimate 𝐽 as 𝐽 = 2𝑀𝑉flat𝑅∗ (Romanowsky & Fall 2012), where 𝑉flat is the rotational velocity of the flat part of the rotation curve (130 km s−1) and 𝑅∗ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0 kpc (values from Fraternali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The time derivative of the angular momentum d𝐽/d𝑡 is given by d𝐽 d𝑡 = d𝐽in d𝑡 − d𝐽out d𝑡 = 2𝜋 ∫ R 0 𝑅′2Fin(𝑅′)𝑉in(𝑅′) d𝑅′ −2𝜋 ∫ R 0 𝑅′2Fout(𝑅′)𝑉out(𝑅′) d𝑅′, (8) where Fin (Fout) is the inflow (outflow) surface density rate given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='3, 𝑉in(𝑅′) (𝑉out(𝑅′)) is the average rotational velocity of all cloud particles falling onto (ejected from) the disc at radius 𝑅′, obtained from our model by tracking the outflow and inflow radius and velocity of all fountain clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The time derivative of the mass, d𝑀/d𝑡, is by definition the accretion rate of new gas given by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Implementing the above equation to our best-fitting model, we have d𝑗/d𝑡 = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='6 × 10−8 km s−1 kpc yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This would indicate that the gas accreted through the fountain cannot be solely responsible for the observed inside-out growth of the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Part of this growth should MNRAS 000, 1–13 (2023) Fountain-driven corona accretion in NGC 2403 11 then be ascribed to gas that is already present in the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This is a viable option, as the gas in the disc is known to be located, on average, at larger radii compared to the stellar component (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Fraternali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This solution is, however, only partly satisfactory, as the gas reservoir at these large radii would, without replacement, be consumed on a relatively short timescale (a few Gyr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Fraternali & Tomassetti 2012), implying that the growth of the disc would not be sustainable in the long term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' With these considerations in mind, we stress that our calculation of d𝑗/d𝑡, presented above, very much depends on the value that we are assuming for the rotational speed of the corona, which is, as we discussed above, very uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Interestingly, when assuming the rotational lag between the fountain and the hot gas is 45 km s−1 (the third model in Table 4), we have d𝑗/d𝑡 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5×10−8 km s−1 kpc yr−1, which indicates an inside-out growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Combining the current value of the specific angular momentum 𝑗 and its derivative d𝑗/d𝑡, we can easily derive the specific angular momentum growth rate, which we define (following Pezzulli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2015) as 𝜈 𝑗 ≡ (1/𝑗) × d𝑗/d𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We find a value of 𝜈 𝑗 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='88 × 10−2 Gyr−1, in excellent agreement with the specific radial growth rate 𝜈𝑅 = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='16) × 10−2 Gyr−1 measured by Pezzulli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2015) for NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The two quantities 𝜈 𝑗 and 𝜈R are comparable and are in fact expected to be equal, as long as the rotation curve of the galaxy can be considered approximately stationary with time3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We have therefore found that our model with a reduced rotational lag is in remarkable quantitative agreement with the galactic fountain being the main source of the observed inside-out growth in NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' It is important to note that in the absence of triggered condensa- tion, a galactic corona will be expected to cool in the very inner parts, where its density tends to be higher, thus producing the accretion of low angular momentum gas that then would need to be expelled via strong feedback (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Brook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Instead, when the cooling is triggered by the fountain, the location of the bulk of the gas ac- cretion is naturally shifted to outer radii for the reasons described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This phenomenon had been indicated as plausibly compatible with the inside-out growth of discs (Pezzulli & Fraternali 2016), but this is the first time that quantitative evidence is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 6 CONCLUSION In this work, we have modelled the distribution and kinematics of the neutral extra-planar gas (EPG) in the late-type nearby galaxy NGC 2403 using a dynamical model of galactic fountain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' In this model, stellar feedback activities continuously eject gas from the galaxy disc, which travels through the halo and falls back to the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This gas cycle brings metal-rich and cold/warm gas to mix and interact with the hot corona, significantly reducing its cooling time, and leading to condensation and accretion of some coronal gas onto the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Due to angular momentum exchange between the fountain clouds and the corona, this interaction is expected to leave a signature in the kinematics of the H i gas at the disc–halo interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The application of our models to the data leverage this signature to infer, along with other parameters, the efficiency of the condensation process and the accretion rate of coronal gas onto the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' While these models have been applied extensively to the EPG of the Milky Way (Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2012, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Fraternali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2013, 3 This is immediately seen by taking the time derivative of the equation 𝑗 = 2𝑉flat𝑅∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2015), so far applications to external galaxies were limited to the pre- liminary studies of FB06 and FB08, which did not include a rotating corona nor a statistically meaningful exploration of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' This study presents the first detailed application of the current fountain accretion framework to an external galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Our results are summarised as follows: (i) The galactic fountain framework can reproduce most of the neutral EPG features in NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' A model where the fountain clouds interact with the hot corona is statistically preferred compared to a pure fountain model without interaction with the hot CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (ii) The best-fitting model requires a fountain with a characteristic outflow velocity of 50±10 km s−1, with the gas being ionised for some time after ejection and then recombining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Recombination appears to occur on average when its vertical velocity has been reduced by about 40 per cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (iii) The H i EPG in NGC 2403 inferred from the best-fitting model has a total EPG mass of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='7+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='9 × 108 M⊙, with an average scale height of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='003 kpc and a vertical gradient in rotational velocity of −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='0±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='7 km s−1 kpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Our values are compatible with a previous estimate of Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2019), which was derived with simpler phenomenological approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (iv) Our model predicts a condensation rate of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='4 Gyr−1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2 Gyr−1 ) for the hot CGM, leading to a total accretion rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='8 M⊙ yr−1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1 M⊙ yr−1) when assuming the rotational lag be- tween the fountain and the hot gas is 75 km s−1 (45 km s−1), similar to the star formation rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='6 M⊙ yr−1 of NGC 2403, suggesting corona accretion as a viable mechanism to maintain the star-formation rate in this galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (v) The accretion rate surface density profile predicted by our model is radially more extended than the star-formation-rate surface density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' We have also shown that, if the rotation velocity of the corona is larger than a certain threshold, the specific angular momentum growth rate predicted by our model is in excellent agreement with the observed inside-out growth rate in NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' The fountain- driven accretion process can therefore be responsible for the inside- out growth of its stellar disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors would like to thank an anonymous referee for helpful comments and Cecilia Bacchini for collecting and providing the H i, H2, and star-formation-rate data of NGC 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' AL was supported by the Netherlands Research School for Astronomy (Nederlandse Onderzoekschool voor Astronomie, NOVA), Network 1, Project 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='9 WEAVE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' GP acknowledges support from the Netherlands Research School for Astronomy (Nederlandse Onderzoekschool voor Astronomie, NOVA) through project 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' DATA AVAILABILITY The data underlying this article were obtained by Fraternali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' (2002) with the CS configuration of the VLA and were later included in the HALOGAS survey, which is available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='nl/halogas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' REFERENCES Afruni A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Pezzulli G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2021, MNRAS, 501, 5575 Anderson M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Bregman J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2011, ApJ, 737, 22 MNRAS 000, 1–13 (2023) 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Anderson M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Churazov E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Bregman J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2016, MNRAS, 455, 227 Armillotta L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Marinacci F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2016, MNRAS, 462, 4157 Aumer M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Binney J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2009, MNRAS, 397, 1286 Bacchini C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Iorio G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Pezzulli G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2019, A&A, 622, A64 Bacchini C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Pezzulli G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Marasco A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2020, A&A, 644, A125 Barbieri C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Oosterloo T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Bertin G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Boomsma R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Sancisi R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2005, A&A, 439, 947 Bigiel F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Leroy A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Walter F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Blitz L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Brinks E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', de Blok W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Madore B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2010, AJ, 140, 1194 Binney J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Nipoti C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2009, MNRAS, 397, 1804 Boomsma R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Oosterloo T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', van der Hulst J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Sancisi R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2008, A&A, 490, 555 Bregman J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 1980, ApJ, 236, 577 Brook C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Stinson G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Gibson B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Roškar R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Wadsley J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Quinn T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2012, MNRAS, 419, 771 Ciotti L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Ostriker J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2012, in Kim D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Pellegrini S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', eds, Astrophysics and Space Science Library Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 378, Astrophysics and Space Science Library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 83 (arXiv:1104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='2238), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1007/978-1-4614-0580-1_4 Crain R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Eke V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Frenk C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Jenkins A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', McCarthy I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Navarro J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Pearce F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2007, MNRAS, 377, 41 Dettmar R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 1990, A&A, 232, L15 Di Teodoro E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2014, A&A, 567, A68 Di Teodoro E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Peek J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2021, ApJ, 923, 220 El-Badry K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2018, MNRAS, 473, 1930 Faerman Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Sternberg A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', McKee C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2020, ApJ, 893, 82 Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2017, in Fox A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Davé R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', eds, ASSL, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 430, Gas Ac- cretion onto Galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Springer International Publishing AG, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 323, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1007/978-3-319-52512-9_14 Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Binney J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2006, MNRAS, 366, 449 Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Binney J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2008, MNRAS, 386, 935 Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Tomassetti M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2012, MNRAS, 426, 2166 Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', van Moorsel G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Sancisi R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Oosterloo T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2002, AJ, 123, 3124 Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Marasco A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Marinacci F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Binney J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2013, ApJ, 764, L21 Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Marasco A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Armillotta L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Marinacci F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2015, MNRAS, 447, L70 French D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2021, ApJ, 923, 50 Fukugita M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Peebles P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2006, ApJ, 639, 590 Gatto A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Read J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Marinacci F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Lux H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Walch S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2013, MNRAS, 433, 2749 Grand R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2017, MNRAS, 467, 179 Grand R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2019, MNRAS, 490, 4786 Gronke M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Oh S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2018, MNRAS, 480, L111 Heald G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Rand R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Benjamin R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Bershady M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Collins J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Bland-Hawthorn J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2005, in AAS Meeting Abstracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='05 Heald G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2011, A&A, 526, A118 Heald G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2012, The Westerbork Hydrogen Accretion in LOcal GAlax- ieS (HALOGAS) survey (Corrigendum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Survey description and pilot observations, Astronomy & Astrophysics, Volume 544, id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='C1, 1 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='1051/0004-6361/201015938e Heckman T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Borthakur S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Wild V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Schiminovich D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Bordoloi R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2017, ApJ, 846, 151 Hess K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Pisano D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Wilcots E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Chengalur J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2009, ApJ, 699, 76 Hodges-Kluck E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Miller M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Bregman J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2016, ApJ, 822, 21 Hopkins A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', McClure-Griffiths N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Gaensler B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2008, ApJ, 682, L13 Hsu W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Putman M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Heitsch F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Stanimirović S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Peek J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Clark S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2011, AJ, 141, 57 Iorio G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Nipoti C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Di Teodoro E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Read J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Battaglia G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2017, MNRAS, 466, 4159 Joung M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Bryan G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Putman M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2012, ApJ, 745, 148 Kamphuis P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Peletier R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Dettmar R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', van der Hulst J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', van der Kruit P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Allen R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2007, A&A, 468, 951 Keller B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Wadsley J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Benincasa S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Couchman H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2014, MN- RAS, 442, 3013 Kennicutt Robert C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 1989, ApJ, 344, 685 Kennicutt Robert C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Tamblyn P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Congdon C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 1994, ApJ, 435, 22 Kennicutt Robert C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2009, ApJ, 703, 1672 Kereš D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Katz N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Weinberg D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Davé R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2005, MNRAS, 363, 2 Kereš D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Katz N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fardal M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Davé R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Weinberg D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2009, MNRAS, 395, 160 Kim C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Ostriker E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2018, ApJ, 853, 173 Kooij R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Grønnow A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2021, MNRAS, 502, 1263 Lagos C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Theuns T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Stevens A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Cortese L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Padilla N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Davis T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Contreras S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Croton D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2017, MNRAS, 464, 3850 Lehner N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Howk J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Thom C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fox A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Tumlinson J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Tripp T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Meiring J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2012, MNRAS, 424, 2896 Lehner N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Howk J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Marasco A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2022, MNRAS, Leitner S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Kravtsov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2011, ApJ, 734, 48 Leroy A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Walter F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Brinks E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Bigiel F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', de Blok W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Madore B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Thornley M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2008, AJ, 136, 2782 Levy R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2019, ApJ, 882, 84 Li J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Bregman J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Wang Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Crain R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Anderson M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2018, ApJ, 855, L24 Li A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Marasco A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Trager S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Verheijen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2021, MN- RAS, 504, 3013 Mac Low M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', McCray R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 1988, ApJ, 324, 776 Mancera Piña P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Posti L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Adams E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Oosterloo T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2021, A&A, 647, A76 Mancera Piña P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Oosterloo T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Adams E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', di Teodoro E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Bacchini C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Iorio G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2022, MNRAS, 514, 3329 Marasco A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2011, A&A, 525, A134 Marasco A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Binney J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2012, MNRAS, 419, 1107 Marasco A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Marinacci F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2013, MNRAS, 433, 1634 Marasco A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', van der Hulst J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Oosterloo T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2017, A&A, 607, A106 Marasco A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2019, A&A, 631, A50 Marasco A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Lehner N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Howk J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2022, MNRAS, 515, 4176 Marinacci F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Binney J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Nipoti C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Ciotti L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Londrillo P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2010, MNRAS, 404, 1464 Marinacci F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Nipoti C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Binney J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Ciotti L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Londrillo P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2011, MNRAS, 415, 1534 Mo H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Mao S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', White S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 1998, MNRAS, 295, 319 Navarro J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Frenk C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', White S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 1997, ApJ, 490, 493 Nelson D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Vogelsberger M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Genel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Sijacki D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Kereš D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Springel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Hernquist L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2013, MNRAS, 429, 3353 Nipoti C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2010, MNRAS, 406, 247 Oosterloo T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Sancisi R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2007, AJ, 134, 1019 Pezzulli G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2016, MNRAS, 455, 2308 Pezzulli G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Boissier S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Muñoz-Mateos J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2015, MNRAS, 451, 2324 Pezzulli G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Binney J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2017, MNRAS, 467, 311 Pointon S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Kacprzak G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Nielsen N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Muzahid S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Murphy M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Churchill C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Charlton J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2019, ApJ, 883, 78 Posti L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Marasco A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Famaey B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2019, A&A, 629, A59 Putman M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Zheng Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Price-Whelan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Grcevich J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Johnson A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Tollerud E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Peek J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2021, ApJ, 913, 53 Romanowsky A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fall S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2012, ApJS, 203, 17 Rubin K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Prochaska J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Koo D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Phillips A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Weiner B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2010, ApJ, 712, 574 Rubin K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Diamond-Stanic A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Coil A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Crighton N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Moustakas J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2018, ApJ, 853, 95 Sancisi R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fraternali F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Oosterloo T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', van der Hulst T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2008, A&ARv, 15, 189 Sandage A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 1986, A&A, 161, 89 Schaye J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2010, MNRAS, 402, 1536 Schroetter I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2019, MNRAS, 490, 4368 Schwarz G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 1978, Annals of Statistics, 6, 461 Shapiro P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Field G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 1976, ApJ, 205, 762 Stinson G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Brook C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Macciò A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Wadsley J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Quinn T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Couchman H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2013, MNRAS, 428, 129 Tacconi L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2018, ApJ, 853, 179 Trapp C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2022, MNRAS, 509, 4149 Tumlinson J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2011, Science, 334, 948 Vijayan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Kim C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Armillotta L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Ostriker E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Li M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2020, ApJ, 894, 12 Voit G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Bryan G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', O’Shea B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Donahue M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2015, ApJ, 808, L30 MNRAS 000, 1–13 (2023) Fountain-driven corona accretion in NGC 2403 13 Wakker B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2001, ApJS, 136, 463 Walker S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Bagchi J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Fabian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2015, MNRAS, 449, 3527 Wang J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2011, MNRAS, 412, 1081 White S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Frenk C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 1991, ApJ, 379, 52 Zahedy F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Chen H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Johnson S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Pierce R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Rauch M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Huang Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Weiner B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Gauthier J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2019, MNRAS, 484, 2257 Zschaechner L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Rand R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Heald G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Gentile G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Kamphuis P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2011, ApJ, 740, 35 de Blok W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2014, A&A, 569, A68 van der Kruit P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', Freeman K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2011, ARA&A, 49, 301 van der Wel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=', 2014, ApJ, 788, 28 APPENDIX A: 2D MARGINALISED POSTERIOR PROBABILITY DISTRIBUTION 2D marginalised posterior probability distribution maps and contours of the three(four)-dimensional grids of free parameters (summarised in Table 3) for pure fountain (fountain + corona accretion) models, shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' A1 and A2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023) 14 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' 2D marginalised posterior probability distribution for our pure fountain models onto different 2D spaces: upper left – (ℎ𝑣, 𝑓ion), upper right – (ℎ𝑣,Norm), lower-left – ( 𝑓ion,Norm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE2T4oBgHgl3EQfCgZi/content/2301.03614v1.pdf'} +page_content=' Iso-probability contours (in yellow) correspond to 2.' metadata={'source': 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inserted by the editor) +Time-of-flight spectroscopy of ultracold neutrons at the PSI +UCN source +G. Bison1, W. Chen1,2, P.-J. Chiu1,2,*, M. Daum1, C. B. Doorenbos1,2, +K. Kirch1,2, V. Kletzl1,2, B. Lauss1, D. Pais1,2, I. Rienäcker1,a, +P. Schmidt-Wellenburg1, G. Zsigmond1,b +1Paul Scherrer Institut, CH-5232 Villigen-PSI, Switzerland +2ETH Zürich, CH-8092 Zürich, Switzerland +*current address: Universität Zürich, CH-8057 Zürich, Switzerland +Received: date / Accepted: date +Abstract The ultracold neutron (UCN) source at the +Paul Scherrer Institute (PSI) provides high intensities +of storable neutrons for fundamental physics experi- +ments. The neutron velocity spectrum parallel to the +beamline axis was determined by time-of-flight spec- +troscopy using a neutron chopper. In particular, the +temporal evolution of the spectrum during neutron pro- +duction and UCN storage in the source storage volume +was investigated and compared to Monte Carlo simula- +tion results. A softening of the measured spectrum from +a mean velocity of 7.7(2) m s−1 to 5.1(1) m s−1 occurred +within the first 30 s after the proton beam pulse had im- +pinged on the spallation target. A spectral hardening +was observed over longer time scales of one measure- +ment day, consistent with the effect of surface degrada- +tion of the solid deuterium moderator. +1 Introduction +Neutrons are defined to be ultracold if they can be re- +flected under all angles of incidence from suitable ma- +terial surfaces [1,2,3]. Such materials have neutron op- +tical potentials of a few hundred neV, corresponding to +critical velocities of a few m s−1. The total reflection +from surfaces allows the confinement of ultracold neu- +trons (UCNs) in storage bottles for hundreds of seconds, +a technique used for the measurement of fundamental +properties of the neutron [4,5]. These experiments are +often statistics limited and it is therefore a world-wide +effort [6] to improve the output of UCN sources. A soft +energy spectrum and a high number of storable neu- +trons is crucial to achieve high UCN densities and long +acorresponding author: ingo.rienaecker@psi.ch +bcorresponding author: geza.zsigmond@psi.ch +observation times in storage experiments. +The Paul Scherrer Institute (PSI) operates a spal- +lation-driven, solid deuterium-based source for UCN [7, +8]. The source hosts worldwide leading experiments to +measure the neutron electric dipole moment [9,10], as +well as to search for dark matter candidates, such as +axion-like particles [11,12] and oscillations of neutrons +into sterile states [13,14]. The thermal moderation [15], +UCN production [16], as well as UCN storage and trans- +port [17] in the source were characterized previously. In +this paper we report on the measurement of the distri- +bution of the longitudinal velocity component v = L/T +of UCNs by time-of-flight T spectroscopy along a flight +path L behind a neutron chopper mounted at beamport +West-1 [7]. The spectrum obtained from this measure- +ment can be used for further calibrations of the UCN +source simulation model and will be valuable input for +future experiments. +In section 2 we present the concept and parameters +of the measurement setup, consisting of a UCN detec- +tion system and a neutron chopper. We describe the +time-of-flight (TOF) data analysis including our back- +ground subtraction technique in section 3. In section 4, +we discuss the deduced velocity spectrum and its evo- +lution in time during UCN storage in the UCN source +volume [7]. The influence of UCN source operational +procedures on the spectrum is investigated. +2 Measurement +The measurement setup, consisting of a neutron chop- +per, a L = 100 cm flight path, and a neutron detector1 +is depicted in Fig. 1 a). The chopper is based on the +arXiv:2301.11668v1 [physics.ins-det] 27 Jan 2023 + +2 +G. Bison et. al. +Fig. 1 a) Drawing of the measurement setup, starting from the electro-polished stainless steel guide that was attached to the beamport +shutter, up to the CASCADE1 detector. b) A picture of the chopper gratings aligned in the open position, seen from a perspective +looking through the short steel guide in front of the chopper towards the detector in the back (behind the chopper). The picture was +taken while the whole setup was removed from the beamport shutter. +design described in [18] and uses the opposing linear +motion of two titanium gratings to achieve short open- +ing times. Each grating has 25, s = 3 mm wide, vertical +slits, separated by 5 mm, as shown in Fig. 1 b). The +first grating is moved by a piston from the left, con- +nected to a linear motor in the vacuum housing of the +chopper. The second grating is connected to a second +motor on the right and slides directly behind the first +grating with a minimal gap between the gratings. The +chopper time resolution is determined by its opening +function Ω(t), i.e. the fraction of the cross section of +the neutron guide that is unblocked by the gratings as +a function of time. When aligned, the 3 mm wide slits +Fig. 2 Plot of the chopper trigger and the subsequent photo +diode signal to measure the opening function Ω(t) of the chopper. +of the two gratings open a maximum of approximately +Ω(0) = 38 % of the cross section of the guide. The accel- +eration of the linear motors for the gratings was set to +a = 100 m s−2 to achieve a full width at half maximum +opening time of ∆FWHM = 2 +� +s/(2a) = 7.7 ms. This is +consistent with a measurement of the opening time (see +Fig. 2) performed with a photo diode located behind the +gratings and a diffuse light source in front. The time +offset, ts = 59.2(3) ms, between the maximum of the +opening function and the chopper trigger, an electronic +signal transmitted by the chopper controller shortly be- +fore each opening operation, was also determined by +this method. Additionally, a calibration measurement +[19] of the time offset was performed by measuring the +UCN count rates, N1(v) and N2(v), with two different +flight path lengths, L1 = 100 cm, L2 = 200 cm, us- +ing two identical guides of 100 cm length. It was found +that dN1 +dv ( +L1 +t1−ts ) = dN2 +dv ( +L2 +t2−ts ) at the respective maxima +of the normalized velocity distributions, confirming the +measured time offset within uncertainties. +During our spectroscopy measurements we recorded +time-of-flight data during approximately one day of un- +interrupted standard operation of the PSI UCN source +[7,8]. A new measurement cycle was started after each +proton beam pulse onto the source’s spallation target. +The average proton beam current during a pulse was +approximately 2.0 mA and the duration was set to 8 s. +The internal clock of the UCN detector was synchro- +nized to the chopper by receiving a coincidence signal +between the electronic trigger sent by the chopper and a +1 www.n-cdt.com/cascade-2d-200 + +detector +a) +b) +L = 100 cm +Ni/Mo 85/15 +chopper +coated glass guide +WCN +vacuum housing +motor +electro-polished +gratings aligned +stainless steel guide +chopper gratings +in open positionchopper trigger +opening function Q(t) +1.0 +t. = 59.2(3) ms +arb. units +0.5 +ms +0.0 +20 +40 +60 +80 +0 +time after trigger (ms)Time-of-flight spectroscopy of ultracold neutrons at the PSI UCN source +3 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 11 12 +0 +10 +20 +counts / ms +time (s) +frame number +0 1 2 3 ... +tp = (0.75 ± 0.25) s +start proton beam pulse +Fig. 3 Raw UCN time spectrum recorded with the CASCADE1 +detector during the first 12 s of one cycle with the chopper oper- +ating at 2 Hz. On the time axis, t = 0 s coincides with the first +chopper trigger signal. Relative to that, the 8 s long proton beam +pulse starts at tp = (0.75±0.25) s, as indicated in the plot. Each +cycle is divided into 600 frames, i.e. time intervals [k tf, (k+1) tf] +with k = 0, 1, ..., 599 which contain TOF spectra with respect to +the time of the corresponding chopper openings. The UCN count +rate increases during the proton beam pulse and slowly decreases +afterwards. +signal from the proton beam control that rises 1 s before +the start of the proton beam pulse. The chopper was +running continuously with a duty cycle of 1.5 %, open- +ing the neutron guide at a frequency of approximately +2 Hz. The exact time between two consecutive chopper +triggers was measured to be tf = 499.855(5) ms, where +the uncertainty denotes the observed stability of the sig- +nal and the accuracy of a measurement over multiple +days. Therefore, the timing of the recorded TOF spec- +tra for each cycle was fixed with respect to the chopper +operation, while the timing with respect to the proton +beam pulse might have varied by the time between two +chopper trigger signals, i.e. by up to ± tf/2 ≈ ± 250 ms, +as shown in Fig. 3. +Several measurements were performed to determine +the leakage rate of UCNs through small gaps between +the chopper housing and the neutron guides or the grat- +ings in the closed position. Taking the ratio of the count +rate measured by the detector with closed chopper ver- +sus (permanently) open chopper, one finds an average +UCN leakage of (4 ± 1) · 10−3. The UCN leakage, and +other effects like electronic noise and secondary radia- +tion during the proton beam pulse, cause background in +the measured TOF spectra, which was fitted and sub- +tracted as discussed in section 3. +The measurement setup was connected to the West-1 +beamport of the PSI UCN source with a 20 cm long +electro-polished stainless steel guide with an inner di- +ameter of 135 mm. The guide was connected to the +beamport shutter via a stainless steel flange. Similar +flanges were used to connect the stainless steel guide +to the chopper on one side, and a L = 100 cm glass +guide with inner diameter of 130 mm on the other side +between chopper and detector. The glass guide was +sputter-coated with Nickel/Molybdenum 85/15 [7] to +obtain a high Fermi potential. A CASCADE 2D U-200 +UCN detector1 was attached at the end of the 100 cm +glass guide. The time resolution of the detector was set +to dt = 1 ms. The energy acceptance of the detector +has a lower cut-off Emin = 54 neV due to its 100 µm +thick AlMg3 entrance window [7], corresponding to a +lower bound on the velocity component orthogonal to +the window of vmin = 3.2 m s−1. +3 Analysis +The measured count rate in each time bin dt at time t +was averaged over all 286 measurement cycles, recorded +in a period of approximately one day, to obtain the +average count rate +dN +dt (t). The data was divided into +600 individual frames, as indicated in Fig. 3. The TOF +spectrum of frame k represents the average spectrum +at time tk = k tf −tp after the start of the proton beam +pulse. Each frame contains one TOF spectrum +dN +dt +��� +tk +(T), +T = mod(t, tf) − ts ∈ [−59, 441] ms, +(1) +where the TOF axis was shifted by the chopper time +offset ts ≈ 59 ms. The timing uncertainties of δtf = +0.005 ms and δts = 0.3 ms of the chopper trigger signal +discussed in section 2 lead to a small systematic uncer- +tainty of δT = k δtf + δts on the time of flight in frame +k and to an associated uncertainty LδT +T 2 on the velocity +bins of the deduced velocity spectrum (see section 4). +Figure 4 shows the evolution of the TOF spectrum, +its maximum and the mean time of flight during and +after the proton beam pulse. The temporary hardening +of the spectrum at approximately 8 s was found to be +correlated to the closing of the neutron shutter at the +bottom of the UCN source storage volume [19]. The +hardening is likely due to reflections or scattering of +very cold neutrons (VCN) with v ≳ 10 m s−1 on the +partially closed neutron valve into the West-1 beamline. + +4 +G. Bison et. al. +Fig. 4 Evolution of the TOF spectrum during the first 60 s after +the start of the 8 s long proton beam pulse. A constant back- +ground rate Rk, Eq. (3), was separately determined and removed +from the spectrum for each frame k at time tk. The mean time +of flight ⟨T⟩ (red dashed line) and the maximum T max (black +line) of the TOF distribution are indicated in the plot. +The combination of a number of n frames yields the +average TOF spectrum +dN +dt +��� +tkn +tk0 +(T) = 1 +n +kn +� +k=k0 +�dN +dt +��� +tk +(T) − Rk +� +(2) +in the corresponding time interval [tk0, tkn] after the +start of the proton beam pulse. As a first correction, a +subtraction of the constant background was performed +by removing a rate Rk of typically around 80 Hz per +frame from the TOF spectra for each frame k individ- +ually. The residual background rate after subtraction +was minimized by choosing Rk such that it is the min- +imum of a 30 ms running average of the original TOF +spectrum, +Rk = min +T +� +1 +30 +30 +� +l=0 +dN +dt +��� +tk +(T + l dt) +� +. +(3) +The further background fitting and subtraction rou- +tine follows the method presented in [20]. In addition +to a constant leakage rate of UCNs through the closed +chopper as discussed in section 2, the spectrum also +contains UCNs that are reflected non-specularly from +surfaces between chopper and detector. After non-specular +reflection, the time of flight is no longer a valid measure +for the initial velocity parallel to the guide axis. In ad- +dition, non-specular reflections and back-scattering of +UCNs with velocity components parallel to the guide +axis below vmin can lead to an accumulation of UCNs +that are quasi-stored with a short storage time con- +stant τ in the neutron guide between chopper and de- +tector. Eventually, these quasi-stored UCNs may be +deflected towards the detector and counted at times +T > +L +vmin = 0.31 s, i.e. later than the nominal maxi- +mum time of flight . +Figure 6 shows the average spectrum after subtrac- +tion of the constant rate Rk on a logarithmic scale. +The exponential rate of detected quasi-stored UCNs is +clearly visible at times later than the maximum time of +flight of specularly reflected UCNs. In order to estimate +the rate of detected quasi-stored UCNs, we fit the aver- +age spectrum at times T > 0.33 s with an exponential +function +b(T) = A e−(T −0.33s)/τ, +(4) +while simultaneously fitting the same model but shifted +by one frame time b(T + tf) for T < 0.03 s, well before +the rising edge of the TOF spectrum. Starting the fit +interval at T > 0.33 s, i.e. slightly later than at the nom- +inal maximum time of flight, was required to obtain a +low χ2 for a single exponential fit. We confirmed that +including a constant offset parameter R in the fit to +the average TOF spectra, without previously removing +a constant rate Rk for each frame individually, leads to +a similar result. +We assume that at the beginning of the frame for +T < T max, i.e. before the maximum of the time-of- +flight distribution, the detected rate of non-specularly +reflected UCNs follows the same time distribution as +the specularly reflected UCNs. The systematic implica- +tions of this assumption are discussed below. Thus, the +Fig. 5 Average TOF spectra per cycle and frame, Eq. (2), during +the first approximately 12 s (dashed line) and between 12.25 s +and 30.25 s (solid line) after the start of the proton beam pulse. +The Poisson errors are smaller than the line width. The open and +solid markers indicate the mean of the respective distribution. +In the first time interval, the detection of very cold neutrons +(VCN) that are produced during the proton beam pulse is clearly +visible by the tail of counts at low time of flight. + +10 +0.4 +mean +max +8 +0.3 +time of flight (s) +count rate (kHz) +6 +0.2 +4 +0.1 +2 +0.0 +0 +10 +20 +30 +40 +50 +60 +time after start of proton beam pulse (s)6 +t = 0.25 s - 12.25 s +286 cycles +n = 24 frames +average count rate (kHz) +total cts: 4.8 × 106 +5 +t = 12.25 s - 30.25 s +4 +n = 36 frames +total cts: 6.1 × 106 +3 +VCN +2 +1 +0 +0.0 +0.1 +0.2 +0.3 +0.4 +time of flight (s)Time-of-flight spectroscopy of ultracold neutrons at the PSI UCN source +5 +Fig. 6 Plot of the average TOF spectrum between 12.25 s and +30.25 s after the start of the proton beam pulse, demonstrating +the background subtraction method. The black diamonds are +the average of the spectra with previously subtracted constant +rate Rk per frame k, Eq (3). The red dashed line indicates the +fitted background B(T), Eq. (5), according to the procedure ex- +plained in the text. The average TOF spectrum with the back- +ground removed is the difference (blue squares) of the above. +function b(T) with best fit parameters (A, τ) is extrap- +olated to the region T ≥ T max. For T < T max we add a +fraction ρ of the average TOF spectrum to the contri- +bution b(T +tf) of quasi-stored UCNs from the previous +frame to obtain a smooth transition of the background +function B(T) (Fig. 6 dashed red line) before and after +the maximum of the TOF spectrum, +B|tkn +tk0 (T) = += +� +(1 − ρ) b(T + tf) + ρ dN +dt |tkn +tk0 (T) +; T < T max +b(T) +; T ≥ T max +(5) +with +ρ = +b(T max) − b(T max + tf) +dN +dt |tkn +tk0 (T max) − b(T max + tf) +. +(6) +For the time range depicted in Fig. 6, the ratio of the +integrated background, including the constant offsets +Rk, to the integral of the final TOF spectrum (Fig. 6, +blue markers) is 13 %. We checked the impact of our as- +sumption about the time distribution of non-specularly +reflected UCNs on the final velocity spectrum by inves- +tigating the possibility that there is a significant delay +in the arrival of non-specularly reflected UCNs at the +detector. To do so, we shifted the maximum of the back- +ground curve (Fig. 6, red dashed line) by 30 ms, half of +the width of the distribution. As a consequence, the +mean of the resulting velocity distribution was found +to be shifted by at most 0.05 m s−1 (which we included +in our uncertainty in Fig. 8) and the maximum of the +distribution by up to 0.3 m s−1. Similarly, an extreme +change of the background model in the signal region to +a linear interpolation from T = 0.1 s to T = 0.33 s was +found to shift the mean velocity by less than 0.06 m s−1. +4 Results and discussion +Figure 7 shows the final results of the axial velocity +spectra +dN +dv +��� +tkn +tk0 +(v) = L +v2 +�dN +dt +��� +tkn +tk0 +(L/v) − B|tkn +tk0 (L/v) +� +(7) +for two time intervals after the proton beam pulse. The +background B was previously subtracted from the TOF +spectra as described in section 3. We investigated whether +a discrete deconvolution [21] with the resolution func- +tion Ω(T), approximated by a box function of ∆FWHM ≈ +8 ms width (see section 2), has an effect on the de- +duced velocity spectrum. We found that the deconvo- +lution only has a marginal effect on the falling edge for +larger velocities of the final velocity spectrum. Hence, +no deconvolution of the TOF spectra with the chopper +resolution function was applied. +The evolution of the mean velocity parallel to the +guide axis in 5 s intervals during 200 s after the start of +the proton beam pulse, is shown in Fig. 8. The verti- +cal errorbars include the systematic uncertainties due +to our model of the background from non-specularly +reflected UCNs, as well as the contribution from the +uncertainty δT on the time of flight, as discussed in +section 3. The mean velocity of the neutrons reduces +Fig. 7 Normalized velocity spectra during the first approxi- +mately 12 s (dashed line) and between 12.25 s and 30.25 s (solid +line) after the start of the proton beam pulse. The open and +solid marker indicate the mean of the velocity distributions of +6.5(1) m s−1 and 5.2(1) m s−1 in the corresponding time inter- +vals. + +101 +t = 12.25 s - 30.25 s +constant rate +Rk subtracted +average count rate (kHz) +background +fit parameters: +100 +difference +A = 116(1)Hz +t = 0.077(1) s +p = 0.137(5) +x2/ndf = 1.70 +10-1 +10-2 +10-3 +0.0 +0.1 +0.2 +0.3 +0.4 +time of flight (s)0.5 +norm. velocity distribution (arb.) +t = 0.25 s - 12.25 s +t = 12.25 s - 30.25 s +0.4 +0.3 +0 +0.1 +0.0 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +velocity (m/s)6 +G. Bison et. al. +from approximately 7.7(2) m s−1 to 5.4(1) m s−1 within +the first 12 s after the start of the proton beam pulse. +The initial softening of the spectrum during the proton +beam pulse correlates with the increase of UCN den- +sity in the source storage volume. This leads to a dom- +inance of UCNs over non-storable VCN produced at an +approximately constant rate. After approximately 12 s +the spectrum consists mostly of UCNs (see Fig. 7) and +the mean velocity decreases slowly to 5.1(1) m s−1 at +30.25 s, comparable to the typical filling times of storage +experiments. Indeed, a gradual softening of the spec- +trum during the storage of UCNs is expected due to +velocity dependent losses in the source storage volume +and neutron guides [7]. +We compared our results to the UCN velocity com- +ponent parallel to the beamline axis obtained from a +MCUCN [22] Monte Carlo simulation. We used the sim- +ulation model of the source described and calibrated in +[7] and [17] and adjusted the geometry to include the +stainless steel and 1 m long glass guide of the chop- +per setup. The UCN velocity component parallel to the +guide axis was tallied at the detector position, behind +the AlMg3 entrance window. The surface parameters +of the guides were set to those found in [17] for the +UCN source beamlines. The evolution of the mean ve- +locity obtained from the simulation is shown in Fig. 8 +and reproduces the observed softening of the velocity +spectrum during and after the proton beam pulse well. +Fig. 8 Evolution of the measured mean velocity (black mark- +ers and red errorbars) parallel to the guide axis during 200 s +after the start of the 8 s proton beam pulse. For the first 12 s, +i.e. during and shortly after the pulse, the mean velocity is sig- +nificantly higher than at later times, where the velocity slowly +reduces from about 5.4 m s−1 to 5.0 m s−1. The blue line and +shaded region denote the mean and statistical uncertainty of +the velocity component parallel to the guide axis at the detector +position, obtained from a Monte Carlo simulation with a model +of the source described and calibrated in [7,17]. +Fig. 9 Ratio of the average velocity spectra between 0.25 and +30.25 s after the proton beam pulse of the last 50 measurement +cycles (cycle 236 - 286) and the first 50 cycles (cycle 1 - 50). The +error bars indicate the propagated Poisson counting errors. +There is also good agreement between the maximum +vmax,meas = 5.2(1) m s−1 of the measured (see Fig. 7) +and simulated velocity distributions with vmax,sim = +5.3(1) m s−1 in the time interval of 12.25 s to 30.25 s +(and also for earlier or later time intervals). This indi- +cates that the velocity dependent loss mechanisms in +the source storage volume and neutron guides are mod- +elled correctly by the simulation. However, the simula- +tion appears to systematically underestimate the mean +velocity by around 5 %, due to an excess of UCN in +the simulated velocity spectrum between 3.2 m s−1 and +4.5 m s−1. It was found that this can be explained by +simplifications in the UCN surface roughness model +used for the AlMg3 detector entrance and beamline vac- +uum separation windows. +In reference [23] it was reported that heat deposi- +tion by radiation from the spallation target can cause a +built-up of solid deuterium (sD2) frost on the surface of +the sD2 moderator over time. Simulations showed that +frost layers cause a hardening of the UCN energy spec- +trum (Fig. 20 of [23]) due to velocity-dependent back- +scattering of UCNs from the optical potential of isotrop- +ically distributed sD2 disks. A conditioning procedure +was developed at the PSI UCN source to restore the +surface quality. We studied the evolution of the average +velocity spectrum on long time scales by performing +our analysis on subsets of measurement cycles. Figure +9 shows the ratio of the velocity spectrum between 3 +and 10 m s−1 obtained from the last 50 cycles, and the +first 50 cycles, recorded during roughly one day (i.e. 286 +cycles). A conditioning procedure was applied shortly +before the first cycle of our measurements. While the +mean velocity of the first 50 cycles and the last 50 cy- + +9 +9 +measurement +S +2 +simulation +8 +8 +mean velocity (m/s) +T +t +7 +7 +TI +6 +6 +5 +5 +4 +4 +3 +3 +0 +25 +50 +75 +100 +125 +150 +175 +200 +time after start of proton beam pulse (s)t = 0.25 s - 30.25 S +1.0 +0.9 +tio +0.8 +0.7 +last 50 cycles / first 50 cycles +0.6 +3 +4 +5 +6 +7 +8 +9 +10 +velocity (m/s)Time-of-flight spectroscopy of ultracold neutrons at the PSI UCN source +7 +cles differs only by about 0.1 m s−1, the ratio shown in +Fig. 9 reduces quickly from close to one for velocities +above approximately 7.2 m s−1 to about 75 % for lower +velocities. This indicates a hardening of the spectrum +by additional losses of slower UCN, consistent with the +observed effects of surface degradation of the solid deu- +terium moderator due to the built-up of sD2 frost layers +[23]. +5 Conclusion +The UCN velocity spectrum parallel to the guide axis at +beamport West-1 was measured by time-of-flight spec- +troscopy using a neutron chopper. A systematic back- +ground subtraction was necessary due to contributions +of a constant leakage of UCNs through the closed chop- +per and non-specular reflections on surfaces between +chopper and detector. We measured the decrease of +the mean UCN velocities during and after the proton +beam pulse from about 7.7 m s−1 to 5 m s−1. We have +also determined the evolution of the UCN energy spec- +trum during source operation, where sD2 frost build-up +on the solid deuterium surface causes additional losses, +predominantly of slower UCN. +The measurement of the velocity spectra at beam- +port West-1 is another step towards the full characteri- +zation [15,7,17,16] of the PSI UCN source. Our results +show good overall agreement with simulations and will +be used for further refinements of the simulation model +and in the study of possible improvements of the UCN +source. +Acknowledgments +We acknowledge the PSI proton accelerator operations +section, all colleagues who have been contributing to +the UCN source operation at PSI, and especially the +BSQ group which has been operating the PSI UCN +source during the measurements, namely B. Blau, P. +Erisman and also S. Grünberger. Excellent technical +support by M. Meier, L. Noorda and M. Schaufelbühl +is acknowledged. This work was supported by the Swiss +National Science Foundation Projects 163413, 169596, +172626, 178951, 188700 and 200441. +References +1. Y.B. Zeldovich, Sov. Phys. JETP-9 1389-90 (1959) +2. V. Lushchikov, Y. Pokotilovskii, A. Strelkov, F. Shapiro, +JETP Lett. 9, 23 (1969) +3. A. Steyerl, Phys. Lett. 29b(1), 33 (1969). doi:10.1016/0370- +2693(69)90127-0 +4. V. Ignatovich, The Physics of Ultracold Neutrons (Claren- +don, Oxford, 1990) +5. R. Golub, D. Richardson, S. Lamoreaux, Ultra-Cold Neu- +trons (Adam Hilger, Bristol, Philadelphia, and New York, +1991) +6. G. +Bison, +M. +Daum, +K. +Kirch, +B. +Lauss, +D. +Ries, +P. Schmidt-Wellenburg, G. Zsigmond, T. Brenner, P. Gel- +tenbort, +T. +Jenke, +O. +Zimmer, +M. +Beck, +W. +Heil, +J. Kahlenberg, J. Karch, K. Ross, K. Eberhardt, C. Gep- +pert, S. Karpuk, T. Reich, C. Siemensen, Y. Sobolev, +N. +Trautmann, +Phys. +Rev. +C +95(4), +045503 +(2017). +doi:10.1103/PhysRevC.95.045503 +7. G. Bison, B. Blau, M. Daum, L. Göltl, R. Henneck, +K. Kirch, B. Lauss, D. Ries, P. Schmidt-Wellenburg, G. Zsig- +mond, The European Physical Journal A 56(2) (2020). +doi:10.1140/epja/s10050-020-00027-w +8. B. Lauss, B. Blau, SciPost Phys. Proc. 5, 004 (2021). +doi:10.21468/scipostphysproc.5.004 +9. C. Abel, S. Afach, N. Ayres, C. Baker, G. Ban, G. Bison, +K. Bodek, V. Bondar, M. Burghoff, E. Chanel, Z. Chowd- +huri, P.J. Chiu, B. Clement, C. Crawford, M. Daum, +S. Emmenegger, L. Ferraris-Bouchez, M. Fertl, P. Flaux, +B. Franke, A. Fratangelo, P. Geltenbort, K. Green, W. Grif- +fith, M. van der Grinten, Z. Grujić, P. Harris, L. Hayen, +W. Heil, R. Henneck, V. Hélaine, N. Hild, Z. Hodge, M. Hor- +ras, P. Iaydjiev, S. Ivanov, M. Kasprzak, Y. Kermaidic, +K. Kirch, A. Knecht, P. Knowles, H.C. Koch, P. Koss, +S. Komposch, A. Kozela, A. Kraft, J. Krempel, M. Kuź- +niak, B. Lauss, T. Lefort, Y. Lemière, A. Leredde, P. Mohan- +murthy, A. Mtchedlishvili, M. Musgrave, O. Naviliat-Cuncic, +D. Pais, F. Piegsa, E. Pierre, G. Pignol, C. Plonka-Spehr, +P. Prashanth, G. Quéméner, M. Rawlik, D. Rebreyend, +I. Rienäcker, D. Ries, S. Roccia, G. Rogel, D. Rozpedzik, +A. Schnabel, P. Schmidt-Wellenburg, N. Severijns, D. Shiers, +R.T. Dinani, J. Thorne, R. Virot, J. Voigt, A. Weis, +E. Wursten, G. Wyszynski, J. Zejma, J. Zenner, G. Zsig- +mond, Physical Review Letters 124(8), 081803 (2020). +doi:10.1103/physrevlett.124.081803 +10. N.J. Ayres, G. Ban, L. Bienstman, G. Bison, K. Bodek, +V. Bondar, T. Bouillaud, E. Chanel, J. Chen, P.J. Chiu, +B. Clément, C.B. Crawford, M. Daum, B. Dechenaux, C.B. +Doorenbos, S. Emmenegger, L. Ferraris-Bouchez, M. Fertl, +A. Fratangelo, P. Flaux, D. Goupillière, W.C. Griffith, +Z.D. Grujic, P.G. Harris, K. Kirch, P.A. Koss, J. Krem- +pel, B. Lauss, T. Lefort, Y. Lemière, A. Leredde, M. Meier, +J. Menu, D.A. Mullins, O. Naviliat-Cuncic, D. Pais, F.M. +Piegsa, G. Pignol, G. Quéméner, M. Rawlik, D. Rebreyend, +I. Rienäcker, D. Ries, S. Roccia, K.U. Ross, D. Rozpedzik, +W. Saenz, P. Schmidt-Wellenburg, A. Schnabel, N. Severi- +jns, B. Shen, T. Stapf, K. Svirina, R.T. Dinani, S. Touati, +J. Thorne, R. Virot, J. Voigt, E. Wursten, N. Yazdandoost, +J. Zejma, G. Zsigmond, The European Physical Journal C +81(6) (2021). doi:10.1140/epjc/s10052-021-09298-z +11. S. Afach, G. Ban, G. Bison, K. Bodek, M. Burghoff, +M. Daum, M. Fertl, B. Franke, Z. Grujić, V. Hélaine, +M. Kasprzak, Y. Kermaïdic, K. Kirch, P. Knowles, H.C. +Koch, S. Komposch, A. Kozela, J. Krempel, B. Lauss, +T. Lefort, Y. Lemière, A. Mtchedlishvili, O. Naviliat-Cuncic, +F. Piegsa, G. Pignol, P. Prashanth, G. Quéméner, D. Re- +breyend, D. Ries, S. Roccia, P. Schmidt-Wellenburg, A. Schn- +abel, N. Severijns, J. Voigt, A. Weis, G. Wyszynski, J. Zejma, +J. Zenner, G. Zsigmond, Physics Letters B 745, 58 (2015). +doi:10.1016/j.physletb.2015.04.024 +12. C. Abel, N. Ayres, G. Ban, G. Bison, K. Bodek, V. Bon- +dar, M. Daum, M. Fairbairn, V. Flambaum, P. Geltenbort, + +8 +G. Bison et. al. +K. Green, W. Griffith, M. van der Grinten, Z. Grujić, +P. Harris, N. Hild, P. Iaydjiev, S. Ivanov, M. Kasprzak, +Y. Kermaidic, K. Kirch, H.C. Koch, S. Komposch, P. Koss, +A. Kozela, J. Krempel, B. Lauss, T. Lefort, Y. Lemière, +D. Marsh, P. Mohanmurthy, A. Mtchedlishvili, M. Musgrave, +F. Piegsa, G. Pignol, M. Rawlik, D. Rebreyend, D. Ries, +S. Roccia, D. Rozpędzik, P. Schmidt-Wellenburg, N. Sev- +erijns, D. Shiers, Y. Stadnik, A. Weis, E. Wursten, J. Ze- +jma, G. Zsigmond, Physical Review X 7(4), 041034 (2017). +doi:10.1103/physrevx.7.041034 +13. C. Abel, N. Ayres, G. Ban, G. Bison, K. Bodek, V. Bon- +dar, +E. +Chanel, +P.J. +Chiu, +C. +Crawford, +M. +Daum, +R. Dinani, S. Emmenegger, P. Flaux, L. Ferraris-Bouchez, +W. Griffith, Z. Grujić, N. Hild, K. Kirch, H.C. Koch, +P. Koss, A. Kozela, J. Krempel, B. Lauss, T. Lefort, +A. Leredde, P. Mohanmurthy, O. Naviliat-Cuncic, D. Pais, +F. Piegsa, G. Pignol, M. Rawlik, D. Rebreyend, I. Rienäcker, +D. Ries, S. Roccia, D. Rozpedzik, P. Schmidt-Wellenburg, +N. Severijns, J. Thorne, A. Weis, E. Wursten, J. Ze- +jma, G. Zsigmond, Physics Letters B 812, 135993 (2021). +doi:10.1016/j.physletb.2020.135993 +14. N.J. Ayres, Z. Berezhiani, R. Biondi, G. Bison, K. Bodek, +V. +Bondar, +P.J. +Chiu, +M. +Daum, +R.T. +Dinani, +C.B. +Doorenbos, S. Emmenegger, K. Kirch, V. Kletzl, J. Krem- +pel, B. Lauss, D. Pais, I. Rienäcker, D. Ries, N. Rossi, +D. Rozpedzik, P. Schmidt-Wellenburg, K.S. Tanaka, J. Ze- +jma, N. Ziehl, G. Zsigmond, Symmetry 14(3), 503 (2022). +doi:10.3390/sym14030503 +15. H. Becker, G. Bison, B. Blau, Z. Chowdhuri, J. Eikenberg, +M. Fertl, K. Kirch, B. Lauss, G. Perret, D. Reggiani, D. Ries, +P. +Schmidt-Wellenburg, +V. +Talanov, +M. +Wohlmuther, +G. Zsigmond, Nucl. Instrum. Methods A 777(0), 20 (2015). +doi:10.1016/j.nima.2014.12.091 +16. G. Bison, B. Blau, W. Chen, M. Daum, C.B. Doorenbos, +N. Hild, K. Kirch, V. Kletzl, B.Lauss, I. Rienäcker, D. Ries, +P. Schmidt-Wellenburg, V. Talanov, G. Zsigmond, arXiv +2211.12126 (2023). doi:10.48550/arXiv.2211.12126 +17. G. Bison, M. Daum, K. Kirch, B. Lauss, D. Ries, P. Schmidt- +Wellenburg, G. Zsigmond, The European Physical Journal A +58(6) (2022). doi:10.1140/epja/s10050-022-00747-1 +18. T. Lauer, Investigation of a superthermal ultracold neutron +source based on a solid deuterium converter for the TRIGA +mainz reactor. +Ph.D. thesis, Universität Mainz, Germany +(2010) +19. I. Rienäcker, Improving ultracold neutron yields and the +search for mirror neutrons at the PSI UCN source. Ph.D. +thesis, ETH Zurich (2022). doi:10.3929/ethz-b-000579840 +20. M. Daum, B. Franke, P. Geltenbort, E. Gutsmiedl, S. Ivanov, +J. Karch, M. Kasprzak, K. Kirch, A. Kraft, T. Lauer, +B. Lauss, A. Mueller, S. Paul, P. Schmidt-Wellenburg, +T. Zechlau, G. Zsigmond, Nucl. Instrum. Methods A 741(0), +71 (2014). doi:10.1016/j.nima.2013.12.050 +21. I. Altarev, M. Daum, A. Frei, E. Gutsmiedl, G. Ham- +pel, F.J. Hartmann, W. Heil, A. Knecht, J.V. Kratz, +T. Lauer, M. Meier, S. Paul, U. Schmidt, Y. Sobolev, +N. Wiehl, G. Zsigmond, Eur. Phys. J. A 37(1), 9 (2008). +doi:10.1140/epja/i2008-10604-8 +22. G. Zsigmond, Nucl. Instrum. Methods A 881, 16 (2018). +doi:10.1016/j.nima.2017.10.065 +23. A. Anghel, T.L. Bailey, G. Bison, B. Blau, L.J. Brous- +sard, S.M. Clayton, C. Cude-Woods, M. Daum, A. Hawari, +N. Hild, P. Huffman, T.M. Ito, K. Kirch, E. Korobkina, +B. Lauss, K. Leung, E.M. Lutz, M. Makela, G. Medlin, +C.L. Morris, R.W. Pattie, D. Ries, A. Saunders, P. Schmidt- +Wellenburg, V. Talanov, A.R. Young, B. Wehring, C. White, +M. Wohlmuther, G. Zsigmond, The European Physical Jour- +nal A 54(9), 148 (2018). doi:10.1140/epja/i2018-12594-2 + diff --git a/W9FJT4oBgHgl3EQf5C1I/content/tmp_files/load_file.txt b/W9FJT4oBgHgl3EQf5C1I/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6125a5e3cf9c915780b0145b332843f1a41995d2 --- /dev/null +++ b/W9FJT4oBgHgl3EQf5C1I/content/tmp_files/load_file.txt @@ -0,0 +1,931 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf,len=930 +page_content='EPJ manuscript No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' (will be inserted by the editor) Time-of-flight spectroscopy of ultracold neutrons at the PSI UCN source G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Bison1, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Chen1,2, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Chiu1,2,*, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Daum1, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Doorenbos1,2, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Kirch1,2, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Kletzl1,2, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Lauss1, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Pais1,2, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Rienäcker1,a, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Schmidt-Wellenburg1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Zsigmond1,b 1Paul Scherrer Institut, CH-5232 Villigen-PSI, Switzerland 2ETH Zürich, CH-8092 Zürich, Switzerland current address: Universität Zürich, CH-8057 Zürich, Switzerland Received: date / Accepted: date Abstract The ultracold neutron (UCN) source at the Paul Scherrer Institute (PSI) provides high intensities of storable neutrons for fundamental physics experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The neutron velocity spectrum parallel to the beamline axis was determined by time-of-flight spec- troscopy using a neutron chopper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' In particular, the temporal evolution of the spectrum during neutron pro- duction and UCN storage in the source storage volume was investigated and compared to Monte Carlo simula- tion results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' A softening of the measured spectrum from a mean velocity of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='7(2) m s−1 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='1(1) m s−1 occurred within the first 30 s after the proton beam pulse had im- pinged on the spallation target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' A spectral hardening was observed over longer time scales of one measure- ment day, consistent with the effect of surface degrada- tion of the solid deuterium moderator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 1 Introduction Neutrons are defined to be ultracold if they can be re- flected under all angles of incidence from suitable ma- terial surfaces [1,2,3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Such materials have neutron op- tical potentials of a few hundred neV, corresponding to critical velocities of a few m s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The total reflection from surfaces allows the confinement of ultracold neu- trons (UCNs) in storage bottles for hundreds of seconds, a technique used for the measurement of fundamental properties of the neutron [4,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' These experiments are often statistics limited and it is therefore a world-wide effort [6] to improve the output of UCN sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' A soft energy spectrum and a high number of storable neu- trons is crucial to achieve high UCN densities and long acorresponding author: ingo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='rienaecker@psi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='ch bcorresponding author: geza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='zsigmond@psi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='ch observation times in storage experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The Paul Scherrer Institute (PSI) operates a spal- lation-driven, solid deuterium-based source for UCN [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The source hosts worldwide leading experiments to measure the neutron electric dipole moment [9,10], as well as to search for dark matter candidates, such as axion-like particles [11,12] and oscillations of neutrons into sterile states [13,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The thermal moderation [15], UCN production [16], as well as UCN storage and trans- port [17] in the source were characterized previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' In this paper we report on the measurement of the distri- bution of the longitudinal velocity component v = L/T of UCNs by time-of-flight T spectroscopy along a flight path L behind a neutron chopper mounted at beamport West-1 [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The spectrum obtained from this measure- ment can be used for further calibrations of the UCN source simulation model and will be valuable input for future experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' In section 2 we present the concept and parameters of the measurement setup, consisting of a UCN detec- tion system and a neutron chopper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' We describe the time-of-flight (TOF) data analysis including our back- ground subtraction technique in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' In section 4, we discuss the deduced velocity spectrum and its evo- lution in time during UCN storage in the UCN source volume [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The influence of UCN source operational procedures on the spectrum is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 2 Measurement The measurement setup, consisting of a neutron chop- per, a L = 100 cm flight path, and a neutron detector1 is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 1 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The chopper is based on the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='11668v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='ins-det] 27 Jan 2023 2 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Bison et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 1 a) Drawing of the measurement setup, starting from the electro-polished stainless steel guide that was attached to the beamport shutter, up to the CASCADE1 detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' b) A picture of the chopper gratings aligned in the open position, seen from a perspective looking through the short steel guide in front of the chopper towards the detector in the back (behind the chopper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The picture was taken while the whole setup was removed from the beamport shutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' design described in [18] and uses the opposing linear motion of two titanium gratings to achieve short open- ing times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Each grating has 25, s = 3 mm wide, vertical slits, separated by 5 mm, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 1 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The first grating is moved by a piston from the left, con- nected to a linear motor in the vacuum housing of the chopper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The second grating is connected to a second motor on the right and slides directly behind the first grating with a minimal gap between the gratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The chopper time resolution is determined by its opening function Ω(t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' the fraction of the cross section of the neutron guide that is unblocked by the gratings as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' When aligned, the 3 mm wide slits Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 2 Plot of the chopper trigger and the subsequent photo diode signal to measure the opening function Ω(t) of the chopper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' of the two gratings open a maximum of approximately Ω(0) = 38 % of the cross section of the guide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The accel- eration of the linear motors for the gratings was set to a = 100 m s−2 to achieve a full width at half maximum opening time of ∆FWHM = 2 � s/(2a) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='7 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' This is consistent with a measurement of the opening time (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 2) performed with a photo diode located behind the gratings and a diffuse light source in front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The time offset, ts = 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='2(3) ms, between the maximum of the opening function and the chopper trigger, an electronic signal transmitted by the chopper controller shortly be- fore each opening operation, was also determined by this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Additionally, a calibration measurement [19] of the time offset was performed by measuring the UCN count rates, N1(v) and N2(v), with two different flight path lengths, L1 = 100 cm, L2 = 200 cm, us- ing two identical guides of 100 cm length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' It was found that dN1 dv ( L1 t1−ts ) = dN2 dv ( L2 t2−ts ) at the respective maxima of the normalized velocity distributions, confirming the measured time offset within uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' During our spectroscopy measurements we recorded time-of-flight data during approximately one day of un- interrupted standard operation of the PSI UCN source [7,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' A new measurement cycle was started after each proton beam pulse onto the source’s spallation target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The average proton beam current during a pulse was approximately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='0 mA and the duration was set to 8 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The internal clock of the UCN detector was synchro- nized to the chopper by receiving a coincidence signal between the electronic trigger sent by the chopper and a 1 www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='n-cdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='com/cascade-2d-200 detector a) b) L = 100 cm Ni/Mo 85/15 chopper coated glass guide WCN vacuum housing motor electro-polished gratings aligned stainless steel guide chopper gratings in open positionchopper trigger opening function Q(t) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='0 t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' = 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='2(3) ms arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' units 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='5 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='0 20 40 60 80 0 time after trigger (ms)Time-of-flight spectroscopy of ultracold neutrons at the PSI UCN source 3 0 1 2 3 4 5 6 7 8 9 10 11 12 0 10 20 counts / ms time (s) frame number 0 1 2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' tp = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25) s start proton beam pulse Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 3 Raw UCN time spectrum recorded with the CASCADE1 detector during the first 12 s of one cycle with the chopper oper- ating at 2 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' On the time axis, t = 0 s coincides with the first chopper trigger signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Relative to that, the 8 s long proton beam pulse starts at tp = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25) s, as indicated in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Each cycle is divided into 600 frames, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' time intervals [k tf, (k+1) tf] with k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=', 599 which contain TOF spectra with respect to the time of the corresponding chopper openings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The UCN count rate increases during the proton beam pulse and slowly decreases afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' signal from the proton beam control that rises 1 s before the start of the proton beam pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The chopper was running continuously with a duty cycle of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='5 %, open- ing the neutron guide at a frequency of approximately 2 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The exact time between two consecutive chopper triggers was measured to be tf = 499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='855(5) ms, where the uncertainty denotes the observed stability of the sig- nal and the accuracy of a measurement over multiple days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Therefore, the timing of the recorded TOF spec- tra for each cycle was fixed with respect to the chopper operation, while the timing with respect to the proton beam pulse might have varied by the time between two chopper trigger signals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' by up to ± tf/2 ≈ ± 250 ms, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Several measurements were performed to determine the leakage rate of UCNs through small gaps between the chopper housing and the neutron guides or the grat- ings in the closed position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Taking the ratio of the count rate measured by the detector with closed chopper ver- sus (permanently) open chopper, one finds an average UCN leakage of (4 ± 1) · 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The UCN leakage, and other effects like electronic noise and secondary radia- tion during the proton beam pulse, cause background in the measured TOF spectra, which was fitted and sub- tracted as discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The measurement setup was connected to the West-1 beamport of the PSI UCN source with a 20 cm long electro-polished stainless steel guide with an inner di- ameter of 135 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The guide was connected to the beamport shutter via a stainless steel flange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Similar flanges were used to connect the stainless steel guide to the chopper on one side, and a L = 100 cm glass guide with inner diameter of 130 mm on the other side between chopper and detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The glass guide was sputter-coated with Nickel/Molybdenum 85/15 [7] to obtain a high Fermi potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' A CASCADE 2D U-200 UCN detector1 was attached at the end of the 100 cm glass guide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The time resolution of the detector was set to dt = 1 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The energy acceptance of the detector has a lower cut-off Emin = 54 neV due to its 100 µm thick AlMg3 entrance window [7], corresponding to a lower bound on the velocity component orthogonal to the window of vmin = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='2 m s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 3 Analysis The measured count rate in each time bin dt at time t was averaged over all 286 measurement cycles, recorded in a period of approximately one day, to obtain the average count rate dN dt (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The data was divided into 600 individual frames, as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The TOF spectrum of frame k represents the average spectrum at time tk = k tf −tp after the start of the proton beam pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Each frame contains one TOF spectrum dN dt ��� tk (T), T = mod(t, tf) − ts ∈ [−59, 441] ms, (1) where the TOF axis was shifted by the chopper time offset ts ≈ 59 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The timing uncertainties of δtf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='005 ms and δts = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='3 ms of the chopper trigger signal discussed in section 2 lead to a small systematic uncer- tainty of δT = k δtf + δts on the time of flight in frame k and to an associated uncertainty LδT T 2 on the velocity bins of the deduced velocity spectrum (see section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Figure 4 shows the evolution of the TOF spectrum, its maximum and the mean time of flight during and after the proton beam pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The temporary hardening of the spectrum at approximately 8 s was found to be correlated to the closing of the neutron shutter at the bottom of the UCN source storage volume [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The hardening is likely due to reflections or scattering of very cold neutrons (VCN) with v ≳ 10 m s−1 on the partially closed neutron valve into the West-1 beamline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 4 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Bison et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 4 Evolution of the TOF spectrum during the first 60 s after the start of the 8 s long proton beam pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' A constant back- ground rate Rk, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' (3), was separately determined and removed from the spectrum for each frame k at time tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The mean time of flight ⟨T⟩ (red dashed line) and the maximum T max (black line) of the TOF distribution are indicated in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The combination of a number of n frames yields the average TOF spectrum dN dt ��� tkn tk0 (T) = 1 n kn � k=k0 �dN dt ��� tk (T) − Rk � (2) in the corresponding time interval [tk0, tkn] after the start of the proton beam pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' As a first correction, a subtraction of the constant background was performed by removing a rate Rk of typically around 80 Hz per frame from the TOF spectra for each frame k individ- ually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The residual background rate after subtraction was minimized by choosing Rk such that it is the min- imum of a 30 ms running average of the original TOF spectrum, Rk = min T � 1 30 30 � l=0 dN dt ��� tk (T + l dt) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' (3) The further background fitting and subtraction rou- tine follows the method presented in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' In addition to a constant leakage rate of UCNs through the closed chopper as discussed in section 2, the spectrum also contains UCNs that are reflected non-specularly from surfaces between chopper and detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' After non-specular reflection, the time of flight is no longer a valid measure for the initial velocity parallel to the guide axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' In ad- dition, non-specular reflections and back-scattering of UCNs with velocity components parallel to the guide axis below vmin can lead to an accumulation of UCNs that are quasi-stored with a short storage time con- stant τ in the neutron guide between chopper and de- tector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Eventually, these quasi-stored UCNs may be deflected towards the detector and counted at times T > L vmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='31 s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' later than the nominal maxi- mum time of flight .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Figure 6 shows the average spectrum after subtrac- tion of the constant rate Rk on a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The exponential rate of detected quasi-stored UCNs is clearly visible at times later than the maximum time of flight of specularly reflected UCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' In order to estimate the rate of detected quasi-stored UCNs, we fit the aver- age spectrum at times T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='33 s with an exponential function b(T) = A e−(T −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='33s)/τ, (4) while simultaneously fitting the same model but shifted by one frame time b(T + tf) for T < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='03 s, well before the rising edge of the TOF spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Starting the fit interval at T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='33 s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' slightly later than at the nom- inal maximum time of flight, was required to obtain a low χ2 for a single exponential fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' We confirmed that including a constant offset parameter R in the fit to the average TOF spectra, without previously removing a constant rate Rk for each frame individually, leads to a similar result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' We assume that at the beginning of the frame for T < T max, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' before the maximum of the time-of- flight distribution, the detected rate of non-specularly reflected UCNs follows the same time distribution as the specularly reflected UCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The systematic implica- tions of this assumption are discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Thus, the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 5 Average TOF spectra per cycle and frame, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' (2), during the first approximately 12 s (dashed line) and between 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 s and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 s (solid line) after the start of the proton beam pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The Poisson errors are smaller than the line width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The open and solid markers indicate the mean of the respective distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' In the first time interval, the detection of very cold neutrons (VCN) that are produced during the proton beam pulse is clearly visible by the tail of counts at low time of flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='4 mean max 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='3 time of flight (s) count rate (kHz) 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='0 0 10 20 30 40 50 60 time after start of proton beam pulse (s)6 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 s - 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 s 286 cycles n = 24 frames average count rate (kHz) total cts: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='8 × 106 5 t = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 s - 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 s 4 n = 36 frames total cts: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='1 × 106 3 VCN 2 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='4 time of flight (s)Time-of-flight spectroscopy of ultracold neutrons at the PSI UCN source 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 6 Plot of the average TOF spectrum between 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 s and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 s after the start of the proton beam pulse, demonstrating the background subtraction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The black diamonds are the average of the spectra with previously subtracted constant rate Rk per frame k, Eq (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The red dashed line indicates the fitted background B(T), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' (5), according to the procedure ex- plained in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The average TOF spectrum with the back- ground removed is the difference (blue squares) of the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' function b(T) with best fit parameters (A, τ) is extrap- olated to the region T ≥ T max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' For T < T max we add a fraction ρ of the average TOF spectrum to the contri- bution b(T +tf) of quasi-stored UCNs from the previous frame to obtain a smooth transition of the background function B(T) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 6 dashed red line) before and after the maximum of the TOF spectrum, B|tkn tk0 (T) = = � (1 − ρ) b(T + tf) + ρ dN dt |tkn tk0 (T) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' T < T max b(T) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' T ≥ T max (5) with ρ = b(T max) − b(T max + tf) dN dt |tkn tk0 (T max) − b(T max + tf) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' (6) For the time range depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 6, the ratio of the integrated background, including the constant offsets Rk, to the integral of the final TOF spectrum (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 6, blue markers) is 13 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' We checked the impact of our as- sumption about the time distribution of non-specularly reflected UCNs on the final velocity spectrum by inves- tigating the possibility that there is a significant delay in the arrival of non-specularly reflected UCNs at the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' To do so, we shifted the maximum of the back- ground curve (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 6, red dashed line) by 30 ms, half of the width of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' As a consequence, the mean of the resulting velocity distribution was found to be shifted by at most 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='05 m s−1 (which we included in our uncertainty in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 8) and the maximum of the distribution by up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='3 m s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Similarly, an extreme change of the background model in the signal region to a linear interpolation from T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='1 s to T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='33 s was found to shift the mean velocity by less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='06 m s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 4 Results and discussion Figure 7 shows the final results of the axial velocity spectra dN dv ��� tkn tk0 (v) = L v2 �dN dt ��� tkn tk0 (L/v) − B|tkn tk0 (L/v) � (7) for two time intervals after the proton beam pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The background B was previously subtracted from the TOF spectra as described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' We investigated whether a discrete deconvolution [21] with the resolution func- tion Ω(T), approximated by a box function of ∆FWHM ≈ 8 ms width (see section 2), has an effect on the de- duced velocity spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' We found that the deconvo- lution only has a marginal effect on the falling edge for larger velocities of the final velocity spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Hence, no deconvolution of the TOF spectra with the chopper resolution function was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The evolution of the mean velocity parallel to the guide axis in 5 s intervals during 200 s after the start of the proton beam pulse, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The verti- cal errorbars include the systematic uncertainties due to our model of the background from non-specularly reflected UCNs, as well as the contribution from the uncertainty δT on the time of flight, as discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The mean velocity of the neutrons reduces Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 7 Normalized velocity spectra during the first approxi- mately 12 s (dashed line) and between 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 s and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 s (solid line) after the start of the proton beam pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The open and solid marker indicate the mean of the velocity distributions of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='5(1) m s−1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='2(1) m s−1 in the corresponding time inter- vals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 101 t = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 s - 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 s constant rate Rk subtracted average count rate (kHz) background fit parameters: 100 difference A = 116(1)Hz t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='077(1) s p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='137(5) x2/ndf = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='70 10-1 10-2 10-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='4 time of flight (s)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='5 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' velocity distribution (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=') t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 s - 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 s t = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 s - 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='0 3 4 5 6 7 8 9 10 11 12 13 14 15 velocity (m/s)6 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Bison et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' from approximately 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='7(2) m s−1 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='4(1) m s−1 within the first 12 s after the start of the proton beam pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The initial softening of the spectrum during the proton beam pulse correlates with the increase of UCN den- sity in the source storage volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' This leads to a dom- inance of UCNs over non-storable VCN produced at an approximately constant rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' After approximately 12 s the spectrum consists mostly of UCNs (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 7) and the mean velocity decreases slowly to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='1(1) m s−1 at 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 s, comparable to the typical filling times of storage experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Indeed, a gradual softening of the spec- trum during the storage of UCNs is expected due to velocity dependent losses in the source storage volume and neutron guides [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' We compared our results to the UCN velocity com- ponent parallel to the beamline axis obtained from a MCUCN [22] Monte Carlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' We used the sim- ulation model of the source described and calibrated in [7] and [17] and adjusted the geometry to include the stainless steel and 1 m long glass guide of the chop- per setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The UCN velocity component parallel to the guide axis was tallied at the detector position, behind the AlMg3 entrance window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The surface parameters of the guides were set to those found in [17] for the UCN source beamlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The evolution of the mean ve- locity obtained from the simulation is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 8 and reproduces the observed softening of the velocity spectrum during and after the proton beam pulse well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 8 Evolution of the measured mean velocity (black mark- ers and red errorbars) parallel to the guide axis during 200 s after the start of the 8 s proton beam pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' For the first 12 s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' during and shortly after the pulse, the mean velocity is sig- nificantly higher than at later times, where the velocity slowly reduces from about 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='4 m s−1 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='0 m s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The blue line and shaded region denote the mean and statistical uncertainty of the velocity component parallel to the guide axis at the detector position, obtained from a Monte Carlo simulation with a model of the source described and calibrated in [7,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 9 Ratio of the average velocity spectra between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 s after the proton beam pulse of the last 50 measurement cycles (cycle 236 - 286) and the first 50 cycles (cycle 1 - 50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The error bars indicate the propagated Poisson counting errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' There is also good agreement between the maximum vmax,meas = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='2(1) m s−1 of the measured (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 7) and simulated velocity distributions with vmax,sim = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='3(1) m s−1 in the time interval of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 s to 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 s (and also for earlier or later time intervals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' This indi- cates that the velocity dependent loss mechanisms in the source storage volume and neutron guides are mod- elled correctly by the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' However, the simula- tion appears to systematically underestimate the mean velocity by around 5 %, due to an excess of UCN in the simulated velocity spectrum between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='2 m s−1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='5 m s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' It was found that this can be explained by simplifications in the UCN surface roughness model used for the AlMg3 detector entrance and beamline vac- uum separation windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' In reference [23] it was reported that heat deposi- tion by radiation from the spallation target can cause a built-up of solid deuterium (sD2) frost on the surface of the sD2 moderator over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Simulations showed that frost layers cause a hardening of the UCN energy spec- trum (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 20 of [23]) due to velocity-dependent back- scattering of UCNs from the optical potential of isotrop- ically distributed sD2 disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' A conditioning procedure was developed at the PSI UCN source to restore the surface quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' We studied the evolution of the average velocity spectrum on long time scales by performing our analysis on subsets of measurement cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Figure 9 shows the ratio of the velocity spectrum between 3 and 10 m s−1 obtained from the last 50 cycles, and the first 50 cycles, recorded during roughly one day (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 286 cycles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' A conditioning procedure was applied shortly before the first cycle of our measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' While the mean velocity of the first 50 cycles and the last 50 cy- 9 9 measurement S 2 simulation 8 8 mean velocity (m/s) T t 7 7 TI 6 6 5 5 4 4 3 3 0 25 50 75 100 125 150 175 200 time after start of proton beam pulse (s)t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 s - 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='25 S 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='9 tio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='7 last 50 cycles / first 50 cycles 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='6 3 4 5 6 7 8 9 10 velocity (m/s)Time-of-flight spectroscopy of ultracold neutrons at the PSI UCN source 7 cles differs only by about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='1 m s−1, the ratio shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 9 reduces quickly from close to one for velocities above approximately 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='2 m s−1 to about 75 % for lower velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' This indicates a hardening of the spectrum by additional losses of slower UCN, consistent with the observed effects of surface degradation of the solid deu- terium moderator due to the built-up of sD2 frost layers [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 5 Conclusion The UCN velocity spectrum parallel to the guide axis at beamport West-1 was measured by time-of-flight spec- troscopy using a neutron chopper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' A systematic back- ground subtraction was necessary due to contributions of a constant leakage of UCNs through the closed chop- per and non-specular reflections on surfaces between chopper and detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' We measured the decrease of the mean UCN velocities during and after the proton beam pulse from about 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='7 m s−1 to 5 m s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' We have also determined the evolution of the UCN energy spec- trum during source operation, where sD2 frost build-up on the solid deuterium surface causes additional losses, predominantly of slower UCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' The measurement of the velocity spectra at beam- port West-1 is another step towards the full characteri- zation [15,7,17,16] of the PSI UCN source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Our results show good overall agreement with simulations and will be used for further refinements of the simulation model and in the study of possible improvements of the UCN source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Acknowledgments We acknowledge the PSI proton accelerator operations section, all colleagues who have been contributing to the UCN source operation at PSI, and especially the BSQ group which has been operating the PSI UCN source during the measurements, namely B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Blau, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Erisman and also S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Grünberger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Excellent technical support by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Meier, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Noorda and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Schaufelbühl is acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' This work was supported by the Swiss National Science Foundation Projects 163413, 169596, 172626, 178951, 188700 and 200441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Zeldovich, Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' JETP-9 1389-90 (1959) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Lushchikov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Pokotilovskii, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Strelkov, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Shapiro, JETP Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 9, 23 (1969) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Steyerl, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 29b(1), 33 (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='1016/0370- 2693(69)90127-0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Ignatovich, The Physics of Ultracold Neutrons (Claren- don, Oxford, 1990) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Golub, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Richardson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Lamoreaux, Ultra-Cold Neu- trons (Adam Hilger, Bristol, Philadelphia, and New York, 1991) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Bison, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Daum, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Kirch, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Lauss, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Ries, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Schmidt-Wellenburg, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Zsigmond, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Brenner, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Gel- tenbort, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Jenke, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Zimmer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Beck, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Heil, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Kahlenberg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Karch, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Ross, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Eberhardt, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Gep- pert, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Karpuk, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Reich, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Siemensen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Sobolev, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Trautmann, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' C 95(4), 045503 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='1103/PhysRevC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='045503 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Bison, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Blau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Daum, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Göltl, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Henneck, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Kirch, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Lauss, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Ries, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Schmidt-Wellenburg, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Zsig- mond, The European Physical Journal A 56(2) (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='1140/epja/s10050-020-00027-w 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Lauss, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Blau, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 5, 004 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='21468/scipostphysproc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='004 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Abel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='024 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Abel, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Ayres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Ban, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Bison, K.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Lauer, Investigation of a superthermal ultracold neutron source based on a solid deuterium converter for the TRIGA mainz reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' thesis, Universität Mainz, Germany (2010) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Rienäcker, Improving ultracold neutron yields and the search for mirror neutrons at the PSI UCN source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' thesis, ETH Zurich (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='3929/ethz-b-000579840 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Daum, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Franke, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Geltenbort, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Gutsmiedl, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Ivanov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Karch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Kasprzak, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Kirch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Kraft, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Lauer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Lauss, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Mueller, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Paul, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Schmidt-Wellenburg, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Zechlau, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Zsigmond, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Methods A 741(0), 71 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='nima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='050 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Altarev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Daum, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Frei, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Gutsmiedl, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Ham- pel, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Hartmann, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Heil, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Knecht, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Kratz, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Lauer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Meier, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Paul, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Schmidt, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Sobolev, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Wiehl, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Zsigmond, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' A 37(1), 9 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='1140/epja/i2008-10604-8 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Zsigmond, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Methods A 881, 16 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='nima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='065 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Anghel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Bailey, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Bison, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Blau, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Brous- sard, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FJT4oBgHgl3EQf5C1I/content/2301.11668v1.pdf'} +page_content=' Clayton, C.' metadata={'source': 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100644 index 0000000000000000000000000000000000000000..a65d883f740115731b134fb10b9598685854b99c --- /dev/null +++ b/_9AzT4oBgHgl3EQfhfxv/content/tmp_files/2301.01485v1.pdf.txt @@ -0,0 +1,357 @@ +arXiv:2301.01485v1 [math.DG] 4 Jan 2023 +Restriction of Donaldson’s functional to diagonal +metrics on Higgs bundles with non-holomorphic +Higgs fields +Natsuo Miyatake +Abstract +We consider a Higgs bundle over a compact K¨ahler manifold with a +smooth, non-holomorphic Higgs field. We assume that the holomorphic +vector bundle decomposes into a direct sum of holomorphic line bundles. +Under an assumption on the non-holomorphic Higgs field, we provide some +necessary and sufficient conditions for Donaldson’s functional to attain a +minimum at the set of diagonal Hermitian metrics for the decomposition of +the vector bundle. In particular, we show that we can solve the Hermitian- +Einstein equation under a strong assumption even if the Higgs field is +non-holomorphic. +1 +Introduction +A Higgs bundle over a K¨ahler manifold (X, ωX) is a pair (E, Φ) consisting of +a holomorphic vector bundle E and a holomorphic section Φ of EndE ⊗ �1,0 +satisfying Φ ∧ Φ = 0. The holomorphic section Φ is called a Higgs field. In this +paper, we consider Higgs bundles with non-holomorphic Higgs fields, which is +not usually assumed in the definition of Higgs bundles. +The Hermitian-Einstein equation and Donaldson’s functional can be defined +even if the Higgs field is non-holomorphic (see Section 2). Therefore, two ques- +tions arise: when can the Hermitian-Einstein equation be solved, and does it +makes sense to try to solve the equation? Neither issue seems easy to resolve, +but with regard to the latter question, it is easy to observe the following: Sup- +pose that dimC X = 1. Let (E, Φ) be a Higgs bundle with a non-holomorphic +Higgs field Φ, and h a solution to the Hermitian-Einstein equation. Then, for +every open subset U ⊆ X, if ¯∂Φ = 0 on U, then (E |U , Φ |U , h |U) is a harmonic +bundle on U. For example, let (E, Φ) → X be a usual Higgs bundle with a holo- +morphic Higgs field Φ and t : X → C a smooth function. Then, in general, tΦ is +not a globally holomorphic Higgs field and (E, tΦ, h) is not a harmonic bundle +for a solution h to the Hermitian-Einstein equation of (E, tΦ). However, on an +open subset U such that t is constant on U, (E |U , tΦ |U , h |U) is a harmonic +bundle. +1 + +We consider the problem of determining when the Hermitian-Einstein equa- +tion can be solved. We suppose that X is compact and that the holomorphic +vector bundle decomposes into a direct sum of holomorphic line bundles. In this +paper, by applying [6, Theorem 1], under an assumption on the non-holomorphic +Higgs field, we give some necessary and sufficient conditions for Donaldson’s +functional to attain a minimum at the set of diagonal Hermitian metrics con- +cerning the decomposition of the vector bundle. In particular, we show that we +can solve the Hermitian-Einstein equation under a strong assumption even if +the Higgs field is non-holomorphic. In order to state the theorem precisely, we +introduce some notations: +• Let E → (X, ωX) be a holomorphic vector bundle over a compact K¨ahler +manifold and Φ a smooth section of EndE ⊗ �1,0 satisfying Φ ∧ Φ = 0. +Note that we do not use the integrability condition in the following. We +also suppose that c1(E) = 0 for simplicity. +• We denote by ΛωX the adjoint of ωX∧. +• Suppose that the holomorphic vector bundle E decomposes as E = L1 ⊕ +· · · ⊕ Lr with holomorphic line bundles L1, . . . , Lr → X. We decompose +Φ as Φ = Φ0 + � +i,j=1,...,r Φi,j, where Φ0 is the diagonal part and Φi,j is +a (1,0)-form which takes values in L−1 +j Li. +• Let V be an r − 1-dimensional real vector space defined as V := {x = +(x1, . . . , xr) ∈ Rr | x1 + · · · + xr = 0}. +• We define vectors vi,j ∈ V (i, j = 1, . . . , r) as vi,j := ui − uj, where we +denote by u1, . . . , ur the canonical basis of Rr. +• For each j = 1, . . . , r, we define a real number γj as γj := degωX(Lj) and +we also define a vector γ ∈ V as γ := (γ1, . . . , γr). +• We define a space HE of Hermitian metrics as +HE := {h | h is a smooth Hermitian metric on E such that det(h) = 1}. +We also define a subset diagL(HE) ⊆ HE as +diagL(HE) := {h ∈ HE | h splits into h = (h1, . . . , hr) for the decomposition}. +• For an h ∈ HE, we denote by Fh the curvature of h, and by Φ∗h the +adjoint of Φ with respect to the metric h. +• We fix a metric K = (K1, . . . , Kr) ∈ diagL(HE). +Then the following holds: +Theorem 1. Suppose that for each i, j = 1, . . . , r, if Φi,j ̸= 0, then log |Φi,j|2 +K,ωX +is integrable. Then the following are equivalent: +2 + +(i) Donaldson’s functional M(·, K) attains a minimum on diagL(HE); +(ii) For any geodesic (ht)t∈R such that ht ∈ diagL(HE) for all t ∈ R, if +M(ht, K) is not a constant, then limt→∞ M(ht, K) = ∞. +(iii) There exists an h ∈ diag(HE) such that +pr(ΛωX(Fh + [Φ ∧ Φ∗h])) = 0, +(1) +where pr : EndE → EndE is the projection to the diagonal part. +(iv) Donaldson’s functional M(·, K) is bounded below at diagL(HE) and there +exist constants C, C′, C′′ such that +|ξ|L2 ≤ (M(h, K) + C)2 + C′M(h, K) + C′′ for all h ∈ diagL(HE), +where we denote by ξ the pair (f1, . . . , fr) of functions such that h = +(ef1K1, . . . , efrKr). +(v) The following holds: +−γ ∈ +� +i,j=1,...,r, +Φi,j̸=0 +R>0vi,j. +(2) +Moreover, if one of the above conditions is satisfied, Donaldson’s functional +M(·, K) has a critical point on HE if and only if for an h ∈ diagL(HE) +the off-diagonal part of ΛωX[Φ ∧ Φ∗h] vanishes. +Corollary 2. Suppose that the non-holomorphic Higgs field is of the following +form: +Φ = + + + + + +0 +Φr +Φ1 +0 +... +... +Φr−1 +0 + + + + + . +(3) +Then, if Φi ̸= 0 and log |Φi|2 +K,ωX is integrable for all i = 1, . . . , r, then the +Hermitian-Einstein equation has a unique diagonal solution. +Remark 3. Higgs bundles whose Higgs fields have the form given in (3) are +called cyclic Higgs bundles (see [6, 7, 8] and the references therein). +Remark 4. For the relationship between the stability condition and condition +(2), we refer the reader to [7]. +Remark 5. The author is not aware of any work that extends Hitchin and +Simpson’s theorem [4, 9] to cases where the Higgs field is non-holomorphic. +However, there is a work [1] that studies the extension to the case where the +base manifolds are not complex when the Higgs field is trivial. +3 + +Remark 6. For the proof of the implications (i) ⇒ (ii), (ii) ⇔ (v), (i) ⇔ (iii), +(iv) ⇒ (i), we need not assume that for each i, j = 1, . . . , r, if Φi,j ̸= 0, then +log |Φi,j|2 +K,ωX is integrable. +Remark 7. Even if the Higgs field is holomorphic, the author does not know +how to show from the definition of stability for a more general geodesic and for +a more general Higgs bundle that the functional diverges when the time of the +geodesic goes to infinity (for the case where the Higgs field is trivial, see [3, 5]). +2 +Hermitian-Einstein equation and Donaldson’s +functional +In order to clarify the meaning of the terms we use and in order to avoid con- +fusion, this section gives precise definitions of each concept. Let (X, ωX) be a +K¨ahler manifold. We suppose that X is compact for simplicity. Let E → X be +a holomorphic vector bundle and Φ a smooth section of EndE ⊗ �1,0 satisfying +Φ ∧ Φ = 0. +Definition 8 (Hermitian-Einstein equation). We call the following PDE for a +Hermitian metric h on E the Hermitian-Einstein equation: +ΛωX(F ⊥ +h + [Φ ∧ Φ∗h]) = 0, +where we denote by F ⊥ +h the trace-free part of the curvature. +Definition 9 (Donaldson’s functional). For smooth Hermitian metrics h and K +on E such that det(h) = det(K), we define a real number M(h, K) as follows: +M(h, K) := +� 1 +0 +dt +� +X +√ +−1Tr(ΛωX(F ⊥ +ht + [Φ ∧ Φ∗ht])g−1 +t +∂tgt), +where (ht)0≤t≤1 is a piecewise smooth family of Hermitian metrics such that +det(ht) = det(K) and gt : E → E is a unique Hermitian endmorphism with +respect to K and ht satisfying K(gt·, ·) = ht(·, ·). As with the usual Donalson’s +functional [2, 9], M(h, K) does not depend on the choice of the path (ht)0≤t≤1. +We call M(·, ·) Donalson’s functional. +As with the usual Donaldson’s functional, the following holds: +Proposition 10. Let K be a smooth metric on E and (ht)t∈R a smooth family +of Hermitian metrics such that det(ht) = det(K) for all t ∈ R. We denote +by gt : E → E the unique Hermitian endmorphism with respect to K and ht +satisfying K(gt·, ·) = ht(·, ·). Then the following holds: +1. The following holds: +d +dtM(ht, K) = +� +X +√ +−1Tr(ΛωX(F ⊥ +ht + [Φ ∧ Φ∗ht])g−1 +t +∂tgt). +4 + +2. Suppose that there exists a Hermitian endmorophism s : E → E with +respect to K such that gt = ets for all t ∈ R. Then the following holds: +d2 +dt2 M(ht, K) = +� +X +|(¯∂ + ad(Φ))s|2 +ht, +where ad(Φ) denotes [Φ, ·]. +3 +Proof +Proof of Theorem 1. Proof of Theorem 1 is obtained by applying [6, Theo- +rem 1]. +Consider condition (iii) of Theorem 1. +Equation (1) for a metric +(ef1K1, . . . , efrKr) is the following: +∆ωXξ + +r +� +j=1 +4|Φi,j|2 +K,ωXe(vi,j,ξ)vi,j = −2 +√ +−1ΛωXFK, +(4) +where ∆ωX denotes the geometric Laplacian, and ξ is defined as ξ := (f1, . . . , fr). +Equation (4) is a special case of equations considered in [6], and Donalson’s +functional restricted to diagL(HE) coincides with the functional introduced in +[6]. Then from [6, Theorem 1] and its proof, one can check that conditions in +Theorem 1 are equivalent. +4 +Non-holomorphic splittings +We discuss a little about non-holomorphic splittings of the vector bundle E. We +introduce some notations: +• Let M1, . . . , Mr be smooth sublinebundles of E such that E = M1 ⊕ · · ·⊕ +Mr. We also fix a holomorphic splitting of E: E = L1 ⊕ · · · ⊕ Lr. +• Let diagL(HE) (resp. diagM(HE)) be the space of diagonal metrics con- +cerning the decomposition E = L1 ⊕ · · · ⊕ Lr (resp. E = M1 ⊕ · · · ⊕ Mr). +• We fix an initial metric K = (K1, . . . , Kr) ∈ diagM(HE). We also fix a +metric href ∈ diagL(HE). +• We denote by Φ = Φ0 + � +i,j=1,...,r Φi,j the decomposition of Φ corre- +sponding to the splitting E = M1 ⊕ · · · ⊕ Mr. +• We normalize the volume of X as 1. +Then the following holds: +Proposition 11. Let h ∈ diagM(HE). Then the following holds: +M(h, K) = +� +X +(Ψ(s)(¯∂s), ¯∂s)href + +√ +−1 +� +X +Tr((δ(ξ) − δ(ξ))ΛωXFhref ) ++ +� +i,j=1,...,r +� +X +2|Φi,j|2 +K,ωXe(vi,j,ξ) + (2πγ, δ(ξ)) + C, +(5) +5 + +where: +• ξ = (f1, . . . , fr) is a pair of functions satisfying h = (ef1K1, . . . , efrKr), +• s is the unique endomorphism satisfying h = href(es·, ·) and δ(ξ) the pro- +jection of s to the diagonal part of EndE of the decomposition induced +from E = L1 ⊕ · · · ⊕ Lr. +• δ(ξ) is the average � +X δ(ξ) of δ(ξ). +• γ := (degωX(L1), . . . , degωX(Lr)) is concerned to be a diagonal matrix. +• The definition of Ψ : R × R → R is the same as that of [9, P.882] and the +definition of Ψ(s)(¯∂s) is the same as that of [9, pp.879-882]. +• C is a constant which is independent of ξ. +Proof. We decompose Donaldson’s functional (see Definition 9) M(h, K) as +follows: +M(h, K) = M1(h, K) + M2(h, K), +M1(h, K) := +� 1 +0 +dt +� +X +√ +−1Tr(ΛωXFhtg−1 +t +∂tgt), +M2(h, K) := +� 1 +0 +dt +� +X +√ +−1Tr(ΛωX([Φ ∧ Φ∗ht])g−1 +t +∂tgt). +Then the second term M2(h, K) coincides with � +i,j=1,...,r +� +X 2|Φi,j|2 +K,ωXe(vi,j,ξ). +The first term M1(h, K) is further decomposed as follows: +M1(h, K) = M1(h, href) + M1(href, K). +We set C := M1(href, K). We decompose M1(h, href) as +M1(h, href) = +� +X +(Ψ(s)¯∂s, ¯∂s)href + +√ +−1 +� +X +Tr(sΛωXFhref ) += +� +X +(Ψ(s)¯∂s, ¯∂s)href + +√ +−1 +� +X +Tr((δ(ξ) − δ(ξ))ΛωXFhref ) + (2πγ, δ(ξ)). +Then we have (5). +Remark 12. For a non-holomorphic splitting, even if the Higgs field is holo- +morphic and Φi,j ̸= 0, log |Φi,j|2 +K,ωX is not integrable in general. +Acknowledgements. I would like to express my gratitude to Ryushi Goto and +Hisashi Kasuya for their valuable discussions and many supports. I am very +grateful to Yoshinori Hashimoto for his valuable discussions, many supports, +helpful comments on this paper, and for informing me of the paper [1]. I would +also like to express my gratitude to Qiongling Li for answering my questions. +6 + +References +[1] P. De Bartolomeis and G. Tian, Stability of complex vector bundles, Journal +of Differential Geometry 43.2 (1996): 231-275. +[2] S.K. Donaldson, Anti self-dual Yang-Mills connections over complex alge- +braic surfaces and stable vector bundles, Proceedings of the London Math- +ematical Society 3.1 (1985): 1-26. +[3] Y. Hashimoto and J. Keller, Quot-scheme limit of Fubini-Study metrics +and Donaldson’s functional for vector bundles, ´Epijournal de G´eom´etrie +Alg´ebrique 5 (2022). +[4] N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. Lon- +don Math. Soc. (3) 55 (1987), no. 1, 59–126. +[5] M. Jonsson, N. McCleerey, and S. Shivaprasad, Geodesic Rays in the +Donaldson-Uhlenbeck-Yau Theorem, arXiv:2210.09246 (2022). +[6] N. Miyatake, Generalized Kazdan-Warner equations associated with a lin- +ear action of a torus on a complex vector space, Geom Dedicata 214, +651–669 (2021). +[7] N. Miyatake, On diagonal pluriharmonic metrics of G-Higgs bundles, +arXiv:2111.07330 (2021). +[8] N. Miyatake, Generalizations of Hermitian-Einstein equation of cyclic +Higgs bundles, their heat equation, and inequality estimates, preprint. +[9] C.T. Simpson, Constructing variations of Hodge structure using Yang-Mills +theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), +no. 4, 867–918. +E-mail address 1: natsuo.m.math@gmail.com +E-mail address 2: n-miyatake@imi.kyushu-u.ac.jp +Institute of Mathematics for Industry, Kyushu University 744 Motooka, Fukuoka +819-0395, Japan +7 + diff --git a/_9AzT4oBgHgl3EQfhfxv/content/tmp_files/load_file.txt b/_9AzT4oBgHgl3EQfhfxv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..601f153b3ebc8cecc1712d3658b129a3caad40f2 --- /dev/null +++ b/_9AzT4oBgHgl3EQfhfxv/content/tmp_files/load_file.txt @@ -0,0 +1,232 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf,len=231 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='01485v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='DG] 4 Jan 2023 Restriction of Donaldson’s functional to diagonal metrics on Higgs bundles with non-holomorphic Higgs fields Natsuo Miyatake Abstract We consider a Higgs bundle over a compact K¨ahler manifold with a smooth, non-holomorphic Higgs field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' We assume that the holomorphic vector bundle decomposes into a direct sum of holomorphic line bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Under an assumption on the non-holomorphic Higgs field, we provide some necessary and sufficient conditions for Donaldson’s functional to attain a minimum at the set of diagonal Hermitian metrics for the decomposition of the vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' In particular, we show that we can solve the Hermitian- Einstein equation under a strong assumption even if the Higgs field is non-holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' 1 Introduction A Higgs bundle over a K¨ahler manifold (X, ωX) is a pair (E, Φ) consisting of a holomorphic vector bundle E and a holomorphic section Φ of EndE ⊗ �1,0 satisfying Φ ∧ Φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' The holomorphic section Φ is called a Higgs field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' In this paper, we consider Higgs bundles with non-holomorphic Higgs fields, which is not usually assumed in the definition of Higgs bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' The Hermitian-Einstein equation and Donaldson’s functional can be defined even if the Higgs field is non-holomorphic (see Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Therefore, two ques- tions arise: when can the Hermitian-Einstein equation be solved, and does it makes sense to try to solve the equation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Neither issue seems easy to resolve, but with regard to the latter question, it is easy to observe the following: Sup- pose that dimC X = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Let (E, Φ) be a Higgs bundle with a non-holomorphic Higgs field Φ, and h a solution to the Hermitian-Einstein equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Then, for every open subset U ⊆ X, if ¯∂Φ = 0 on U, then (E |U , Φ |U , h |U) is a harmonic bundle on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' For example, let (E, Φ) → X be a usual Higgs bundle with a holo- morphic Higgs field Φ and t : X → C a smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Then, in general, tΦ is not a globally holomorphic Higgs field and (E, tΦ, h) is not a harmonic bundle for a solution h to the Hermitian-Einstein equation of (E, tΦ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' However, on an open subset U such that t is constant on U, (E |U , tΦ |U , h |U) is a harmonic bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' 1 We consider the problem of determining when the Hermitian-Einstein equa- tion can be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' We suppose that X is compact and that the holomorphic vector bundle decomposes into a direct sum of holomorphic line bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' In this paper, by applying [6, Theorem 1], under an assumption on the non-holomorphic Higgs field, we give some necessary and sufficient conditions for Donaldson’s functional to attain a minimum at the set of diagonal Hermitian metrics con- cerning the decomposition of the vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' In particular, we show that we can solve the Hermitian-Einstein equation under a strong assumption even if the Higgs field is non-holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' In order to state the theorem precisely, we introduce some notations: Let E → (X, ωX) be a holomorphic vector bundle over a compact K¨ahler manifold and Φ a smooth section of EndE ⊗ �1,0 satisfying Φ ∧ Φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Note that we do not use the integrability condition in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' We also suppose that c1(E) = 0 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' We denote by ΛωX the adjoint of ωX∧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Suppose that the holomorphic vector bundle E decomposes as E = L1 ⊕ · · ⊕ Lr with holomorphic line bundles L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' , Lr → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' We decompose Φ as Φ = Φ0 + � i,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=',r Φi,j, where Φ0 is the diagonal part and Φi,j is a (1,0)-form which takes values in L−1 j Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Let V be an r − 1-dimensional real vector space defined as V := {x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' , xr) ∈ Rr | x1 + · · · + xr = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' We define vectors vi,j ∈ V (i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' , r) as vi,j := ui − uj, where we denote by u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' , ur the canonical basis of Rr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' For each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' , r, we define a real number γj as γj := degωX(Lj) and we also define a vector γ ∈ V as γ := (γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' , γr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' We define a space HE of Hermitian metrics as HE := {h | h is a smooth Hermitian metric on E such that det(h) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' We also define a subset diagL(HE) ⊆ HE as diagL(HE) := {h ∈ HE | h splits into h = (h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' , hr) for the decomposition}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' For an h ∈ HE, we denote by Fh the curvature of h, and by Φ∗h the adjoint of Φ with respect to the metric h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' We fix a metric K = (K1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' , Kr) ∈ diagL(HE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Then the following holds: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Suppose that for each i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' , r, if Φi,j ̸= 0, then log |Φi,j|2 K,ωX is integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Then the following are equivalent: 2 (i) Donaldson’s functional M(·, K) attains a minimum on diagL(HE);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' (ii) For any geodesic (ht)t∈R such that ht ∈ diagL(HE) for all t ∈ R, if M(ht, K) is not a constant, then limt→∞ M(ht, K) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' (iii) There exists an h ∈ diag(HE) such that pr(ΛωX(Fh + [Φ ∧ Φ∗h])) = 0, (1) where pr : EndE → EndE is the projection to the diagonal part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' (iv) Donaldson’s functional M(·, K) is bounded below at diagL(HE) and there exist constants C, C′, C′′ such that |ξ|L2 ≤ (M(h, K) + C)2 + C′M(h, K) + C′′ for all h ∈ diagL(HE), where we denote by ξ the pair (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' , fr) of functions such that h = (ef1K1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' , efrKr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' (v) The following holds: −γ ∈ � i,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=',r, Φi,j̸=0 R>0vi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' (2) Moreover, if one of the above conditions is satisfied, Donaldson’s functional M(·, K) has a critical point on HE if and only if for an h ∈ diagL(HE) the off-diagonal part of ΛωX[Φ ∧ Φ∗h] vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Suppose that the non-holomorphic Higgs field is of the following form: Φ = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed 0 Φr Φ1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Φr−1 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' (3) Then, if Φi ̸= 0 and log |Φi|2 K,ωX is integrable for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' , r, then the Hermitian-Einstein equation has a unique diagonal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Higgs bundles whose Higgs fields have the form given in (3) are called cyclic Higgs bundles (see [6, 7, 8] and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' For the relationship between the stability condition and condition (2), we refer the reader to [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' The author is not aware of any work that extends Hitchin and Simpson’s theorem [4, 9] to cases where the Higgs field is non-holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' However, there is a work [1] that studies the extension to the case where the base manifolds are not complex when the Higgs field is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' 3 Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' For the proof of the implications (i) ⇒ (ii), (ii) ⇔ (v), (i) ⇔ (iii), (iv) ⇒ (i), we need not assume that for each i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' , r, if Φi,j ̸= 0, then log |Φi,j|2 K,ωX is integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Even if the Higgs field is holomorphic, the author does not know how to show from the definition of stability for a more general geodesic and for a more general Higgs bundle that the functional diverges when the time of the geodesic goes to infinity (for the case where the Higgs field is trivial, see [3, 5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' 2 Hermitian-Einstein equation and Donaldson’s functional In order to clarify the meaning of the terms we use and in order to avoid con- fusion, this section gives precise definitions of each concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Let (X, ωX) be a K¨ahler manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' We suppose that X is compact for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Let E → X be a holomorphic vector bundle and Φ a smooth section of EndE ⊗ �1,0 satisfying Φ ∧ Φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Definition 8 (Hermitian-Einstein equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' We call the following PDE for a Hermitian metric h on E the Hermitian-Einstein equation: ΛωX(F ⊥ h + [Φ ∧ Φ∗h]) = 0, where we denote by F ⊥ h the trace-free part of the curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Definition 9 (Donaldson’s functional).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' For smooth Hermitian metrics h and K on E such that det(h) = det(K), we define a real number M(h, K) as follows: M(h, K) := � 1 0 dt � X √ −1Tr(ΛωX(F ⊥ ht + [Φ ∧ Φ∗ht])g−1 t ∂tgt), where (ht)0≤t≤1 is a piecewise smooth family of Hermitian metrics such that det(ht) = det(K) and gt : E → E is a unique Hermitian endmorphism with respect to K and ht satisfying K(gt·, ·) = ht(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' As with the usual Donalson’s functional [2, 9], M(h, K) does not depend on the choice of the path (ht)0≤t≤1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' We call M(·, ·) Donalson’s functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' As with the usual Donaldson’s functional, the following holds: Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Let K be a smooth metric on E and (ht)t∈R a smooth family of Hermitian metrics such that det(ht) = det(K) for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' We denote by gt : E → E the unique Hermitian endmorphism with respect to K and ht satisfying K(gt·, ·) = ht(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Then the following holds: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' The following holds: d dtM(ht, K) = � X √ −1Tr(ΛωX(F ⊥ ht + [Φ ∧ Φ∗ht])g−1 t ∂tgt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Suppose that there exists a Hermitian endmorophism s : E → E with respect to K such that gt = ets for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Then the following holds: d2 dt2 M(ht, K) = � X |(¯∂ + ad(Φ))s|2 ht, where ad(Φ) denotes [Φ, ·].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' 3 Proof Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Proof of Theorem 1 is obtained by applying [6, Theo- rem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Consider condition (iii) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Equation (1) for a metric (ef1K1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' , efrKr) is the following: ∆ωXξ + r � j=1 4|Φi,j|2 K,ωXe(vi,j,ξ)vi,j = −2 √ −1ΛωXFK, (4) where ∆ωX denotes the geometric Laplacian, and ξ is defined as ξ := (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' , fr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Equation (4) is a special case of equations considered in [6], and Donalson’s functional restricted to diagL(HE) coincides with the functional introduced in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Then from [6, Theorem 1] and its proof, one can check that conditions in Theorem 1 are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' 4 Non-holomorphic splittings We discuss a little about non-holomorphic splittings of the vector bundle E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' We introduce some notations: Let M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' , Mr be smooth sublinebundles of E such that E = M1 ⊕ · · ·⊕ Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' We also fix a holomorphic splitting of E: E = L1 ⊕ · · · ⊕ Lr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Let diagL(HE) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' diagM(HE)) be the space of diagonal metrics con- cerning the decomposition E = L1 ⊕ · · · ⊕ Lr (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' E = M1 ⊕ · · · ⊕ Mr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' We fix an initial metric K = (K1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' , Kr) ∈ diagM(HE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' We also fix a metric href ∈ diagL(HE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' We denote by Φ = Φ0 + � i,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=',r Φi,j the decomposition of Φ corre- sponding to the splitting E = M1 ⊕ · · · ⊕ Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' We normalize the volume of X as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Then the following holds: Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Let h ∈ diagM(HE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Then the following holds: M(h, K) = � X (Ψ(s)(¯∂s), ¯∂s)href + √ −1 � X Tr((δ(ξ) − δ(ξ))ΛωXFhref ) + � i,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=',r � X 2|Φi,j|2 K,ωXe(vi,j,ξ) + (2πγ, δ(ξ)) + C, (5) 5 where: ξ = (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' , fr) is a pair of functions satisfying h = (ef1K1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' , efrKr), s is the unique endomorphism satisfying h = href(es·, ·) and δ(ξ) the pro- jection of s to the diagonal part of EndE of the decomposition induced from E = L1 ⊕ · · · ⊕ Lr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' δ(ξ) is the average � X δ(ξ) of δ(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' γ := (degωX(L1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' , degωX(Lr)) is concerned to be a diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' The definition of Ψ : R × R → R is the same as that of [9, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='882] and the definition of Ψ(s)(¯∂s) is the same as that of [9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='879-882].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' C is a constant which is independent of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' We decompose Donaldson’s functional (see Definition 9) M(h, K) as follows: M(h, K) = M1(h, K) + M2(h, K), M1(h, K) := � 1 0 dt � X √ −1Tr(ΛωXFhtg−1 t ∂tgt), M2(h, K) := � 1 0 dt � X √ −1Tr(ΛωX([Φ ∧ Φ∗ht])g−1 t ∂tgt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Then the second term M2(h, K) coincides with � i,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=',r � X 2|Φi,j|2 K,ωXe(vi,j,ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' The first term M1(h, K) is further decomposed as follows: M1(h, K) = M1(h, href) + M1(href, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' We set C := M1(href, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' We decompose M1(h, href) as M1(h, href) = � X (Ψ(s)¯∂s, ¯∂s)href + √ −1 � X Tr(sΛωXFhref ) = � X (Ψ(s)¯∂s, ¯∂s)href + √ −1 � X Tr((δ(ξ) − δ(ξ))ΛωXFhref ) + (2πγ, δ(ξ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Then we have (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Remark 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' For a non-holomorphic splitting, even if the Higgs field is holo- morphic and Φi,j ̸= 0, log |Φi,j|2 K,ωX is not integrable in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' I would like to express my gratitude to Ryushi Goto and Hisashi Kasuya for their valuable discussions and many supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' I am very grateful to Yoshinori Hashimoto for his valuable discussions, many supports, helpful comments on this paper, and for informing me of the paper [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' I would also like to express my gratitude to Qiongling Li for answering my questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' 6 References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' De Bartolomeis and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Tian, Stability of complex vector bundles, Journal of Differential Geometry 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='2 (1996): 231-275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Donaldson, Anti self-dual Yang-Mills connections over complex alge- braic surfaces and stable vector bundles, Proceedings of the London Math- ematical Society 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='1 (1985): 1-26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' [3] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Hashimoto and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Keller, Quot-scheme limit of Fubini-Study metrics and Donaldson’s functional for vector bundles, ´Epijournal de G´eom´etrie Alg´ebrique 5 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' [4] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Hitchin, The self-duality equations on a Riemann surface, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Lon- don Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' (3) 55 (1987), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' 1, 59–126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Jonsson, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' McCleerey, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Shivaprasad, Geodesic Rays in the Donaldson-Uhlenbeck-Yau Theorem, arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='09246 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' [6] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Miyatake, Generalized Kazdan-Warner equations associated with a lin- ear action of a torus on a complex vector space, Geom Dedicata 214, 651–669 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' [7] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Miyatake, On diagonal pluriharmonic metrics of G-Higgs bundles, arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='07330 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' [8] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Miyatake, Generalizations of Hermitian-Einstein equation of cyclic Higgs bundles, their heat equation, and inequality estimates, preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' [9] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' 1 (1988), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' 4, 867–918.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content=' E-mail address 1: natsuo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='math@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='com E-mail address 2: n-miyatake@imi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='kyushu-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} +page_content='jp Institute of Mathematics for Industry, Kyushu University 744 Motooka, Fukuoka 819-0395, Japan 7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AzT4oBgHgl3EQfhfxv/content/2301.01485v1.pdf'} diff --git a/_tAzT4oBgHgl3EQfFvq5/content/tmp_files/2301.01017v1.pdf.txt b/_tAzT4oBgHgl3EQfFvq5/content/tmp_files/2301.01017v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a05ff4eaf8fa2fa28d3bdedd3b5d589b7968024a --- /dev/null +++ b/_tAzT4oBgHgl3EQfFvq5/content/tmp_files/2301.01017v1.pdf.txt @@ -0,0 +1,1297 @@ +Through-life Monitoring of Resource-constrained Systems and +Fleets +Felipe Montana1, Adam Hartwell1, Will Jacobs1, Visakan Kadirkamanathan1, Andrew R +Mills1, and Tom Clark2 +1Department of Automatic Control and Systems Engineering, University of Sheffield, UK +2Rolls-Royce Plc, UK +Abstract +A Digital Twin (DT) is a simulation of a physical +system that provides information to make decisions +that add economic, social or commercial value. The +behaviour of a physical system changes over time, a +DT must therefore be continually updated with data +from the physical systems to reflect its changing be- +haviour. +For resource-constrained systems, updat- +ing a DT is non-trivial because of challenges such +as on-board learning and the off-board data transfer. +This paper presents a framework for updating data- +driven DTs of resource-constrained systems geared +towards system health monitoring. The proposed so- +lution consists of: (1) an on-board system running a +light-weight DT allowing the prioritisation and par- +simonious transfer of data generated by the physical +system; and (2) off-board robust updating of the DT +and detection of anomalous behaviours. +Two case +studies are considered using a production gas turbine +engine system to demonstrate the digital represen- +tation accuracy for real-world, time-varying physical +systems. +1 +Introduction +Digital Twin (DT) is an emerging technology that +has recently gained attention due to the rapid de- +velopment of simulations, data acquisition and data +communication that trigger interaction between the +physical and virtual spaces [1]. A DT is a simulation +of an as-built system, e.g., a vehicle or factory, that +accurately represents its corresponding twin [2]. In +contrast to a simple model or simulation, a DT is a +living and evolving model that follows the lifecycle of +its physical twin [3]. This is done by integrating data +collected from the system’s sensors, environment, his- +toric maintenance data, and available system knowl- +edge. +Moreover, DTs can contain a description of +the structure, functions, behaviour and control of the +physical system [4]. The information provided by the +DT facilitates the processes of making decisions that +will affect the physical asset [5] and adds economic, +social or commercial value to stakeholders. +The DT paradigm is used in a variety of applica- +tions and sectors, including product design and fac- +tory optimisation [6], prediction of aircraft structural +life [7], and monitoring of automotive braking sys- +tems [8] among others. Another example is the man- +agement and monitoring of a fleet of assets. In this +case, information based on the fleet distribution is +inadequate to assess individual systems due to the +variability in the usage, manufacturing and material +properties of different assets [9]. The availability of +individually tailored DTs is therefore desirable to ac- +curately predict future performance against require- +ments and detect deviation from the current system +behaviour driven by emerging faults. +The behaviour of a physical system is expected to +change over time due to normal degradation, mechan- +1 +arXiv:2301.01017v1 [cs.LG] 3 Jan 2023 + +ical modifications, etc. The result is a drift between +the real system and its twin. Another cause of devia- +tion between a real system and its twin, particularly +in data-driven DTs, is operation in conditions previ- +ously unseen by the twin. Both scenarios motivate +the update of the DT using data acquired from the +physical system to provide a better representation of +the current state of the system. +For applications such as predictive maintenance +and monitoring, updates to the DT mainly consist +of updating model parameters. Different methods for +parameter updating have been proposed such as solv- +ing an optimisation problem [10] or using Gaussian +processes regression [11]. +However, little attention +has been paid to the detection of anomalous data +during the updating process, despite such data be- +ing ubiquitous in real-world applications. Anomalous +data should be detected, then removed from the up- +date data set, before the DT update. This is essential +to stop the DT adapting itself to represent emerging +faults which would otherwise make the detection of +anomalous behaviours impossible [12]. +The process to update a model is typically many- +fold more computationally demanding than its exe- +cution. +In many real engineering systems, compu- +tational power to perform these updates is severely +limited “on-board”, e.g., in the proximity of a gas +turbine engine. Not only is the computational power +directly connected to asset limiting for on-line learn- +ing, but there is also limited computational storage +capacity, low data transmission capacity, or high cost +of transmission limits the volume of data that may be +transferred for remote updating of the DT. In addi- +tion to the data required to update the DT, data con- +taining possible anomalous system behaviour must +also be collected for the purpose of health monitoring, +e.g., to identify faults in the physical system. There is +therefore a joint challenge of continuously monitoring +the asset to detect undesirable changes in behaviour, +while also updating the model to reflect expected or +nominal changes in behaviour, these challenges are +illustrated in Fig. 1. +To address these challenges, we propose a data- +driven approach to develop DTs for groups of com- +putationally resource-limited systems. We limit the +scope of the DT to applications such as predictive +Figure 1: Data flow in a Digital Twin. A Digital +Twin must be routinely updated, with data recorded +from the physical system, to reflect changes due to +degradation, build variance, etc. +Anomalous data +and constraints in data collection are some of the +challenges faced when updating the Digital Twin. +The Digital Twin provides knowledge about the phys- +ical system and the means to simulate the system +under different conditions. +maintenance and system monitoring. Nevertheless, +the updating framework is applicable to other appli- +cations of DT. Physical systems, which exhibit non- +linear dynamic behaviour, are modelled using a deep +neural network trained with data from each individ- +ual asset. In contrast to model-based methods that +require physical knowledge of the system, data-driven +DTs only rely on data to accurately represent their +physical twins. Therefore, our framework can be used +to represent a variety complex systems. To update +the DT over time, segments of data that are not well +understood, i.e., are predicted with low confidence, +by the digital model, e.g., data generated by the sys- +tem in new, previously unseen, operating conditions +are selected and used to update the model. This ap- +proach avoids the costly transmission of data with +low additional information content. +In addition to +novel data, the most anomalous data, as determined +by the prediction error of the DT, are collected dur- +ing the operation of the physical asset. +The data +are automatically labelled by comparison to the fleet +data set and made available for expert evaluation to +aid root-cause analysis. Anomalous data are removed +from the update data set to avoid learning emerging +faults. +The main contributions of the proposed approach +are: (1) a light-weight DT capable of running online +in constrained systems, (2) a DT-based data priori- +2 + +Fleet data +Physical +Digital +system +Twintisation to collect the most relevant data, and (3) a +robust anomaly detection for individual systems that +takes advantage of the fleet information. +Two key features of the proposed approach are the +selection of appropriate data to update the digital +model and the detection of anomalous data. These +topics are discussed in the remainder of this section. +1.1 +Data selection +The DT should be routinely updated with data gen- +erated by the physical system in order to keep the +twin up-to-date and be able to identify unusual be- +haviours. For resource-limited systems, such an up- +date cannot be performed online; the data must be +stored or transmitted to update the DT on a more +powerful computing system. Several factors can limit +the amount of data collected by resource-limited sys- +tems, e.g., data transmission cost, limited bandwidth, +limited storage or energy capacity. This problem re- +quires the development of methods to select high- +quality data that are information-rich for the desired +task. +The data selection problem is present in the field of +active learning. Active learning considers the prob- +lem of selecting unlabelled data to be labelled for the +purpose of model training. +The main challenge is +how to select the most informative data, such that +the performance of a model is maximised. Research +on this problem has resulted in different strategies +to select data such as models’ disagreement [13], ex- +pected model change maximization [14] and uncer- +tainty sampling [15]. In uncertainty sampling, sam- +ples whose class assignment or prediction are the +most uncertain are selected. Due to the high memory +and computational requirements of the first two ap- +proaches, these are not suitable for resource-limited +systems. Therefore, the proposed solution in this pa- +per selects segments of data based on the model un- +certainty to update the DT. +1.2 +Anomaly detection +An anomaly, also called an outlier, is a pattern in the +data that deviates from a defined notion of normal- +ity. At a high level, the problem of anomaly detection +consists of learning a region or representation of nor- +mal behaviour and identifying data that does not be- +long to that region. There are several challenges that +make this task difficult [16]. First, a clear distinction +between anomalies and normal behaviour is not well +defined in many situations. Second, depending on the +approach used to define normality, a large amount of +labelled data, which can be difficult and expensive to +collect, is required. Finally, when the system under +analysis changes over time, e.g., concept drift, previ- +ous definitions of normality might not be adequate to +identify anomalies. Depending on the application or +domain, additional challenges may be presented. +Multiple solutions have been proposed to solve +the problem of anomaly detection [17]. +A consid- +erable amount of available anomaly detection meth- +ods assume that the data are stored and signifi- +cant computational resources are available. +How- +ever, for many real-world systems such as stream- +ing data systems, where data grows infinitely, or for +resource-constrained systems, such methods cannot +be executed [18]. +Such limitations have motivated +the development of new methods to detect anomalies +within resource-constrained systems. +Efficient on- +device anomaly detectors based on model comparison +[19] and remote processing [20] have been proposed. +Anomaly detection has also been studied in groups +of systems, e.g., a fleet of vehicles. In this scenario, +although the systems all have similar behaviour, each +system is unique due to variations in manufacturing, +usage and degradation. Learning a single model of +normality with the average fleet behaviour can result +in a poor overall detection performance due to the +variability of individual systems. An alternative is to +learn a representation of normality for each system +in the group. Although anomalies can be detected +with such an approach, considering data from all the +systems in the group helps to mitigate the problem +of unusual data and data availability. For instance, +unknown operating conditions for a particular sys- +tem might be seen as normal when data from other +systems are considered. +The solution proposed in this paper uses a combi- +nation of local and centralised anomaly detection in +order to exploit the benefits of fleet data while min- +imising handling data costs. At the system level, the +3 + +digital model is used to identify potentially anoma- +lous segments of data in each system. At the group +level, a centralised system with access to historical +data from the group is used to refine the anomaly +detection performed by each system. In contrast to +other available methods, the approach proposed here +allows the update of individual models and the iden- +tification of anomalies without collecting or transmit- +ting all data, and hence facilitates a practical imple- +mentation. +The rest of the paper is organised as follows. Sec- +tion 2 presents the problem solved in this paper and +introduces two case studies. Section 3 explains in de- +tail the proposed solution. Finally, results and con- +clusions are given in sections 4 and 5, respectively. +2 +Problem Definition +In this paper, we focus on the development of a data- +driven DT for groups of resource-limited systems. +Specifically, we consider systems with the following +characteristics. The system is not capable of execut- +ing tasks with a high computational load such as run- +ning full-physics models, updating digital models or +executing complex anomaly detection methods. The +system cannot store or transmit all the data to be +processed by a system with more resource availability. +Moreover, data cannot be continuously transmitted +to another system. +We consider complex non-linear and dynamic sys- +tems subject to slow nominal intrinsic degradation, +rare but acceptable extrinsic disturbances and the +risk of abnormal intrinsic anomalies / faults. Given +these characteristics, the challenge is to develop a +framework to compute and update a DT that is ca- +pable of accurately simulating a varying physical sys- +tem throughout its lifetime. +To maintain a digital +representation of the system, the problems of select- +ing, managing and analysing collected data must be +addressed. +2.1 +Case studies +The solution presented in this paper is demonstrated +with two real data case studies, both using data from +aerospace gas turbine engines. The two case studies +are presented below: +Anomaly detection. The first case study con- +siders an in-service gas turbine engine with a known +fault. Data recorded during 116 consecutive flights +are used to train the DT and test its ability to per- +form anomaly detection. +The dataset consists of +1Hz time series data with 12 channels - each chan- +nel records data from a different sensor placed on the +engine. Fault symptoms are observed in one flight, +see Fig. 2. This figure shows a permanent change +in the relationship between signals at the moment +the fault occurs. The detection of this fault is used +to illustrate the capacity of the proposed solution to +identify anomalous data. +Figure 2: Case study 1, observed anomaly. Top: +normalised multivariate time series data recorded +during flight. Bottom: anomaly observed in a pres- +sure signal (highlighted in top figure). +Digital Twin update. +The second case study +demonstrates the ability of the proposed solution +to update the DT to accurately simulate a physi- +cal system when its behaviour changes through time. +Data were recorded from an aerospace gas turbine +engine over multiple runs on a testbed over a pe- +riod of several months. During this period the en- +gine showed signs of nominal degradation and several +maintenance actions were performed, see Fig. 3. The +degradation and maintenance caused the dynamic be- +haviour of the system to change continuously over +time. As in the previous case study, the data con- +sist of 1Hz time series data, but here over 80 data +channels are available. +4 + +4 +Ban[BA +2 +-2 +7500 +8500 +00b6 +0056 +000 +14500 +11000 +1150 +Pressure +BanJBA +2. +Temperature +1 +Speed +-1 +Ott6 +9460 +oab6 +9500 +DZ56 +dasaFigure 3: +Case study 2, observed degrada- +tion and maintenance. Mean engine temperature +recorded at a fixed shaft speed. As the engine de- +grades over time it becomes less efficient and more +energy is required to achieve a demanded shaft speed. +As a result, for a given shaft speed, a higher engine +temperature is recorded. +Highlighted regions show +periods where a major maintenance action was per- +formed. +3 +Solution +Digital Twins should represent the real system +throughout its life cycle. +In general, a model pre- +diction can deviate from the observed data due to +the following reasons: (1) degradation: the system +behaviour changes due to expected degradation, (2) +unseen operating conditions: the system operates un- +der conditions not previously observed, and (3) sys- +tem fault: unexpected behaviour is observed caused +by a system fault. To update the DT, data must be +constantly collected from the physical system. More- +over, to stop a DT from tracking (learning) emerg- +ing faults, the data used to update the DT must be +analysed. This is a challenge faced in on-line learn- +ing where a model is constantly updated from a data +stream. Here, a model must learn changes in data dis- +tribution but avoid learning anomalous behaviours. +Solutions based on changes in the data distribution +have been proposed [21]. In these approaches, when +a model prediction deviates from the observed data +over a significant period of time, it is assumed that +the data distribution has changed. However, as pre- +sented in Section 4, depending on the problem, a +permanent change of the system behaviour can be +considered as an anomaly. +We propose a closed-loop approach: selected data +collected from the physical system are sent to an +anomaly detector, which also has access to group +data. +The selection is based on the model’s un- +certainty and data anomaly. This allows us to col- +lect data that are not well understood by the model, +e.g., data from new operating conditions. The non- +anomalous uncertain data are then used to update +the DT. By improving the capability of the DT to +predict behaviours in different operation conditions, +the DT increases its capability to identify anomalous +behaviours with greater accuracy as explained in the +next section. In addition, access to the group’s data +allows us to identify behaviours caused by degrada- +tion or previously unseen operating conditions, for a +particular system, that could be flagged as anomalous +even if such behaviour is normal at the group level. +The combination of the DT update with selected data +and the access to the group’s data increases the ro- +bustness of the anomaly detection. An overview of +the proposed approach is shown in Fig. 4. +Figure 4: Overview of proposed solution. Uncer- +tain and potentially anomalous data are selected on- +board the physical system by using the Digital Twin. +Data are sent to the off-board system for analysis, +with access to the group’s historical data. The off- +board system identifies and presents the anomalous +data to a user for further analyses. +Normal (non- +anomalous) data are used to update the Digital Twin. +Since we consider resource-constrained systems, +collecting or transmitting all the data is not feasi- +ble. +We use a light-weight DT capable of running +on-board the physical system. The DT has the abil- +ity to identify and collect both potentially anomalous +5 + +1200 +1150 +Temperature +1100 +1050 +1000 +2017-04 +2017-05 +2017-06 +2017-07 +2017-08 +2017-09 +2017-10 +2017-11 +2017-12 +DateHistorical +data +Physical system +Off-board +Digital +Sensors +system +Twindata, and data that are not well understood, i.e., +data associated with a high prediction uncertainty. +An example of the latter includes data at previously +unseen operating conditions. In this paper, we re- +fer to the combination of on-board DT and software +used for data collection as the on-board system. Data +collected by the on-board system are then analysed +by a centralised system, referred to as off-board sys- +tem, with access to greater computational resources +and the group’s historical data. The off-board system +identifies whether the data returned by the on-board +system are anomalous at the group level. This data +can then be presented to experts for assessment of la- +bel accuracy, root-cause analysis and sanctioning of +alerts. Data not identified as anomalous are used to +update the DT. In the remainder of this section, the +on-board and off-board systems are presented. +3.1 +On-board system +Here, we present the approaches used to model the +physical system and for data selection. Due to the +system’s limited processing and storage capacity, a +custom-designed deep neural network was selected to +model the system’s behaviour. In contrast to other +computationally expensive models, e.g., full-physics +models, deep neural networks can be efficiently im- +plemented to run on systems with limited resources +using techniques such as quantisation and pruning +[22]. +The key elements of the network that allow +the selection of uncertain and potentially anomalous +data are discussed below. +A bespoke Convolution +Neural Network (CNN) has been designed to capture +the dynamic behaviour of multi-input and non-linear +physical systems. +CNNs are attractive in this ap- +plication due to ability to compress the models for +efficient execution on embedded systems [23]. +The +model is trained to represent the physical system be- +haviour with a parsimonious set of features, aiding +fast run-time execution. +At each sample time, k, the deep neural network re- +ceives a window of time-series data, X, (signals from +the system’s sensors) as an input and makes a predic- +tion on the distribution of the output signal, y, which +Figure 5: Overview of the network architecture. +Convolutional layers are used to extract features from +individual signals. To improve performance at mini- +mal computational cost, the network takes advantage +of skip connections. +is assumed to be Gaussian; +y(k) ∼ N (y(k)|f(X, θ), σ(X, θ)) +(1) +where θ are the network weights, +X += +[x1, x2, . . . , xm] ∈ RN×m, +xi += +[xi(k − N), xi(k − N + 1), . . . , xi(k − 1)]T ∈ RN×1 +is a vector containing the previous N data points of +the i’th input data stream and m is the number of +channels in the input data. +The Gaussian distribution is characterised by its +mean, µ = f(X, θ), and variance σ = σ(X, θ), which +are dependant on the input data, X, and are esti- +mated simultaneously by the deep neural network, +see Fig. 5. The model weights θ are trained via min- +imisation of the negative log likelihood given by +L = − ln +� B +� +n=1 +N (y(n)|µ(n), σ(n)) +� +, +(2) +using a stochastic gradient descent based algorithm +which is fed batches of data at each iteration with +6 + +Input layer +Convolutional layers +Convolutional layer +Convolutional layer +Concatenatelayer +Dense layers +Dense layersbatch size B. The details of the neural network ar- +chitecture, training and implementation are omitted +in this paper and the interested reader is referred to +[24]. +The purpose of computing the variance is twofold: +(1) identify data that are uncertain to the model, +and (2) compute a standardised Euclidean distance to +identify anomalous data. Large prediction variance +(low confidence) can be attributed to input data that +is distributed differently to input data given at train- +ing time. To improve the prediction in such regions +of the input space, and hence improve the robustness +of the DT, data in a window around the most un- +certain predictions are collected for analysis by the +off-board system. If the returned segments of data +are not anomalous with respect the fleet data they +are used to update the DT as explained in Section +3.3. This update is critical to avoid the masking of +anomalous behaviours by previously unseen normal +behaviours. +To identify possible anomalies, the mean and vari- +ance are used to calculate the standardised Euclidean +distance: +d(y(k), µ(k)) = +� +(y(k) − µ(k))2 +σ2(k) +. +(3) +Intuitively, the standardised Euclidean distance +penalises prediction errors that are large relative to +the predicted standard deviation. +This avoids la- +belling nominal data in unknown conditions as faults. +Similar to the uncertain data, windows centred at +large standardised Euclidean distance are collected. +Since the physical system memory is limited, the top +N most uncertain and anomalous windows are re- +tained during operation. +These windows are then +sent to the off-board system to be analysed when a +data link is available, avoiding the need for a decision +on anomalous behaviour to be made on-board. Note +that N is chosen to store the maximum amount of +data. Hence, it is determined by the systems data +storage constraints. +3.2 +Off-board system +The objective of the off-board system is to collect +data returned by on-board system to update the dig- +ital model and identify anomalies. Two key elements +differentiate the on-board and off-board systems: (1) +access to more computational resources, and (2) ac- +cess to the individual asset and group’s historical +data. More computational resources mean that the +off-board system can run more computationally de- +manding anomaly detection routines. Moreover, run- +ning time is not crucial in the off-board system in +contrast to the on-board system, where data are con- +stantly received and have to be analysed online. Ro- +bustness in the anomaly detection is achieved by ac- +cessing the group’s historical data to allow the off- +board system to identify data that are anomalous to +a particular system but not to the group. +The selection of the method used off-board is +highly problem dependant. Therefore, we only give +an overview of the method used for the case studies. +As discussed in Section 2, the data analysed in +the case studies are multivariate time series. +Fea- +tures were extracted from the time series of all his- +torical data. Specifically, a combination of features, +obtained by using Kernel PCA [25] that reflects the +correlation between signals and statistical features +from individual time series were used. +These sta- +tistical features can be designed and updated based +on knowledge from anomalous data collected by the +off-board system over time. The extraction of these +features is computationally expensive, hence such a +method cannot be run on-board. The features are +used to train a one-class Support Vector Machine +(SVM). The SVM can be trained in such a way that +rare nominal events not learnt by the on-board sys- +tem, and hence flagged as possible anomalies, are +classified as non-anomalous data. +3.3 +Digital Twin update +When the physical system changes its behaviour due +to factors such as degradation, the DT should be up- +dated to describe the new behaviour. To show the +effects of a change in the system behaviour, consider +the example shown in Fig. 6. The example shows +7 + +the mean squared prediction error of a predicted sig- +nal over several runs from the same engine during a +time period of multiple months. The predictions are +made by a model trained with data from engine runs +performed before the initial date shown in the figure. +As noted in Section 2, the engine undergoes main- +tenance during each large period of inoperation. As +a result, the dynamics of the engine change at each +maintenance event such that the current model is not +able to predict the engine behaviour. +Figure 6: Increase of prediction error due to system +overhaul and degradation. +The example above shows the necessity of updat- +ing the DT with the most recently collected data. In +other words, the deep neural network has to be up- +dated with a relatively small dataset compared to the +dataset initially used to train the model. This prob- +lem has been addressed in transfer learning by using +fine-tuning [26]. Two major problems of using a small +dataset to update a pre-trained model are overfitting +and catastrophic forgetting [27], i.e., a model can for- +get previously learnt knowledge when is trained with +new information. Parameter regularisation methods +have been used to mitigate this problem. By restrict- +ing the ability of the network to learn, the problem of +overfitting can be reduced. Formally, these methods +minimise a loss function of the form: +˜L = L + +� +j +Ωj∥θj − θ∗ +j ∥2 +2, +(4) +where L is the original loss function, here given by +(2), θj is the j’th network weight, θ∗ +j is the j’th +weight of the pre-trained network and Ω is a hyper- +parameter that controls the regularisation strength. +One of the most common types of regularisation is L2 +regularisation, where all the parameters are forced +towards zero, i.e., θ∗ +j = 0 ∀ j. +Other approaches +maintain the values of the original network, i.e., θ∗, +and compute the regularisation strength of individ- +ual weights based on how important they are to the +previous knowledge [28]. +Two different regularisers were considered to up- +date the model when new data are received from the +on-board system: L2 and L2-SP [29]. Results ob- +tained from several tasks show that these regularisers +are competitive compared to more complex and com- +putationally demanding approaches [30]. The L2-SP +regulariser penalises all the parameters with the same +factor, i.e., Ωj = α ∀ j and α ∈ R. All the weights +are hence forced to remain close to those of the pre- +trained network. In contrast to a model updated with +the L2 regulariser, a model updated with the L2-SP +regulariser is expected to remember previously seen +behaviours while learning new information. +4 +Results +In this section, we demonstrate the ability of the pro- +posed solution to model a physical system, identify +anomalous data, and update the digital model. +4.1 +Anomaly detection +The ability to detect anomalous data is demonstrated +with the first case study presented in Section 2. +The dataset was divided into training and testing +sets. The data known to contain fault symptoms was +placed in the testing set. After training the neural +network, the test dataset was used to test the ability +of the network to predict the behaviour of the engine +and to identify anomalies. The on-board system was +run on an ARM Cortex-A7 micro-processor (as used +in a production monitoring system). +The time se- +ries in the testing set were fed to the neural network +to predict the value of the target signal. The stan- +dardised Euclidean distance or score was computed +at each timestep to identify the most anomalous data. +8 + +0.020 +Mean squared error +0.015 +0.010 +0.005 +0.000 +2017-04 +2017-05 +2017-06 +2017-07 +2017-08 +2017-09 +2017-10 +2017-11 +2017-12 +DateFig. 7 shows the prediction and score of a segment +of the dataset containing the known fault. It is pos- +sible to see that once the fault occurs, the behaviour +of the engine changes and therefore the model is not +capable of predicting the new behaviour. This results +in a high score after the fault. +Figure 7: Top: The real and predicted values of the +target signal (pressure). A fault occurs at time step +9476 causing a change in the system behaviour. Bot- +tom: Standardised Euclidean distance. +The stan- +dardised Euclidean distance is used to select the most +anomalous data. +From each time series in the testing set, the most +anomalous and uncertain windows were collected. +The amount of data returned by the on-board sys- +tem is limited by the transmission bandwidth. +To +maximise the amount of data collected, a threshold is +not used by the on-board system to select anomalous +data. Instead, the on-board system uses a ranking +system where the most anomalous data, based on +the standardised Euclidean distance, is prioritised. +In the case study, the top 100 anomalous and un- +certain windows were collected for each flight. The +data were then analysed by the off-board system to +remove non-anomalous data. +Due to limited data +availability, the off-board system anomaly detector +was trained with historical data, collected over a long +period of time from one engine. Such historical data +can contain rare behaviours that the current DT may +not have adequately learned. +Hence, the off-board +system is capable of identifying segments of nomi- +nal data flagged as anomalous by the on-board sys- +tem. Note that although just one engine was consid- +ered, the proposed approach is designed to use data +from a group of systems or fleet. From 1400 anal- +ysed windows, collected by the on-board system over +the preceding 14 flights, the off-board system iden- +tified 10 windows as anomalies, including the known +anomaly. This demonstrates the ability of the pro- +posed solution to reduce the workload of the user. +A further analysis was done in the rest of the win- +dows detected as anomalous. This analysis consists of +comparing the data recorded from one engine to the +data recorded from its sister engine (the other engine +used during flight). A high residual between the data +recorded from both engines indicates an anomalous +behaviour in one of the engines. Of the 10 flagged +windows, 5 windows (including the known anomaly) +presented high residual between engines. This con- +firms the known anomalous behaviour and possible +previously unknown anomalies. +The example above shows the detection of real +anomalies in the predicted signal. To illustrate the +detection of anomalies in the inputs of the on-board +neural network, anomalies with different sizes were +synthetically injected into the input data, see Fig. 8. +The top figure shows a distribution of the prediction +error, i.e., the difference between predicted and real +value, as the size of the anomalies increases. As ex- +pected, an increment of the anomaly size results in +an increment of the prediction error. The figure in +the centre shows the cumulative distribution of the +standardised Euclidean distance. This figure demon- +strates that for a given communication capacity, the +average error corresponding to the returned anoma- +lies will increase. +Finally, the last plot shows the effects of unknown +inputs in the confidence of the prediction. The con- +fidence of the prediction decreases as the inputs be- +come more anomalous. This confidence reduction pe- +nalises prediction errors for previously unknown in- +puts. However, as presented in the results of the first +case study, such a penalty does not avoid detecting +anomalies for sufficiently large anomalous inputs. +9 + +12 +Prediction +Real value +1D +0.B +0.6 +0 +0.2 +0.0 +8500 +8750 +0006 +9250 +0056 +14000 +14250 +14500 +250 +20D +150 +140 +50 +0 +8500 +8750 +94D0 +DSZ6 +0056 +DS26 +14000 +14250 +14501 +Time stepFigure 8: Effects of different anomalies with differ- +ent magnitudes (as a percentage of input range) in +the deep neural network inputs. Top: Histogram of +prediction errors. +Centre: Cumulative distribution +of normalised standardised Euclidean distance. Bot- +tom: Histogram of the predicted variance. +As the +noise levels in the input space increases the confidence +of the network decreases, i.e. there is an increment +in the variance. +4.2 +Digital Twin update +Here we present the results of updating the digital +model with data collected from the physical system. +This corresponds to the second case study presented +in Section 2. Moreover, we compare the regularisa- +tion approaches presented in Section 3.3 in terms of +prediction accuracy and the capacity to remember +previous behaviours. +An iterative update strategy is used to update the +DT. An initial model is trained with all data from +runs performed before April 17, and is used to se- +lect the unusual data from proceeding 10 runs (after +the training data). The collected data were used to +update the DT. Once the model was updated, data +from the next 10 runs were processed and collected to +again update the DT. This procedure was repeated +until all runs were processed. The prediction errors +of the initial model and the updated model are shown +in Fig. 9. The significant increase in error observed +after 2017-09 for the initial model is due to a mainte- +nance action on the engine. The results show that by +updating the DT the behaviour of the physical sys- +tem can be predicted, despite changes in the system. +Figure 9: Mean squared error of predictions made by +the initial and iteratively retrained model. +To illustrate how the incorrect simulation of +the physical system can affect the identification of +anomalies, synthetic anomalies (spikes of fixed length +and amplitude based on the signal’s local standard +deviation) were injected in randomly selected runs. +The runs were divided into two groups: before and +after the September maintenance event. As shown in +Fig. 9, when the DT is not updated, the behaviour +of the system is not predicted accurately. Therefore, +poor performance in the detection of anomalies is +expected, and this is shown in the results. When the +DT is not updated, 22 of the synthetic anomalies +are detected before the maintenance event and +only one after. +In contrast, when it is updated, +28 and 23 anomalies are detected before and after +the overhaul, respectively. +The results show an +improvement in the number of anomalies detected +when the DT is routinely updated. As expected, a +more noticeable improvement is made on the data +after the maintenance event. +The results presented above show that when a +10 + +8000 +Base +1.0% +2.5% +6000 +Count +4000 +2000 +0 +-6 +-4 +-2 +0 +2 +Prediction error +1.0 +Cumulative probability +0.8 + 0.6 + 0.4 +0.2 +Base +1.0% + 0.0 +2.5% +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(Normalised) Standardised Euclidean distance +17500 +Base +1.0% +15000 +2.5% +12500 +Count +10000 +7500 +5000 +2500 +0 +1 +2 +E +4 +5 +6 +Variance (Confidence)Initial model +0.020 +Retrained model +Mean squared error +0.015 +0.010 +0.005 +0.000 +2017-04 +2017-05 +2017-06 +2017-07 +2017-08 +2017-09 +2017-10 +2017-11 +2017-12 +Datemodel is routinely updated, accurate prediction of +engine behaviour is achieved. However, they do not +show how the model is affected in terms of forgetting +previous behaviours. +To see this effect, two differ- +ent manoeuvres or operating profiles, labelled A and +B, were extracted from each engine run, see Fig. 10 +(top). A model was initially trained with manoeuvre +A, and it is assumed that no manoeuvre B data are +available. Then, the model was iteratively retrained +with data from manoeuvre B following the procedure +presented above. +Three different approaches were +used to retrain the model: the L2 regularisation, the +L2 − SP regularisation, and an augmented training +data approach which includes historic data from ma- +noeuvre A. Fig. 10 (bottom) shows the prediction of +manoeuvre A after retraining the model. The pre- +diction of manoeuvre B did not vary significantly be- +tween approaches and is not shown. +Figure 10: Top: manoeuvres representing different +behaviours of the engine. Bottom: Prediction of ma- +noeuvre A after retraining a model with manoeuvre +B using different training approaches. +As expected, the results show that when the model +is updated with the L2 regulariser, the model for- +gets how to predict manoeuvre A. The forgetting is +significantly reduced when using the L2 − SP regu- +lariser. Adding data from manoeuvre A during the +retraining improves the capacity of the model to re- +member manoeuvre A. While this approach does not +represent a challenge in the toy example presented +above, it raises new challenges when data must be +selected from a large pool of historical engine be- +haviours which might not represent the current sys- +tem behaviour. +5 +Conclusion +Digital Twins are used in a wide range of areas for +management, optimisation and decision support of +physical systems. To accurately represent its physi- +cal counterpart, DTs must be routinely updated with +data collected from the physical system to be able to +identify unusual behaviours in the physical system. +This can be a significant challenge for systems with +constrained resources. Moreover, not all the collected +data are suitable to update the DT, e.g., anoma- +lies related to emerging faults must be handled care- +fully, particularly in our chosen application of sys- +tem health monitoring. This paper presents a novel +2-stage framework to keep a data-driven DT synchro- +nised with its the physical system counterpart. The +proposed solution allows real-time asset monitoring, +and the selection of high-quality data for remote up- +date of a DT using fault-free fleet data. Results from +a gas turbine engine case study show the capacity +of the solution to accurately simulate the behaviour +of an engine throughout its life. To further reduce +the computational and data requirements for updat- +ing, interesting future research directions include the +definition of (optimal) criteria on when to perform +model updates and the introduction of physics-based +constraints to the machine learning update process. +References +[1] F. Tao, +H. Zhang, +A. Liu, +and A. Y.C. +Nee. Digital twin in industry: State-of-the-art. +IEEE Transactions on Industrial Informatics, +15(4):2405–2415, 2018. +[2] E. +Glaessgen +and +D. +Stargel. +The +dig- +ital +twin +paradigm +for +future +nasa +and +u.s. +air +force +vehicles. +In +Proc. +of +AIAA/ASME/ASCE/AHS/ASC +structures, +structurual dynamics and materials conference, +page 118, 2012. +11 + +Manoeuvre A +Manoeuvre B +0.B +BNEA +0.6 +paads +E'O +t0 +0.2 +0.2 +0.1 +50 +140 +150 +240 +250 +310 +1 +24 +31 +50 +Timestep +Timestep +0.10 +L2 +0.DB +L2 SP +Initial prediction +Augmented training set +lenbg +0.04 +uBa +20.D2 +2417-05 +2417-06 +Dabe[3] B. R. Barricelli, E. Casiraghi, and D. Fogli. A +survey on digital twin: Definitions, characteris- +tics, applications, and design implications. IEEE +access, 7:167653–167671, 2019. +[4] J. Leng, D. Wang, W. Shen, X. Li, Q. Liu, and +X. Chen. +Digital twins-based smart manufac- +turing system design in industry 4.0: A review. +Journal of manufacturing systems, 60:119–137, +2021. +[5] The DTHub. http://https://digitaltwinhub.co.uk/. +Accessed: 2021-06-15. +[6] Q. Qi and F. Tao. +Digital twin and big data +towards smart manufacturing and industry 4.0: +360 degree comparison. +Ieee Access, 6:3585– +3593, 2018. +[7] E. J. Tuegel, A. R. Ingraffea, T. G. Eason, and +S. M. Spottswood. Reengineering aircraft struc- +tural life prediction using a digital twin. Interna- +tional Journal of Aerospace Engineering, 2011, +2011. +[8] R. +Magargle, +L. +Johnson, +P. +Mandloi, +P. +Davoudabadi, +O. +Kesarkar, +S. +Krish- +naswamy, J. Batteh, and A. Pitchaikani. +A +simulation-based digital twin for model-driven +health monitoring and predictive maintenance +of an automotive braking system. +In Proc. of +the International Modelica Conference, pages +35–46. Link¨oping University Electronic Press, +2017. +[9] C. Li, S. Mahadevan, Y. Ling, S. Choze, and +L. Wang. +Dynamic bayesian network for air- +craft wing health monitoring digital twin. Aiaa +Journal, 55(3):930–941, 2017. +[10] J. Wang, L. Ye, R. X. Gao, C. Li, and L. Zhang. +Digital twin for rotating machinery fault diagno- +sis in smart manufacturing. International Jour- +nal of Production Research, 57(12):3920–3934, +2019. +[11] S. Chakraborty and S. Adhikari. Machine learn- +ing based digital twin for dynamical systems +with multiple time-scales. Computers & Struc- +tures, 243:106410, 2021. +[12] O. Fink, Q. Wang, M. Svensen, P. Dersin, +W. Lee, and M. Ducoffe. +Potential, chal- +lenges and future directions for deep learning +in prognostics and health management applica- +tions. Engineering Applications of Artificial In- +telligence, 92:103678, 2020. +[13] R. Burbidge, J. J. Rowland, and R. D. King. +Active learning for regression based on query by +committee. In Proc. of IDEAL, pages 209–218. +Springer, 2007. +[14] W. Cai, Y. Zhang, and J. Zhou. +Maximizing +expected model change for active learning in re- +gression. In Proc. of ICDM, pages 51–60. IEEE, +2013. +[15] Y. Yang, Z. Ma, F. Nie, X. Chang, and A. G. +Hauptmann. +Multi-class active learning by +uncertainty sampling with diversity maximiza- +tion. International Journal of Computer Vision, +113(2):113–127, 2015. +[16] V. Chandola, A. Banerjee, and V. Kumar. +Anomaly detection: A survey. ACM computing +surveys (CSUR), 41(3):1–58, 2009. +[17] M. A.F. Pimentel, D. A. Clifton, L. Clifton, and +L. Tarassenko. +A review of novelty detection. +Signal Processing, 99:215–249, 2014. +[18] S. Rajasegarar, C. Leckie, and M. Palaniswami. +Anomaly detection in wireless sensor networks. +IEEE Wireless Communications, 15(4):34–40, +2008. +[19] M. +B. +Attia, +C. +Talhi, +A. +Hamou-Lhadj, +B. Khosravifar, +V. Turpaud, +and M. Cou- +ture. On-device anomaly detection for resource- +limited systems. +In Proc. of ACM/SIGAPP, +pages 548–554, 2015. +[20] P. +Amontamavut, +Y. +Nakagawa, +and +E. Hayakawa. +Separated linux process log- +ging mechanism for embedded systems. In Proc. +of RTCSA, pages 411–414. IEEE, 2012. +12 + +[21] S. Saurav, P. Malhotra, V. TV, N. Gugulothu, +L. Vig, P. Agarwal, and G. Shroff. +Online +anomaly detection with concept drift adaptation +using recurrent neural networks. In Proc. of the +ACM India Joint International Conference on +Data Science and Management of Data, pages +78–87, 2018. +[22] B. Reddy, Y. Kim, S. Yun, C. Seo, and J. Jang. +Real-time driver drowsiness detection for em- +bedded system using model compression of deep +neural networks. In Proc. of CVPR, pages 121– +128, 2017. +[23] V. Sze, Y. Chen, T. Yang, and J. S. Emer. Ef- +ficient processing of deep neural networks: A +tutorial and survey. +Proceedings of the IEEE, +105(12):2295–2329, 2017. +[24] Adam Hartwell, Felipe Montana, Will Jacobs, +Visakan Kadirkamanathan, Andrew R Mills, +and Tom Clark. In-flight novelty detection with +convolutional neural networks. +arXiv preprint +arXiv:2112.03765, 2021. +[25] K. Yang and C. Shahabi. A pca-based kernel for +kernel pca on multivariate time series. In Proc. +of ICDM, 2005. +[26] J. Yosinski, J. Clune, Y. Bengio, and H. Lipson. +How transferable are features in deep neural net- +works? In Advances in neural information pro- +cessing systems, pages 3320–3328, 2014. +[27] M. McCloskey and N. J. Cohen. Catastrophic +interference in connectionist networks: The se- +quential learning problem. +In Psychology of +learning and motivation, volume 24, pages 109– +165. Elsevier, 1989. +[28] J. Kirkpatrick, R. Pascanu, N. Rabinowitz, +J. Veness, G. Desjardins, A. A. Rusu, K. Milan, +J. Quan, T. Ramalho, A. Grabska-Barwinska, +et al. Overcoming catastrophic forgetting in neu- +ral networks. Proc. of the national academy of +sciences, 114(13):3521–3526, 2017. +[29] H. Daum´e III. Frustratingly easy domain adap- +tation. arXiv preprint arXiv:0907.1815, 2009. +[30] X. Li, Y. Grandvalet, and F. Davoine. +Ex- +plicit +inductive +bias +for +transfer +learning +with convolutional networks. +arXiv preprint +arXiv:1802.01483, 2018. +13 + diff --git a/_tAzT4oBgHgl3EQfFvq5/content/tmp_files/load_file.txt b/_tAzT4oBgHgl3EQfFvq5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ee443a39a6d27eeeead7c15f10bf37ecd17c9a7c --- /dev/null +++ b/_tAzT4oBgHgl3EQfFvq5/content/tmp_files/load_file.txt @@ -0,0 +1,658 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf,len=657 +page_content='Through-life Monitoring of Resource-constrained Systems and Fleets Felipe Montana1, Adam Hartwell1, Will Jacobs1, Visakan Kadirkamanathan1, Andrew R Mills1, and Tom Clark2 1Department of Automatic Control and Systems Engineering, University of Sheffield, UK 2Rolls-Royce Plc, UK Abstract A Digital Twin (DT) is a simulation of a physical system that provides information to make decisions that add economic, social or commercial value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The behaviour of a physical system changes over time, a DT must therefore be continually updated with data from the physical systems to reflect its changing be- haviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' For resource-constrained systems, updat- ing a DT is non-trivial because of challenges such as on-board learning and the off-board data transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' This paper presents a framework for updating data- driven DTs of resource-constrained systems geared towards system health monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The proposed so- lution consists of: (1) an on-board system running a light-weight DT allowing the prioritisation and par- simonious transfer of data generated by the physical system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' and (2) off-board robust updating of the DT and detection of anomalous behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Two case studies are considered using a production gas turbine engine system to demonstrate the digital represen- tation accuracy for real-world, time-varying physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 1 Introduction Digital Twin (DT) is an emerging technology that has recently gained attention due to the rapid de- velopment of simulations, data acquisition and data communication that trigger interaction between the physical and virtual spaces [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' A DT is a simulation of an as-built system, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=', a vehicle or factory, that accurately represents its corresponding twin [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In contrast to a simple model or simulation, a DT is a living and evolving model that follows the lifecycle of its physical twin [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' This is done by integrating data collected from the system’s sensors, environment, his- toric maintenance data, and available system knowl- edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Moreover, DTs can contain a description of the structure, functions, behaviour and control of the physical system [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The information provided by the DT facilitates the processes of making decisions that will affect the physical asset [5] and adds economic, social or commercial value to stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The DT paradigm is used in a variety of applica- tions and sectors, including product design and fac- tory optimisation [6], prediction of aircraft structural life [7], and monitoring of automotive braking sys- tems [8] among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Another example is the man- agement and monitoring of a fleet of assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In this case, information based on the fleet distribution is inadequate to assess individual systems due to the variability in the usage, manufacturing and material properties of different assets [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The availability of individually tailored DTs is therefore desirable to ac- curately predict future performance against require- ments and detect deviation from the current system behaviour driven by emerging faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The behaviour of a physical system is expected to change over time due to normal degradation, mechan- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='01017v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='LG] 3 Jan 2023 ical modifications, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The result is a drift between the real system and its twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Another cause of devia- tion between a real system and its twin, particularly in data-driven DTs, is operation in conditions previ- ously unseen by the twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Both scenarios motivate the update of the DT using data acquired from the physical system to provide a better representation of the current state of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' For applications such as predictive maintenance and monitoring, updates to the DT mainly consist of updating model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Different methods for parameter updating have been proposed such as solv- ing an optimisation problem [10] or using Gaussian processes regression [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' However, little attention has been paid to the detection of anomalous data during the updating process, despite such data be- ing ubiquitous in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Anomalous data should be detected, then removed from the up- date data set, before the DT update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' This is essential to stop the DT adapting itself to represent emerging faults which would otherwise make the detection of anomalous behaviours impossible [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The process to update a model is typically many- fold more computationally demanding than its exe- cution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In many real engineering systems, compu- tational power to perform these updates is severely limited “on-board”, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=', in the proximity of a gas turbine engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Not only is the computational power directly connected to asset limiting for on-line learn- ing, but there is also limited computational storage capacity, low data transmission capacity, or high cost of transmission limits the volume of data that may be transferred for remote updating of the DT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In addi- tion to the data required to update the DT, data con- taining possible anomalous system behaviour must also be collected for the purpose of health monitoring, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=', to identify faults in the physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' There is therefore a joint challenge of continuously monitoring the asset to detect undesirable changes in behaviour, while also updating the model to reflect expected or nominal changes in behaviour, these challenges are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' To address these challenges, we propose a data- driven approach to develop DTs for groups of com- putationally resource-limited systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' We limit the scope of the DT to applications such as predictive Figure 1: Data flow in a Digital Twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' A Digital Twin must be routinely updated, with data recorded from the physical system, to reflect changes due to degradation, build variance, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Anomalous data and constraints in data collection are some of the challenges faced when updating the Digital Twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The Digital Twin provides knowledge about the phys- ical system and the means to simulate the system under different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' maintenance and system monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Nevertheless, the updating framework is applicable to other appli- cations of DT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Physical systems, which exhibit non- linear dynamic behaviour, are modelled using a deep neural network trained with data from each individ- ual asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In contrast to model-based methods that require physical knowledge of the system, data-driven DTs only rely on data to accurately represent their physical twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Therefore, our framework can be used to represent a variety complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' To update the DT over time, segments of data that are not well understood, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=', are predicted with low confidence, by the digital model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=', data generated by the sys- tem in new, previously unseen, operating conditions are selected and used to update the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' This ap- proach avoids the costly transmission of data with low additional information content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In addition to novel data, the most anomalous data, as determined by the prediction error of the DT, are collected dur- ing the operation of the physical asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The data are automatically labelled by comparison to the fleet data set and made available for expert evaluation to aid root-cause analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Anomalous data are removed from the update data set to avoid learning emerging faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The main contributions of the proposed approach are: (1) a light-weight DT capable of running online in constrained systems, (2) a DT-based data priori- 2 Fleet data Physical Digital system Twintisation to collect the most relevant data, and (3) a robust anomaly detection for individual systems that takes advantage of the fleet information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Two key features of the proposed approach are the selection of appropriate data to update the digital model and the detection of anomalous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' These topics are discussed in the remainder of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='1 Data selection The DT should be routinely updated with data gen- erated by the physical system in order to keep the twin up-to-date and be able to identify unusual be- haviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' For resource-limited systems, such an up- date cannot be performed online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' the data must be stored or transmitted to update the DT on a more powerful computing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Several factors can limit the amount of data collected by resource-limited sys- tems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=', data transmission cost, limited bandwidth, limited storage or energy capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' This problem re- quires the development of methods to select high- quality data that are information-rich for the desired task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The data selection problem is present in the field of active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Active learning considers the prob- lem of selecting unlabelled data to be labelled for the purpose of model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The main challenge is how to select the most informative data, such that the performance of a model is maximised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Research on this problem has resulted in different strategies to select data such as models’ disagreement [13], ex- pected model change maximization [14] and uncer- tainty sampling [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In uncertainty sampling, sam- ples whose class assignment or prediction are the most uncertain are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Due to the high memory and computational requirements of the first two ap- proaches, these are not suitable for resource-limited systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Therefore, the proposed solution in this pa- per selects segments of data based on the model un- certainty to update the DT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='2 Anomaly detection An anomaly, also called an outlier, is a pattern in the data that deviates from a defined notion of normal- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' At a high level, the problem of anomaly detection consists of learning a region or representation of nor- mal behaviour and identifying data that does not be- long to that region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' There are several challenges that make this task difficult [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' First, a clear distinction between anomalies and normal behaviour is not well defined in many situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Second, depending on the approach used to define normality, a large amount of labelled data, which can be difficult and expensive to collect, is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Finally, when the system under analysis changes over time, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=', concept drift, previ- ous definitions of normality might not be adequate to identify anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Depending on the application or domain, additional challenges may be presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Multiple solutions have been proposed to solve the problem of anomaly detection [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' A consid- erable amount of available anomaly detection meth- ods assume that the data are stored and signifi- cant computational resources are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' How- ever, for many real-world systems such as stream- ing data systems, where data grows infinitely, or for resource-constrained systems, such methods cannot be executed [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Such limitations have motivated the development of new methods to detect anomalies within resource-constrained systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Efficient on- device anomaly detectors based on model comparison [19] and remote processing [20] have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Anomaly detection has also been studied in groups of systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=', a fleet of vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In this scenario, although the systems all have similar behaviour, each system is unique due to variations in manufacturing, usage and degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Learning a single model of normality with the average fleet behaviour can result in a poor overall detection performance due to the variability of individual systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' An alternative is to learn a representation of normality for each system in the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Although anomalies can be detected with such an approach, considering data from all the systems in the group helps to mitigate the problem of unusual data and data availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' For instance, unknown operating conditions for a particular sys- tem might be seen as normal when data from other systems are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The solution proposed in this paper uses a combi- nation of local and centralised anomaly detection in order to exploit the benefits of fleet data while min- imising handling data costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' At the system level, the 3 digital model is used to identify potentially anoma- lous segments of data in each system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' At the group level, a centralised system with access to historical data from the group is used to refine the anomaly detection performed by each system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In contrast to other available methods, the approach proposed here allows the update of individual models and the iden- tification of anomalies without collecting or transmit- ting all data, and hence facilitates a practical imple- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The rest of the paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Sec- tion 2 presents the problem solved in this paper and introduces two case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Section 3 explains in de- tail the proposed solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Finally, results and con- clusions are given in sections 4 and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 2 Problem Definition In this paper, we focus on the development of a data- driven DT for groups of resource-limited systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Specifically, we consider systems with the following characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The system is not capable of execut- ing tasks with a high computational load such as run- ning full-physics models, updating digital models or executing complex anomaly detection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The system cannot store or transmit all the data to be processed by a system with more resource availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Moreover, data cannot be continuously transmitted to another system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' We consider complex non-linear and dynamic sys- tems subject to slow nominal intrinsic degradation, rare but acceptable extrinsic disturbances and the risk of abnormal intrinsic anomalies / faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Given these characteristics, the challenge is to develop a framework to compute and update a DT that is ca- pable of accurately simulating a varying physical sys- tem throughout its lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' To maintain a digital representation of the system, the problems of select- ing, managing and analysing collected data must be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='1 Case studies The solution presented in this paper is demonstrated with two real data case studies, both using data from aerospace gas turbine engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The two case studies are presented below: Anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The first case study con- siders an in-service gas turbine engine with a known fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Data recorded during 116 consecutive flights are used to train the DT and test its ability to per- form anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The dataset consists of 1Hz time series data with 12 channels - each chan- nel records data from a different sensor placed on the engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Fault symptoms are observed in one flight, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' This figure shows a permanent change in the relationship between signals at the moment the fault occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The detection of this fault is used to illustrate the capacity of the proposed solution to identify anomalous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Figure 2: Case study 1, observed anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Top: normalised multivariate time series data recorded during flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Bottom: anomaly observed in a pres- sure signal (highlighted in top figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Digital Twin update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The second case study demonstrates the ability of the proposed solution to update the DT to accurately simulate a physi- cal system when its behaviour changes through time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Data were recorded from an aerospace gas turbine engine over multiple runs on a testbed over a pe- riod of several months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' During this period the en- gine showed signs of nominal degradation and several maintenance actions were performed, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The degradation and maintenance caused the dynamic be- haviour of the system to change continuously over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' As in the previous case study, the data con- sist of 1Hz time series data, but here over 80 data channels are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 4 4 Ban[BA 2 2 7500 8500 00b6 0056 000 14500 11000 1150 Pressure BanJBA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Temperature 1 Speed 1 Ott6 9460 oab6 9500 DZ56 dasaFigure 3: Case study 2, observed degrada- tion and maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Mean engine temperature recorded at a fixed shaft speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' As the engine de- grades over time it becomes less efficient and more energy is required to achieve a demanded shaft speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' As a result, for a given shaft speed, a higher engine temperature is recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Highlighted regions show periods where a major maintenance action was per- formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 3 Solution Digital Twins should represent the real system throughout its life cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In general, a model pre- diction can deviate from the observed data due to the following reasons: (1) degradation: the system behaviour changes due to expected degradation, (2) unseen operating conditions: the system operates un- der conditions not previously observed, and (3) sys- tem fault: unexpected behaviour is observed caused by a system fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' To update the DT, data must be constantly collected from the physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' More- over, to stop a DT from tracking (learning) emerg- ing faults, the data used to update the DT must be analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' This is a challenge faced in on-line learn- ing where a model is constantly updated from a data stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Here, a model must learn changes in data dis- tribution but avoid learning anomalous behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Solutions based on changes in the data distribution have been proposed [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In these approaches, when a model prediction deviates from the observed data over a significant period of time, it is assumed that the data distribution has changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' However, as pre- sented in Section 4, depending on the problem, a permanent change of the system behaviour can be considered as an anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' We propose a closed-loop approach: selected data collected from the physical system are sent to an anomaly detector, which also has access to group data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The selection is based on the model’s un- certainty and data anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' This allows us to col- lect data that are not well understood by the model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=', data from new operating conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The non- anomalous uncertain data are then used to update the DT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' By improving the capability of the DT to predict behaviours in different operation conditions, the DT increases its capability to identify anomalous behaviours with greater accuracy as explained in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In addition, access to the group’s data allows us to identify behaviours caused by degrada- tion or previously unseen operating conditions, for a particular system, that could be flagged as anomalous even if such behaviour is normal at the group level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The combination of the DT update with selected data and the access to the group’s data increases the ro- bustness of the anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' An overview of the proposed approach is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Figure 4: Overview of proposed solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Uncer- tain and potentially anomalous data are selected on- board the physical system by using the Digital Twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Data are sent to the off-board system for analysis, with access to the group’s historical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The off- board system identifies and presents the anomalous data to a user for further analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Normal (non- anomalous) data are used to update the Digital Twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Since we consider resource-constrained systems, collecting or transmitting all the data is not feasi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' We use a light-weight DT capable of running on-board the physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The DT has the abil- ity to identify and collect both potentially anomalous 5 1200 1150 Temperature 1100 1050 1000 2017-04 2017-05 2017-06 2017-07 2017-08 2017-09 2017-10 2017-11 2017-12 DateHistorical data Physical system Off-board Digital Sensors system Twindata, and data that are not well understood, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=', data associated with a high prediction uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' An example of the latter includes data at previously unseen operating conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In this paper, we re- fer to the combination of on-board DT and software used for data collection as the on-board system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Data collected by the on-board system are then analysed by a centralised system, referred to as off-board sys- tem, with access to greater computational resources and the group’s historical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The off-board system identifies whether the data returned by the on-board system are anomalous at the group level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' This data can then be presented to experts for assessment of la- bel accuracy, root-cause analysis and sanctioning of alerts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Data not identified as anomalous are used to update the DT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In the remainder of this section, the on-board and off-board systems are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='1 On-board system Here, we present the approaches used to model the physical system and for data selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Due to the system’s limited processing and storage capacity, a custom-designed deep neural network was selected to model the system’s behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In contrast to other computationally expensive models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=', full-physics models, deep neural networks can be efficiently im- plemented to run on systems with limited resources using techniques such as quantisation and pruning [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The key elements of the network that allow the selection of uncertain and potentially anomalous data are discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' A bespoke Convolution Neural Network (CNN) has been designed to capture the dynamic behaviour of multi-input and non-linear physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' CNNs are attractive in this ap- plication due to ability to compress the models for efficient execution on embedded systems [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The model is trained to represent the physical system be- haviour with a parsimonious set of features, aiding fast run-time execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' At each sample time, k, the deep neural network re- ceives a window of time-series data, X, (signals from the system’s sensors) as an input and makes a predic- tion on the distribution of the output signal, y, which Figure 5: Overview of the network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Convolutional layers are used to extract features from individual signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' To improve performance at mini- mal computational cost, the network takes advantage of skip connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' is assumed to be Gaussian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' y(k) ∼ N (y(k)|f(X, θ), σ(X, θ)) (1) where θ are the network weights, X = [x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' , xm] ∈ RN×m, xi = [xi(k − N), xi(k − N + 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' , xi(k − 1)]T ∈ RN×1 is a vector containing the previous N data points of the i’th input data stream and m is the number of channels in the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The Gaussian distribution is characterised by its mean, µ = f(X, θ), and variance σ = σ(X, θ), which are dependant on the input data, X, and are esti- mated simultaneously by the deep neural network, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The model weights θ are trained via min- imisation of the negative log likelihood given by L = − ln � B � n=1 N (y(n)|µ(n), σ(n)) � , (2) using a stochastic gradient descent based algorithm which is fed batches of data at each iteration with 6 Input layer Convolutional layers Convolutional layer Convolutional layer Concatenatelayer Dense layers Dense layersbatch size B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The details of the neural network ar- chitecture, training and implementation are omitted in this paper and the interested reader is referred to [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The purpose of computing the variance is twofold: (1) identify data that are uncertain to the model, and (2) compute a standardised Euclidean distance to identify anomalous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Large prediction variance (low confidence) can be attributed to input data that is distributed differently to input data given at train- ing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' To improve the prediction in such regions of the input space, and hence improve the robustness of the DT, data in a window around the most un- certain predictions are collected for analysis by the off-board system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' If the returned segments of data are not anomalous with respect the fleet data they are used to update the DT as explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' This update is critical to avoid the masking of anomalous behaviours by previously unseen normal behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' To identify possible anomalies, the mean and vari- ance are used to calculate the standardised Euclidean distance: d(y(k), µ(k)) = � (y(k) − µ(k))2 σ2(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' (3) Intuitively, the standardised Euclidean distance penalises prediction errors that are large relative to the predicted standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' This avoids la- belling nominal data in unknown conditions as faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Similar to the uncertain data, windows centred at large standardised Euclidean distance are collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Since the physical system memory is limited, the top N most uncertain and anomalous windows are re- tained during operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' These windows are then sent to the off-board system to be analysed when a data link is available, avoiding the need for a decision on anomalous behaviour to be made on-board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Note that N is chosen to store the maximum amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Hence, it is determined by the systems data storage constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='2 Off-board system The objective of the off-board system is to collect data returned by on-board system to update the dig- ital model and identify anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Two key elements differentiate the on-board and off-board systems: (1) access to more computational resources, and (2) ac- cess to the individual asset and group’s historical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' More computational resources mean that the off-board system can run more computationally de- manding anomaly detection routines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Moreover, run- ning time is not crucial in the off-board system in contrast to the on-board system, where data are con- stantly received and have to be analysed online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Ro- bustness in the anomaly detection is achieved by ac- cessing the group’s historical data to allow the off- board system to identify data that are anomalous to a particular system but not to the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The selection of the method used off-board is highly problem dependant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Therefore, we only give an overview of the method used for the case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' As discussed in Section 2, the data analysed in the case studies are multivariate time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Fea- tures were extracted from the time series of all his- torical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Specifically, a combination of features, obtained by using Kernel PCA [25] that reflects the correlation between signals and statistical features from individual time series were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' These sta- tistical features can be designed and updated based on knowledge from anomalous data collected by the off-board system over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The extraction of these features is computationally expensive, hence such a method cannot be run on-board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The features are used to train a one-class Support Vector Machine (SVM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The SVM can be trained in such a way that rare nominal events not learnt by the on-board sys- tem, and hence flagged as possible anomalies, are classified as non-anomalous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='3 Digital Twin update When the physical system changes its behaviour due to factors such as degradation, the DT should be up- dated to describe the new behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' To show the effects of a change in the system behaviour, consider the example shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The example shows 7 the mean squared prediction error of a predicted sig- nal over several runs from the same engine during a time period of multiple months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The predictions are made by a model trained with data from engine runs performed before the initial date shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' As noted in Section 2, the engine undergoes main- tenance during each large period of inoperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' As a result, the dynamics of the engine change at each maintenance event such that the current model is not able to predict the engine behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Figure 6: Increase of prediction error due to system overhaul and degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The example above shows the necessity of updat- ing the DT with the most recently collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In other words, the deep neural network has to be up- dated with a relatively small dataset compared to the dataset initially used to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' This prob- lem has been addressed in transfer learning by using fine-tuning [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Two major problems of using a small dataset to update a pre-trained model are overfitting and catastrophic forgetting [27], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=', a model can for- get previously learnt knowledge when is trained with new information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Parameter regularisation methods have been used to mitigate this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' By restrict- ing the ability of the network to learn, the problem of overfitting can be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Formally, these methods minimise a loss function of the form: ˜L = L + � j Ωj∥θj − θ∗ j ∥2 2, (4) where L is the original loss function, here given by (2), θj is the j’th network weight, θ∗ j is the j’th weight of the pre-trained network and Ω is a hyper- parameter that controls the regularisation strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' One of the most common types of regularisation is L2 regularisation, where all the parameters are forced towards zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=', θ∗ j = 0 ∀ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Other approaches maintain the values of the original network, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=', θ∗, and compute the regularisation strength of individ- ual weights based on how important they are to the previous knowledge [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Two different regularisers were considered to up- date the model when new data are received from the on-board system: L2 and L2-SP [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Results ob- tained from several tasks show that these regularisers are competitive compared to more complex and com- putationally demanding approaches [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The L2-SP regulariser penalises all the parameters with the same factor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=', Ωj = α ∀ j and α ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' All the weights are hence forced to remain close to those of the pre- trained network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In contrast to a model updated with the L2 regulariser, a model updated with the L2-SP regulariser is expected to remember previously seen behaviours while learning new information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 4 Results In this section, we demonstrate the ability of the pro- posed solution to model a physical system, identify anomalous data, and update the digital model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='1 Anomaly detection The ability to detect anomalous data is demonstrated with the first case study presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The dataset was divided into training and testing sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The data known to contain fault symptoms was placed in the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' After training the neural network, the test dataset was used to test the ability of the network to predict the behaviour of the engine and to identify anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The on-board system was run on an ARM Cortex-A7 micro-processor (as used in a production monitoring system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The time se- ries in the testing set were fed to the neural network to predict the value of the target signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The stan- dardised Euclidean distance or score was computed at each timestep to identify the most anomalous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='020 Mean squared error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='000 2017-04 2017-05 2017-06 2017-07 2017-08 2017-09 2017-10 2017-11 2017-12 DateFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 7 shows the prediction and score of a segment of the dataset containing the known fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' It is pos- sible to see that once the fault occurs, the behaviour of the engine changes and therefore the model is not capable of predicting the new behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' This results in a high score after the fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Figure 7: Top: The real and predicted values of the target signal (pressure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' A fault occurs at time step 9476 causing a change in the system behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Bot- tom: Standardised Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The stan- dardised Euclidean distance is used to select the most anomalous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' From each time series in the testing set, the most anomalous and uncertain windows were collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The amount of data returned by the on-board sys- tem is limited by the transmission bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' To maximise the amount of data collected, a threshold is not used by the on-board system to select anomalous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Instead, the on-board system uses a ranking system where the most anomalous data, based on the standardised Euclidean distance, is prioritised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In the case study, the top 100 anomalous and un- certain windows were collected for each flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The data were then analysed by the off-board system to remove non-anomalous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Due to limited data availability, the off-board system anomaly detector was trained with historical data, collected over a long period of time from one engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Such historical data can contain rare behaviours that the current DT may not have adequately learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Hence, the off-board system is capable of identifying segments of nomi- nal data flagged as anomalous by the on-board sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Note that although just one engine was consid- ered, the proposed approach is designed to use data from a group of systems or fleet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' From 1400 anal- ysed windows, collected by the on-board system over the preceding 14 flights, the off-board system iden- tified 10 windows as anomalies, including the known anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' This demonstrates the ability of the pro- posed solution to reduce the workload of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' A further analysis was done in the rest of the win- dows detected as anomalous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' This analysis consists of comparing the data recorded from one engine to the data recorded from its sister engine (the other engine used during flight).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' A high residual between the data recorded from both engines indicates an anomalous behaviour in one of the engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Of the 10 flagged windows, 5 windows (including the known anomaly) presented high residual between engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' This con- firms the known anomalous behaviour and possible previously unknown anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The example above shows the detection of real anomalies in the predicted signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' To illustrate the detection of anomalies in the inputs of the on-board neural network, anomalies with different sizes were synthetically injected into the input data, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The top figure shows a distribution of the prediction error, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=', the difference between predicted and real value, as the size of the anomalies increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' As ex- pected, an increment of the anomaly size results in an increment of the prediction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The figure in the centre shows the cumulative distribution of the standardised Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' This figure demon- strates that for a given communication capacity, the average error corresponding to the returned anoma- lies will increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Finally, the last plot shows the effects of unknown inputs in the confidence of the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The con- fidence of the prediction decreases as the inputs be- come more anomalous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' This confidence reduction pe- nalises prediction errors for previously unknown in- puts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' However, as presented in the results of the first case study, such a penalty does not avoid detecting anomalies for sufficiently large anomalous inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 9 12 Prediction Real value 1D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='0 8500 8750 0006 9250 0056 14000 14250 14500 250 20D 150 140 50 0 8500 8750 94D0 DSZ6 0056 DS26 14000 14250 14501 Time stepFigure 8: Effects of different anomalies with differ- ent magnitudes (as a percentage of input range) in the deep neural network inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Top: Histogram of prediction errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Centre: Cumulative distribution of normalised standardised Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Bot- tom: Histogram of the predicted variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' As the noise levels in the input space increases the confidence of the network decreases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' there is an increment in the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='2 Digital Twin update Here we present the results of updating the digital model with data collected from the physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' This corresponds to the second case study presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Moreover, we compare the regularisa- tion approaches presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='3 in terms of prediction accuracy and the capacity to remember previous behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' An iterative update strategy is used to update the DT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' An initial model is trained with all data from runs performed before April 17, and is used to se- lect the unusual data from proceeding 10 runs (after the training data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The collected data were used to update the DT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Once the model was updated, data from the next 10 runs were processed and collected to again update the DT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' This procedure was repeated until all runs were processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The prediction errors of the initial model and the updated model are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The significant increase in error observed after 2017-09 for the initial model is due to a mainte- nance action on the engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The results show that by updating the DT the behaviour of the physical sys- tem can be predicted, despite changes in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Figure 9: Mean squared error of predictions made by the initial and iteratively retrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' To illustrate how the incorrect simulation of the physical system can affect the identification of anomalies, synthetic anomalies (spikes of fixed length and amplitude based on the signal’s local standard deviation) were injected in randomly selected runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The runs were divided into two groups: before and after the September maintenance event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 9, when the DT is not updated, the behaviour of the system is not predicted accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Therefore, poor performance in the detection of anomalies is expected, and this is shown in the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' When the DT is not updated, 22 of the synthetic anomalies are detected before the maintenance event and only one after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In contrast, when it is updated, 28 and 23 anomalies are detected before and after the overhaul, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The results show an improvement in the number of anomalies detected when the DT is routinely updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' As expected, a more noticeable improvement is made on the data after the maintenance event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The results presented above show that when a 10 8000 Base 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='0 (Normalised) Standardised Euclidean distance 17500 Base 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='0% 15000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='5% 12500 Count 10000 7500 5000 2500 0 1 2 E 4 5 6 Variance (Confidence)Initial model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='020 Retrained model Mean squared error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='000 2017-04 2017-05 2017-06 2017-07 2017-08 2017-09 2017-10 2017-11 2017-12 Datemodel is routinely updated, accurate prediction of engine behaviour is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' However, they do not show how the model is affected in terms of forgetting previous behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' To see this effect, two differ- ent manoeuvres or operating profiles, labelled A and B, were extracted from each engine run, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 10 (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' A model was initially trained with manoeuvre A, and it is assumed that no manoeuvre B data are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Then, the model was iteratively retrained with data from manoeuvre B following the procedure presented above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Three different approaches were used to retrain the model: the L2 regularisation, the L2 − SP regularisation, and an augmented training data approach which includes historic data from ma- noeuvre A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 10 (bottom) shows the prediction of manoeuvre A after retraining the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The pre- diction of manoeuvre B did not vary significantly be- tween approaches and is not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Figure 10: Top: manoeuvres representing different behaviours of the engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Bottom: Prediction of ma- noeuvre A after retraining a model with manoeuvre B using different training approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' As expected, the results show that when the model is updated with the L2 regulariser, the model for- gets how to predict manoeuvre A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The forgetting is significantly reduced when using the L2 − SP regu- lariser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Adding data from manoeuvre A during the retraining improves the capacity of the model to re- member manoeuvre A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' While this approach does not represent a challenge in the toy example presented above, it raises new challenges when data must be selected from a large pool of historical engine be- haviours which might not represent the current sys- tem behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 5 Conclusion Digital Twins are used in a wide range of areas for management, optimisation and decision support of physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' To accurately represent its physi- cal counterpart, DTs must be routinely updated with data collected from the physical system to be able to identify unusual behaviours in the physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' This can be a significant challenge for systems with constrained resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Moreover, not all the collected data are suitable to update the DT, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=', anoma- lies related to emerging faults must be handled care- fully, particularly in our chosen application of sys- tem health monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' This paper presents a novel 2-stage framework to keep a data-driven DT synchro- nised with its the physical system counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The proposed solution allows real-time asset monitoring, and the selection of high-quality data for remote up- date of a DT using fault-free fleet data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Results from a gas turbine engine case study show the capacity of the solution to accurately simulate the behaviour of an engine throughout its life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' To further reduce the computational and data requirements for updat- ing, interesting future research directions include the definition of (optimal) criteria on when to perform model updates and the introduction of physics-based constraints to the machine learning update process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' References [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Tao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Liu, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Nee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Digital twin in industry: State-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' IEEE Transactions on Industrial Informatics, 15(4):2405–2415, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Glaessgen and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Stargel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' The dig- ital twin paradigm for future nasa and u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' air force vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' of AIAA/ASME/ASCE/AHS/ASC structures, structurual dynamics and materials conference, page 118, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 11 Manoeuvre A Manoeuvre B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='B BNEA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content="6 paads E'O t0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='1 50 140 150 240 250 310 1 24 31 50 Timestep Timestep 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='10 L2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='DB L2 SP Initial prediction Augmented training set lenbg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='04 uBa 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='D2 2417-05 2417-06 Dabe[3] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Barricelli, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Casiraghi, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Fogli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' A survey on digital twin: Definitions, characteris- tics, applications, and design implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' IEEE access, 7:167653–167671, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Leng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Shen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Li, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Liu, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Digital twins-based smart manufac- turing system design in industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='0: A review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Journal of manufacturing systems, 60:119–137, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [5] The DTHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' http://https://digitaltwinhub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='uk/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Accessed: 2021-06-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [6] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Qi and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Tao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Digital twin and big data towards smart manufacturing and industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='0: 360 degree comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Ieee Access, 6:3585– 3593, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [7] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Tuegel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Ingraffea, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Eason, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Spottswood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Reengineering aircraft struc- tural life prediction using a digital twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Interna- tional Journal of Aerospace Engineering, 2011, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [8] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Magargle, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Johnson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Mandloi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Davoudabadi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Kesarkar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Krish- naswamy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Batteh, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Pitchaikani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' A simulation-based digital twin for model-driven health monitoring and predictive maintenance of an automotive braking system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' of the International Modelica Conference, pages 35–46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Link¨oping University Electronic Press, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [9] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Mahadevan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Ling, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Choze, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Dynamic bayesian network for air- craft wing health monitoring digital twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Aiaa Journal, 55(3):930–941, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Ye, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Gao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Li, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Digital twin for rotating machinery fault diagno- sis in smart manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' International Jour- nal of Production Research, 57(12):3920–3934, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Chakraborty and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Adhikari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Machine learn- ing based digital twin for dynamical systems with multiple time-scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Computers & Struc- tures, 243:106410, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [12] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Fink, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Svensen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Dersin, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Lee, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Ducoffe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Potential, chal- lenges and future directions for deep learning in prognostics and health management applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Engineering Applications of Artificial In- telligence, 92:103678, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [13] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Burbidge, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Rowland, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' King.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Active learning for regression based on query by committee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' of IDEAL, pages 209–218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Springer, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [14] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Cai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Zhang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Maximizing expected model change for active learning in re- gression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' of ICDM, pages 51–60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' IEEE, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [15] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Ma, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Nie, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Chang, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Hauptmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Multi-class active learning by uncertainty sampling with diversity maximiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' International Journal of Computer Vision, 113(2):113–127, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [16] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Chandola, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Banerjee, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Kumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Anomaly detection: A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' ACM computing surveys (CSUR), 41(3):1–58, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Pimentel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Clifton, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Clifton, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Tarassenko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' A review of novelty detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Signal Processing, 99:215–249, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Rajasegarar, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Leckie, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Palaniswami.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Anomaly detection in wireless sensor networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' IEEE Wireless Communications, 15(4):34–40, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [19] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Attia, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Talhi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Hamou-Lhadj, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Khosravifar, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Turpaud, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Cou- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' On-device anomaly detection for resource- limited systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' of ACM/SIGAPP, pages 548–554, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [20] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Amontamavut, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Nakagawa, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Hayakawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Separated linux process log- ging mechanism for embedded systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' of RTCSA, pages 411–414.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' IEEE, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 12 [21] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Saurav, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Malhotra, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' TV, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Gugulothu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Vig, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Agarwal, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Shroff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Online anomaly detection with concept drift adaptation using recurrent neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' of the ACM India Joint International Conference on Data Science and Management of Data, pages 78–87, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [22] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Reddy, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Yun, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Seo, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Jang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Real-time driver drowsiness detection for em- bedded system using model compression of deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' of CVPR, pages 121– 128, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [23] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Sze, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Chen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Yang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Emer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Ef- ficient processing of deep neural networks: A tutorial and survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Proceedings of the IEEE, 105(12):2295–2329, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [24] Adam Hartwell, Felipe Montana, Will Jacobs, Visakan Kadirkamanathan, Andrew R Mills, and Tom Clark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In-flight novelty detection with convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' arXiv preprint arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='03765, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [25] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Yang and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Shahabi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' A pca-based kernel for kernel pca on multivariate time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' of ICDM, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [26] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Yosinski, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Clune, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Bengio, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Lipson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' How transferable are features in deep neural net- works?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In Advances in neural information pro- cessing systems, pages 3320–3328, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' McCloskey and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Cohen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Catastrophic interference in connectionist networks: The se- quential learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' In Psychology of learning and motivation, volume 24, pages 109– 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Elsevier, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [28] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Kirkpatrick, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Pascanu, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Rabinowitz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Veness, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Desjardins, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Rusu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Milan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Quan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Ramalho, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Grabska-Barwinska, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Overcoming catastrophic forgetting in neu- ral networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' of the national academy of sciences, 114(13):3521–3526, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [29] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Daum´e III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Frustratingly easy domain adap- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' arXiv preprint arXiv:0907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='1815, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' [30] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Grandvalet, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Davoine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' Ex- plicit inductive bias for transfer learning with convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' arXiv preprint arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content='01483, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} +page_content=' 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAzT4oBgHgl3EQfFvq5/content/2301.01017v1.pdf'} diff --git a/a9A0T4oBgHgl3EQfGP_A/vector_store/index.pkl b/a9A0T4oBgHgl3EQfGP_A/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..f91c888abfd3449be8f39fb76fa5fed2614d2e81 --- /dev/null +++ b/a9A0T4oBgHgl3EQfGP_A/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:cb8ada0903454795c73ae39a700355245c8e8d28a5420735056c0febc7a741bb +size 104249 diff --git a/a9AzT4oBgHgl3EQfLPt-/content/tmp_files/2301.01111v1.pdf.txt b/a9AzT4oBgHgl3EQfLPt-/content/tmp_files/2301.01111v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..add42c8e278d0fa6c694e8148a848ec59648e04d --- /dev/null +++ b/a9AzT4oBgHgl3EQfLPt-/content/tmp_files/2301.01111v1.pdf.txt @@ -0,0 +1,1616 @@ +Emergent complexity in slowly driven stochastic processes +Antonio Carlos Costa∗ and Massimo Vergassola +Laboratoire de Physique de l’Ecole normale sup´erieure, ENS, Universit´e PSL, +CNRS, Sorbonne Universit´e, Universit´e de Paris, F-75005 Paris, France +(Dated: January 4, 2023) +We consider the distribution of first passage time events in the presence of non-ergodic modes that +drive otherwise ergodic dynamics on a potential landscape. We find that in the limit of slow and large +enough fluctuations the distribution of first passage time events, f(t), exhibits heavy tails dominated +by a power law with exponent f(t) ∼ t−2, and corrections that depend on the strength and the +nature of fluctuations. We support our theoretical findings through direct numerical simulations in +illustrative examples. +I. +INTRODUCTION +Complex dynamics are ubiquitous in the natural world. +Despite their intrinsic irregularity and unpredictabil- +ity, they can nonetheless exhibit coherent and univer- +sal emergent properties Of particular importance in the +study of complex systems is the understanding of the +time taken for rare events to occur [1–3]. Notable ex- +amples include natural disasters [4] or the spreading of a +virus [5]. In fact, first passage times are central to many +fields within physics and beyond, with important exam- +ples stemming from chemistry, biology and finance (see, +e.g., [6–12] and references therein). +Biology in partic- +ular is ripe with examples where time is of the essence +[13], such as fertilization [14], intracellular events [15–21], +search processes (e.g., foraging or chemotaxis) [22–24], +neural activity [25, 26] or population dynamics [27]. +We here consider the estimation of first passage time +distributions (FPTDs) from finite-time observations in +an experimental context. In particular, we are interested +in systems with intrinsic time scales comparable to the +observation time, for which weak ergodicity breaking be- +comes evident [28, 29]. Such dynamics can be found for +instance in glassy systems [30–33], where the time scales +of equilibration are so long that one can decompose the +dynamics into a stationary component and an “aging” +component that breaks time-translation invariance. +Our main inspiration comes from the less traditional +branch of the physics of animal behavior [34, 35]. Re- +markably, recent advances in machine vision (see, e.g., +[36–39]) have resulted in an explosion of high spatio- +temporal resolution behavioral data. +Analysis of fine- +scale posture movements shows that, much like the run- +and-tumble behavior of bacteria [40], more complex or- +ganisms also exhibit stereotyped behaviors, albeit with +a more intricate structure [41–48]. The notion of stereo- +typy in behavior inherently stems from the time scale sep- +aration between variations on what is defined as a behav- +ioral state, and the transitions between behavioral states, +much like a particle hopping between wells in a poten- +tial landscape. For example, while foraging for food the +∗ antonio.costa@phys.ens.fr +nematode worm C. elegans transitions between coarse- +grained “runs” and “pirouettes”, which are stereotyped +sequences of finer scale movements [48, 49]. +However, +unlike the particle hopping among potential wells which +has a characteristic exponential distribution of transition +rates, the time spent in a given behavior can be heavy- +tailed (see, e.g. Fig. 4E of [48] or Fig. 3 of [50]). We here +hypothesize that such heavy-tailed distributions reflect +the slow continuous modulation of behavior on longer +time scales, resulting from environmental factors or fluc- +tuating internal states driven by neuromodulation, such +as hunger, stress or arousal (see, e.g., [51–53]). Indeed, +it has been shown that C. elegans continuously modu- +lates its rate of reorientation events to explore larger and +larger arenas in search for food [54]. In order to truly +capture the multiscale nature of behavior, we therefore +need to account for the fact that it can be modulated on +time scales comparable to the observation time. +We introduce a general model of behavior in which +the pose dynamics evolves in potential landscapes that +fluctuate over time. We then study how these dynam- +ics impact the estimation of the distribution of times +spent in a given behavior. +In section II we introduce +our phenomenological description of the behavioral dy- +namics, decomposing it into ergodic dynamics on a po- +tential landscape and the non-ergodic modulation of the +landscape. We then derive a general result for the dis- +tribution of first passage times in section III, which we +illustrate through direct numerical simulations of three +example systems in section IV. +II. +SLOWLY DRIVEN ERGODIC DYNAMICS +Given a set of observations of animal locomotion (e.g. +from video imaging), we consider that the dynamics +can be decomposed into ergodic and non-ergodic com- +ponents. The former are the state-space variables that +mix sufficiently well and define the potential wells that +correspond to the stereotyped behaviors ; the latter non- +ergodic components evolve on time scales comparable to +the observation time and slowly modulate the potential +landscape. The full dynamics is thus given by +arXiv:2301.01111v1 [cond-mat.stat-mech] 3 Jan 2023 + +2 +� ˙⃗X = F( ⃗X,⃗λ) +τλ ˙⃗λ = G( ⃗X,⃗λ) +, +(1) +where ⃗X ∈ RD represents the ergodic components, ⃗λ ∈ +RDλ represents the non-ergodic degrees of freedoms, F +and G are nonlinear, possibly noisy, functions, and τλ +is assumed to be of the order of the measurement time +Texp, τλ = O(Texp), such that the ⃗λ dynamics do not mix. +Given the time scale separation between the dynamics of +⃗X and ⃗λ, we assume that the dynamics of ⃗X is well ap- +proximated by quasi-stationary Fokker-Planck dynamics +˙ρ = Lρ, where L represents the Fokker-Planck operator. +Since we are primarily interested in the long time scale +behavior of the system, we consider a projection of the +dynamics onto the slowest mode of L, yielding a gener- +alized Langevin equation [55, 56] with history-dependent +friction and fluctuations. Assuming that we can sample +the system on a time scale longer than the noise correla- +tion time, we obtain an effective overdamped description : +˙⃗X = F( ⃗X,⃗λ) ⇒ ˙x = −∂xU(x, λ) + +� +2Txηx(t) , +(2) +where Tx captures the effective temperature, ηx is Gaus- +sian white noise, and λ is a slow control parameter that +modulates the effective potential landscape on slow time +scales. Similarly, we consider that λ also obeys an effec- +tive overdamped Langevin equation, +˙λ = −τ −1 +λ ∂λV (λ) + +� +2Tλτ −1 +λ ηλ(t), +(3) +where V is assumed to be uncoupled from the dynamics +of x for simplicity, Tλ captures the degree of fluctuations +in λ and ηλ is Gaussian white noise. +III. +FIRST PASSAGE TIME DISTRIBUTIONS +We are primarily interested in studying the time spent +in a given behavioral state. Within the context of the +Langevin dynamics of Eq. 2, this is given by the first pas- +sage time to reach an energy barrier xf from the bottom +of the potential x0, defined as, +τx0,xf (λ) = inf {τ : x(t + τ, λ) = xf|x(t, λ) = x0} . (4) +Despite the general interest in this concept, finding an- +alytical expressions for the density of first passage time +events is generally a formidable task [57]. Remarkably +few closed-form expressions for the FPTD are known, +with most results concerning only the mean first pas- +sage time (MFPT) which is more tractable (see, e.g., +[1, 6, 9]). +However, the MFPT provides only limited +information on the first passage events, especially when +multiple time scales are involved [15]. Here, we are in- +terested in studying the behavior of the full first passage +time distribution, with particular focus on its long time +behavior in the presence of weakly non-ergodic dynamics +for the general dynamics of Eqs. 2 and 3. +As previously discussed, the measurement time Texp +essentially separates ergodic from non-ergodic dynamics. +In addition, it also sets a lower bound on the slowest +observed hopping rates ωmin ∼ T −1 +exp, such that when τλ = +O(Texp) we can make an adiabatic approximation and +assume that transition events occur within a nearly static +potential. For a given hopping rate ω, the FPTD is given +by +f(t, ω) = ωe−ωt , +where ω(λ) = 1/τx0,xf (λ) is the dominating slow kinetic +transition rate which implicitly depends on the dynamics +of λ. Over multiple realizations of a finite-time experi- +ment, we expect that when τλ ∼ Texp the distribution +of first passage times, f(t), will be given by the expec- +tation value of f(t, ω) over the distribution of ω, p(ω), +weighted by the effective number of transition observed +within Texp, which is proportional to ω. Marginalizing +over ω we get, +f(t) ∼ +� ωmax +ωmin +p(ω)ω2e−ωtdω. +(5) +While the barrier height is going to depend on the dy- +namics of a slow control parameter λ, the tail of the dis- +tribution is going to be dominated by instances in which +the barrier height is the largest, motivating the use of +Kramers approximation (see, e.g., [2]), +ω(λ) = ω0 exp +� +−∆U(λ) +Tx +� +, +(6) +where ∆U(λ) = U(xf, λ)−U(x0, λ) and ω0 is a constant. +Given that λ obeys an overdamped Langevin equation, +Eq. 3, the distribution of λ is given by the Boltzmann +weight +p(λ) ∼ exp +� +−V (λ) +2Tλ +� +. +(7) +Leveraging Eqs. 5,6,7 we can obtain an asymptotic ap- +proximation of the FPTD in the large t limit (see Ap- +pendix A) +f(t) ∼ t−2 exp +� +−V (∆U (−1)(Tx log(ω0t))) +2Tλ +� +, +(8) +where ∆U (−1)(·) represents the inverse function of the +potential difference defined by Eq. 6 and we have kept + +3 +FIG. 1. +Heavy-tailed first passage time distributions for a +slowly-driven overdamped harmonic oscillator. (a) We simu- +late the dynamics of a particle in a harmonic oscillator while +slowly driving the potential landscape, and estimate the dis- +tribution of times it takes to reach xf. The gray line rep- +resents the minimum of potential, x0 = s, and the color +scheme different values of s. See Appendix C for simulation +details. (b) FPTDs obtained from direct numerical simula- +tions of Eq. 9 for different values of the temperature Ts that +controls the level of fluctuations for the parameter driving +the slow variations of the potential landscape. As predicted, +the tail of the distribution behaves as a power law with an +exponent f(t) ∼ t−2−α, with α = +Tx +2Ts . The color scheme rep- +resents different ratios of temperatures, and the black dashed +line the Ts → ∞ limit. +only the dominant order of the approximation detailed +in Appendix A. For very general conditions on V (λ) and +U(x, λ), we thus get f(t) ∼ t−2 for t → ∞ and Tλ ≫ 1. +In the following section we will demonstrate the validity +of this result in three illustrative examples. +IV. +ILLUSTRATIVE EXAMPLES +A. +Slowly driven harmonic oscillator +Consider that x evolves in a harmonic potential, +U(x, s) = (x − sxf)2, that is driven by a slow parame- +ter s that fluctuates within V (s) = s2/2, pushing U(x, s) +closer or further from xf in a time scale τs, Fig. 1(a). The +equations of motion are given by a set of Ito stochastic +differential equation, corresponding to coupled Ornstein- +Uhlenbeck processes, +� +dxt = −(xt − stxf)dt + √2TxdWt +dst = −τ −1 +s +st + +� +2Tsτ −1 +s +dWt +, +(9) +where Tx and Ts captures the degree of fluctuations, dWt +is a Wiener Gaussian white noise process. We are inter- +ested in the density of first passage time events from the +minimum of the potential x0 = s to xf = 1, for which +it is challenging to find a closed form analytical expres- +sion, even when s(t) = s ∈ R [57]. In Appendix B, we +derive the FPTD in Laplace space [58] and leverage it +to estimate the FPTD through numerical inversion [59] +FIG. 2. +Heavy-tailed first passage time distribution of a +slowly-driven double-well potential. (a) Schematic of the vari- +ation in the double-well potential with s (colored from blue +to red; the black line represents s = µs). See Appendix C for +simulation details. (b) FPTDs from direct numerical simu- +lations of Eq. 11 for different values of Ts. As expected, the +tail of the distribution behaves as a power law f(t) ∼ t−2−α, +where α = +Tx +4Ts (colored line). The black dashed line repre- +sents the Ts → ∞ limit. +for varying values of τs (as in Ref. [60]), see Fig. A2. +We find that when s fluctuates fast enough, τs → 0, we +can average out s and get the simpler dynamics dxt = +− (xt − ⟨s⟩xf) dt + √2TxdWt. +In this case, the FPTD +is well approximated by f(t) ≈ f(t, ⟨ω⟩) = ⟨ω⟩e−⟨ω⟩t, +where ⟨ω⟩ is the average hopping rate which is set by +⟨s⟩. Even when τs > 0 but short, it is possible to obtain +a self-consistent Markovian dynamics for x(t) (see +e.g., +[1]). In this case, the distribution of first passage times +is still dominantly exponential, but with a corrected first +passage time which depends on the ratio of temperatures +Ts/Tx and the slow time scale τs. However, as we have +shown in the previous section and in Appendix A, when +τs = O(Texp), the distribution of first passage times be- +comes heavy-tailed. In this limit, we can leverage Eq. 8 +to derive an asymptotic approximation to the distribu- +tion of first passage times. The tail of the distribution +will be dominated by low ω values, which correspond +to |s| >> 1. +In this limit, the barrier height primar- +ily behaves as ∆U(s) = s2/2 + O(s). In addition, since +V (s) = s2/2, we see that V (∆U −1(x)) = x and Eq. 8 +yields (see Appendix A for details), +f(t) ∼ t−2− Tx +2Ts , +(10) +which matches what we obtain from direct numerical sim- +ulations of Eq. 9, see Figs. 1(b), A2 and A3(a). +B. +Slowly-driven double-well potential +We now consider a symmetric double-well potential in +which the barrier height is slowly modulated according +to an Ornstein-Uhlenbeck process, Fig. 2(a), +� +dxt = −4s2 +txt(xt − 1)2dt + √2TxdWt +dst = −τ −1 +s +(st − µs)dt + +� +2Tsτ −1 +s +dWt +, +(11) + +(a) +(b) +sf +0 +10-1 +10-5 +10 +f (t) +Ts +1 +10-9 +T +U(a, s) +0.1 +f +100 +102 +104 +106 +(s) 4a +(b) +s +10-1 +0 +10-5 + 10 +f (t) +2-α +10-9 +Ts +T +U(a, s) +10-13 +0.1 +100 +102 +104 +106 +a +t +(s4 +where all the parameters are the same as in Eq. 9 with an +extra µs that represents the expectation value of s, which +we set as µs = 1. In this case, we have a quartic potential +for x, U(x, s) = s2(x2 − 1)2, which yields ∆U(s) = s2. +Since V (s) = s2/2, we see that V (∆U −1(x)) = x/2 and +Eq. 8 yields (see Appendix A for details), +f(t) ∼ t−2− Tx +4Ts , +(12) +matching what we find through direct numerical simula- +tions of Eq. 11, see Figs. 2(b) and A3(b). +C. +Rugged parabolic potential +Finally, we consider a rugged parabolic potential as +a simple model of the rough energy landscapes found +across complex systems, from glasses to proteins (see, +e.g., [19, 20, 61]). +We construct a rugged landscape +by superimposing a sinusoidal perturbation onto a har- +monic potential [62], U(x, s) = U0(x, s) + U1(x), where +U0(x, s) = (x − s)2/2 and U1(x) = − cos(2πkx)/(kπ). +The corresponding dynamics are given by, +� +dxt = − (xt − st + 2 sin(2πkxt)) dt + √2TxdWt +dst = −τ −1 +s +st + +� +2Tsτ −1 +s +dWt +, +(13) +where k sets the number of smaller barriers between the +global minimum of the potential and xf = 1. We set +k = 10 resulting in a rugged potential as illustrated in +Fig. 3(a). In this case, since U(x, s) is not as simple as be- +fore, it is more challenging to derive the correction terms +to the power law. However, it has been shown [62] that +by spatial averaging of U1(x) = − cos(2πkx)/(kπ) over +one period, the resulting hopping rate is simply corrected +by a constant prefactor ω = I−2 +0 (k−1π−1T −1 +x )ω0, where +I0 is the modified Bessel function and ω0 is the hopping +rate in the absence of the sinusoidal perturbation (from +U0(x, s) = (x − s)2/2). As such, we expect the asymp- +totic behavior of f(t) to be the same as for the slowly +driven harmonic potential, Eq. 10. Indeed, this is what +we observe in Figs. 3(b), A3(c). +V. +DISCUSSION +Inspired by quantitative analysis of animal behavior, +we here examined how the existence of slow non-ergodic +modes impacts the statistics collected experimentally, fo- +cusing on the distribution of first passage time events. +Our results show the emergence of heavy-tailed distribu- +tions. In particular, we find that the distribution asymp- +totes to a power law with an exponent f(t) ∼ t−2 in the +limit of large fluctuations, regardless of the details of the +dynamics. As remarked in the Introduction, our results +have important implications to a wide variety of fields, +and we here discuss some of these in detail. +FIG. 3. Heavy-tailed first passage time distribution in slowly +driven rugged parabolic potential. (a) We estimate the first +passage time to reach xf from the global minimum of a rugged +parabolic potential. See Appendix C for simulation details. +(b) FPTDs from direct numerical simulations of Eq. 13 for +different values of Ts. As expected, the tail of the distribution +behaves as a power law f(t) ∼ t−2−α (colored lines) with +α = +Tx +2Ts . The black dashed line corresponds to the Ts → ∞ +limit. +In the context of animal behavior, heavy-tailed first +passage times with an exponent f(t) ≈ t−2 have been +found extensively across multiple species, from bacteria +[63], termites [64] and rats [65] to marine animals [66, 67], +humans [68] and even fossil records [69]. In the context of +search behaviors (e.g., when foraging for food), such ob- +servations have led researchers to hypothesize that L´evy- +flights (power law distributed run lengths) are efficient +search strategies and thus evolutionarily favorable [70– +74]. +However, we here show that such fat tails may +emerge when the animal is continuously adapting its be- +havior (slowly modulating the potential landscape), even +in the absence of external drives. In fact, even in the ab- +sence of active control by the nervous system, morpholog- +ical changes resulting from maturation and development +may be enough to give rise to the emergent power law +behavior, as was observed in fly larvae with a disrupted +nervous system [75]. +Importantly, power laws have been observed in a wide +variety of systems, from solar flares [76, 77] to the brain +[78] and different hypotheses have been put forward to ex- +plain their emergence (for a review see e.g. [79]). Among +these, work inspired by phase transitions in statistical +mechanics associates power laws to “criticality”, mostly +due to the fact that models inferred from the data appear +to require fine-tuning of the model parameters to a spe- +cial regime between two qualitatively different “phases” +(see, e.g., [80]). However, as we have shown here, power +laws can emerge without fine tuning and far from “crit- +icality”. Indeed, slow modes that evolve on time scales +comparable to the observation time are challenging to in- +fer from data, and can give rise to best-fit models that +appear “critical”. While some of the arguments we have +put forward have also been proposed in other contexts +[22, 81–83], we here place them into the framework of out- +of-equilibrium statistical mechanics, explicitly connect- + +a +(b) +10-1 +10-5 +10 +f (t) +S<0 +Ts +10-9 +T. +0.1 +100 +102 +¥104106 +t(s)5 +ing the long time scale emergent behavior with the un- +derlying effective fluctuations. In addition, unlike other +approaches [82, 84], our framework does not require ex- +plicit external drives, but simply collective modes that +evolve in a weakly non-ergodic fashion. +Our starting point is an effective description of the long +time scale dynamics, and further work will be required to +fully bridge between microscopic collective dynamics, and +the emergent long time behavior of the first passage time +distribution that we uncovered. +For example, Fig. A2 +shows that for intermediate values of 1 ≪ τλ ≪ Texp the +FPTD behaves as a truncated power law with an effective +exponent that is slightly smaller that -2, which goes be- +yond arguments presented here. What are the minimum +τλ and Tλ for power laws to be measurable, and how do +simple exponentials (τλ ≪ Texp) transition to power law +behavior? These are important questions if one hopes to +test our predictions in an experimental context (using for +example a set-up akin to the ones used to test stochas- +tic resonance [85, 86]). Additionally, we note that when +τλ ≫ Texp, the distribution of initial conditions is impor- +tant to define the emergent behavior, Fig. A4. Inspired +by experiments in animal behavior, which are typically +done with multiple animals, we here assume that the ini- +tial condition for the slow mode is sampled according +to its Boltzmann distribution λ(t = 0) ∼ e− V (λ) +2Tλ , re- +flecting the variability across individuals. In this case, +the emergent behavior we have derived holds true from +τλ ∼ Texp to τλ → ∞. However, if the variability across +experiments is smaller than that of the Boltzmann dis- +tribution, the τλ → ∞ limit will differ from the behavior +at τλ ∼ Texp. Indeed, in the limit τλ → ∞ the observed +distribution of λ is directly set by the initial condition. +As such, if the variance of the initial distribution of λ +is smaller than that of the Boltzmann distribution, the +temperature Tλ in our derivation should be changed to +a new effective temperature T 0 +λ < Tλ reflecting the lower +variance of the initial conditions. Making this transfor- +mation we still get a power law distribution of first pas- +sage times, but with a modified exponent that reflects +the lower variance. Naturally, for τλ ∼ Texp our result +is independent of the initial conditions, since λ will be +distributed according to the Boltzmann distribution. +To conclude, we have considered the effect of slow non- +ergodic modulations and theoretically captured their ef- +fects on the distribution of first passage times, a result +that we believe is widely relevant to a range of natural +systems. +ACKNOWLEDGEMENTS +We thank Adrian van Kan, St´ephan Fauve, Feder- +ica Ferretti, Tosif Ahamed and Arghyadip Mukherjee +for comments and lively discussions. +This work was +supported by the LabEx ENS-ICFP: ANR-10-LABX- +0010/ANR-10-IDEX-0001-02 PSL*. +AC also acknowl- +edges useful discussions at the Aspen Center for Physics, +which is supported by National Science Foundation +Grant PHY-1607611. +Appendix A: General derivation +We here derive the expression for the first passage +time distribution in a fluctuating potential landscape. +Since ω += +ω0 exp {−∆U(λ)/Tx}, we have λ(ω) += +∆U (−1)(−Tx log (ω/ω0)), where ω0 is a typical (fast) fre- +quency of the hopping dynamics. The distribution p(ω) +obeys p(ω)dω = p(λ)dλ, and it is thus given by +p(ω) ∼ exp +� +−V (λ(ω)) +2Tλ +� +Tx/ω +∂λ∆U(λ) . +By using the adiabatic limit, Eq. 5, we get +f(t) ∝ +� ωmax +ωmin +exp +� +−V (λ(ω)) +2Tλ +� +Tx +∂λ∆U(λ)ωe−ωtdω. +(A1) +The exponential factor e−ωt restricts the contributions to +ω ∼ 1/t, which motivates the change of variable ω = θ +t . +The above integral is then recast in the form +f(t) ∝ t−2 +� θmax(t) +θmin(t) +exp +� +−θ − V (λ(θ)) +2Tλ ++ log (θ) +� +∂λ∆U(λ(θ)) +dθ, +(A2) +where λ(θ) = ∆U (−1) � +−Tx log +� +θ +ω0t +�� +, θmin(t) = ωmint +and θmax(t) = ωmaxt. +To grasp the structure of the integral, it is convenient +to consider first the special case where V and ∆U can +be written as a power series expansion V (λ) ∼ aλn and +∆U(λ) ∼ bλn, a, b ∈ R with an equal dominant (at large +values of the argument, see below) exponent n. The in- +tegral reduces then to the form +f(t) ∝ t−2− aTx +2bTλ +� θmax +θmin +θ1+ aTx +2bTλ e−θ +� +− log +� +θ +ωot +��1− 1 +n dθ . +The power law factor in t yields the scaling result an- +nounced in the main text. +It remains to verify that +the time dependencies at the denominator of the inte- +grand and the limits of integration do not spoil the be- +havior at large times. +This is verified by noting that +the numerator of the integrand has the structure of an +Euler-Γ function of order 2 + +aTx +2bTλ . The integrand has +its maximum at θ∗ = 1 + +aTx +2bTλ , decays over a range of +values of order unity and vanishes at the origin. In that +range, the argument of the power at the denominator +log (ω0t) − log (θ) ≃ log (ω0t), which yields the final scal- +ing with subdominant logarithmic corrections +f(t) ∼ t−2− aTx +2bTλ × log(ω0t) +1 +n −1. +(A3) + +6 +FIG. A1. Numerical integration of f(t) for different choices of V (λ) and ∆U(λ), compared to the asymptotic approximation +of Eq. A5 (black dashed line) with Tx = 0.1 and Tλ = 0.2. +To complete the argument, we note that the time depen- +dency of θmin is not an issue as long as values θ ∼ O(1) +are in the integration range. In practice, this means that +the minimum hopping rate ωmin should be comparable to +(or larger than) the measurement time, ω−1 +min ∼ O(Texp). +Before moving to the general case, two remarks are in +order. +First, for ω0t ≫ 1 the functions V and ∂λ∆U +that appear in Eq. A2 have their argument λ ≫ 1. The +dominant behavior of the two functions should then be +understood for large values of their arguments. Second, +the denominator ∂λ∆U could a priori be included in the +exponential at the numerator but this does not modify +our conclusion. It is indeed easy to verify that the maxi- +mum θ∗ and the decay range would not be not shifted at +the dominant order (and this holds also for the general +case considered hereafter). +We can now consider the general case with different +dominant exponents V (λ) ∼ aλn and ∆U(λ) ∼ bλk, +a, b ∈ R. The argument of the exponential in Eq. A2 +L (θ) = −θ − V (λ(θ)) +2Tλ ++ log (θ) , +(A4) +has its maximum at θ∗, defined by the implicit equation +θ∗ = 1 + Tx +2Tλ +∂λV (λ (θ∗)) +∂λ∆U(λ (θ∗)) = 1 + Tx +2Tλ +an +bk λn−k , +where we have used +∂θV (λ) = ∂λV (λ) × dλ(θ) +dθ +; +dλ(θ) +dθ += − +Tx/θ +∂λ∆U(λ) . +For n < k, the maximum θ∗ ≃ 1 and the integrand +decays in a range of order unity. Indeed, it is checked that +the dominant order of the derivatives ∂pL (p ≥ 2) at θ = +θ∗ coincide with those of log (θ). It follows that L(θ) − +L(θ∗) ≃ log (θ/θ∗)−(θ − θ∗). The resulting integral over +θ is an Euler Γ-function of order two, which indeed forms +at values O(1). In that range, λ ∼ +� Tx +b log (ω0t) +�1/k and +the integral is then approximated by exp{L (θ∗)} and the +function f(t) in Eq. A2 by +f(t) ∼ t−2 exp +� +−a +� Tx +b log(ω0t) +�n/k +2Tλ +� +. +The +factor +at +the +denominator +in +Eq. +A2 +is +O [exp {(1/k − 1) log[log(ω0t)]}] and thus of the same or- +der as terms that we have discarded in our approximation +so we neglect it as well. Since the integral over θ forms for +values O(1), the constraint on the minimum hopping rate +is the same as for the n = k case, i.e., ω−1 +min ∼ O(Texp). +For n > k, the maximum θ∗ ∼ (log ω0t)n/k−1, which +is now large. The dominant order of the derivatives ∂pL +(p ≥ 2) at θ = θ∗ is given by (−1)p−1 (p − 1)! (θ∗)−p−1, +that is they coincide with those of θ∗ log (θ). It follows +that L(θ) − L(θ∗) ≃ θ∗ [log (θ/θ∗) − (θ − θ∗) /θ∗]. The +resulting integral over θ is an Euler Γ-function of (large) +argument θ∗ + 1 : its value is approximated by Stirling +formula, which yields +� +(θ/θ∗)θ∗ +e−(θ−θ∗) dθ ≃ +√ +θ∗. The +√ +θ∗ reflects the fact that the integral forms around the +maximum at θ∗ of the integrand over a range +√ +θ∗, which +implies that the approximation − log (θ/ω0t) ≃ log (ωot) +still holds, as in the previous cases n ≤ k. The +√ +θ∗, as +well as the log (ω0t)1/k−1 coming from the denominator +in Eq. A2, is subdominant with respect to terms that +we have neglected in the expansion of L. We therefore +discard them from our final approximation for n > k : +f(t) ∼ t−2 exp +� +−a +� Tx +b log(ω0t) +�n/k +2Tλ +� +. +Since +the +integral +over +θ +forms +for +values +O +� +(log ω0t)n/k−1� +≫ 1, the condition ω−1 +min ∼ O(Texp) +ensures a fortiori that the finite value of ωmin does not +affect the above result. +Discarding subdominant terms, in all three cases we + +V() = >2 +V() = ^4 +V() = >2 +10-5 +10-5 +10-5 +△U(V) = >2 +AU() = >2 +△U() = 4 +10-10 +10-10 +10-10 +f(t) +10-15 +10-15 +10-15 +10-20 +10-20 +10-20 +10-25 +10-10 +10-5 +10-20 10-15 10-10 10-5 100 +10-20 +10-15 +10-20 +10-15 +10-10 +10-5 +(Tα log(t)n /k +(Tα log(t)n /k +(Ta log(t)n /k +exp +exp +exp +2Tx +2Tx +2Tx7 +FIG. A2. Emergence of heavy-tails when the driving is sufficiently slow (a) First passage time distribution for different values of +τs (colored from purple to yellow) show the emergence of heavy tails as τs → ∞. We also plot the full FPTDs fL(t, s), Eq. (A4), +for different fixed values of s (color coded from red to blue). From these estimates, we highlight fL(t, ⟨s⟩ = 0) (gray dashed +line), which corresponds to the mean first passage time in the limit τs → 0 (purple line). We also plot fL(t) (black dashed line), +which corresponds to the adiabatic approximation of the FPTD obtained through a weighted average of the FPTDs obtained +for fixed s, Eq. (A5); and f(t), which corresponds to Eq. (10). (b) First passage times for 3 qualitatively different regimes and +the corresponding distribution of s at the first passage event (inset). When τs → 0 (left) the distribution of s at the first +passage time events corresponds to the Boltzmann distribution of s, p(s) ∼ e−V (s)/Ts (gray dashed line in the inset), and the +distribution of first passage times f(t) matches the one obtained from a fixed s = ⟨s⟩ = 0 (dashed line); when τs matches the +time scale of x (middle), we observe that the mean first passage time is reduced, boosted by events in which the harmonic +potential moves closer to the boundary as shown in the inset; finally, when τs ≫ 1, we observe the emergence of a heavy tailed +first passage time accurately predicted by Eq. (10) (black line). +thus get the general expression, +f(t) ∼ t−2 exp +� +−a +� Tx +b log(ω0t) +�n/k +2Tλ +� +. +(A5) +To verify the validity of the above arguments, we +show in Fig.A1 how, to dominant order, asymptotic pre- +dictions agree with a detailed numerical integration of +Eq. A1 for ∆U(λ) = λk and V (λ) = λn. +Appendix B: First-passage time through an +absorbing boundary in a harmonic oscillator +We here derive the first passage distribution for the +harmonic oscillator of Eq. 9 but for fixed s, so for +dxt = −(xt − sxf)dt + +� +2TxdWt, +(A1) +where s ∈ R. The corresponding Fokker-Planck equation +is given by +∂tρ =Lρ ; +L =∂x((x − sxf)•) + Tx∂2 +x(•) . +We are interested in the distribution of first passage times +from x0 to the energy barrier located at xf, derived in the +Supplementary Information of [87]. We here reformulate +this derivation and tune it to our particular case. The +survival probability can be written as +S(xf, t|x0) = +� xf +−∞ +Pxf (x, t|x0)dx , +where Pxf (x, t|x0) is the propagator from t = 0 to t with +the constraint that x < xf. +In other words, we have +an absorbing boundary condition at xf, P(xf, t|x0) = 0. +From the survival probability, the first passage density +can be obtained as +fxf (t|x0) = −∂tSxf (t|x0). +The backward Kolmogorov equation for the propagator +Pxf (x, t|x0) is ∂tPxf (x, t|x0) = L† +x0Pxf (x, t|x0), where +L† +x0 is the adjoint of the generator of the stochastic pro- +cess, L† +x0 = −(x0 − sxf)∂x0 • +Tx∂2 +x0•. Integrating the +backward Kolmogorov equation and then taking the time +derivative we get +∂tSxf (t|x0) = L† +x0Sxf (t|x0) +∂tfxf (t|x0) = L† +x0fxf (t|x0), +which vanishes for x0 > xf, fxf (0|x0) = 0, fxf (t|xf) = +δ(t). +To solve for f we take the Laplace transform, +Lfxf (t|x) = ˆfxf (p|x) +� ∞ +0 +e−pt∂tfxf (t|x0)dt =L† +x0 ˆfxf (p|x0) +e−ptfxf (t|x0)|∞ +0 + p +� ∞ +0 +e−ptfxf (t|x0)dt =L† +x0 ˆfxf (p|x0) +fxt(0|x0) + p ˆfxf (p|x0) =L† +x0 ˆfxf (p|x0) +(L† +x0 − p) ˆfxf (p|x0) =0 +where we performed integration by parts and made use +of the initial condition. The unique solution to the above + +(a) +(b) +10-2s +10ls +106s +-1 +0 +1 +Ts +--- fr(t, (s)) +10-1, +10-2 +1.2 +2.0 +--- fL(t) +T +1.5 +pB(s) +一 f(t) +PDF +106 +1.0 +f(t) 10-4 +0.4 +0.5 +f(t) 10-5 +102 +0.0 +0.0 +0.0. +-1 +0 +1 +-1 +0 +1 +-1 +0 +1 +-10-2 +10-6 +s +s +s +2 +10-9 +4 +6 +8 +2 +4 +6 +8 +2 +4 +6 +8 +10-1 +101 +103 +105 +t(×103s) +t(×103s) +t(×103s) +(s) ↑8 +FIG. A3. Details of the accuracy of the asymptotic prediction for the behavior of the tail of f(t) across temperatures Ts for +the slowly driven harmonic oscillator (a), the slowly driven double well potential (b) and the slowly driven rugged parabolic +potential (c). +problem is given by ˆfxf (p|x0) = vp(x0)/vp(xf), where +vp(•) is the unique increasing positive solution of the +equation (L† +x0−p)vp = 0 [58]. For the harmonic oscillator +we get +−(x − sxf)∂xvp(x) + Tx∂2 +xvp(x) − pvp(x) = 0, +which we can solve by rewriting vp(x) as +vp(x) = exp +�x2 − 2sxfx +4D +� +Zp(x), +yielding, +Tx∂2 +xZp(x) + Zp(x) +� +−p + 1 +2 − (x − sxf)2 +4Tx +� += 0 +Rescaling x = √Txy + sxf, and using the chain rule, +∂2 +xZp(x) = T −1 +x ∂2 +yZp(y), we get, +d2Zp(y) +dy2 ++ Zp(y) +� +−y2 +4 + 1 +2 − p +� += 0 +which is Weber’s parabolic cylinder differential equation, +with solution, +Zp(x) = D−p +� +− +� +T −1 +x +(x − sxf)2 +� +, +where Dα is the parabolic cylinder function. Thus, +vp(x) = exp +�x2 − 2sxfx +4Tx +� +D−p +� +− +� +T −1 +x +(x − sxf)2 +� +(A2) +From this we can write an expression for the Laplace +transform of the distribution of times for a particle to go +from x0 = 0 to xf = 1, + +a +100 +10-1 +10-1 +10-1 +10-4 +10-5, +10-5, +10-5 +f(t) +10-8 +10-9 +10-9. +10-9, +100 +102 +104 +100 +102 +104 +100 +102 +104 +100 +102 +104 +106 +(b) +100 +100 +10-1] +10-1 +10-4 +10-4 +10-5 +10-5. +f(t) +10-8 +10-8. +10-9 +10-9, +10-12 +10-12 +100 +102 +104 +106 +100 +102 +104 +106 +100 +102 +104 +106 +100 +102 +104 +106 +(c) +100 +10-1 +10-1 +10-1 +10-4. +10-5 +10-5 +10-5 +f(t) +10-8 +10-9 +10-9/ +10-9, +100 +102 +104 +100 +102 +104 +106 +100 +102 +104 +106 +100 +102 +104 +106 +t(s) +t(s) +t(s) +t(s) +Ts +10-1.2 +10-0.8 +10-0.4 +1009 +FIG. A4. First passage time distribution for different values of +τs (colored from purple to yellow) when the initial condition +is a narrower Boltzmann distribution with T 0 +s = Ts/4. As +discussed in the main text, when τs → ∞ the FPTD exhibits +a deeper power law exponent. When τs ∼ Texp we recover the +asymptotic behavior derived in Eq. 10. +ˆfL(p, s) = ˆfxf =1(p|x0 = 0) = vp(x0 = 0) +vp(xf = 1) += e− 1−2s +4Tx +D−p +� +− +� +s2/Tx +� +D−p +� +− +� +(1 − s)2/Tx +�, +(A3) +which we can invert numerically to evaluate the first pas- +sage time distribution (see e.g. [88]), +fL(t, s) = L(−1) � +ˆfL(p, s) +� +. +(A4) +In Fig. A2(a) we used the method of de Hoog et al. [59] +to numerically invert the Laplace transform and obtain +the FPTDs for each fixed s. In addition, we estimate the +adiabatic approximation of the full FPTD through, +fL(t) = +� +p(s)fL(t, s)ds, +(A5) +where p(s) is the distribution of the values of s during +first passage time events. +Appendix C: Numerics +Simulations +Driven harmonic oscillator : We generate 1000 simu- +lations of the dynamics of Eq. 9 through an Euler-scheme +with a sampling time of ∆t = 10−4 s for Texp = 107 s and +with initial condition x(0) = 0 and s(0) ∼ N(0, √Ts) +is sampled according to the Boltzmann distribution. As +for the parameter values, we take Tx = 0.1, Ts = 0.1 and +vary τs in Figs. A2, A4, and fix τs = 103 × Texp and vary +Ts in Figs. 1, A3(a). +Driven double well potential: We generate 1000 sim- +ulations of the dynamics of Eq. 11 through an Euler- +scheme with a sampling time of ∆t = 10−4 s for Texp = +107 s, τs = 103 × Texp and with initial condition x(0) +which is randomly chosen as x(0) = 1 and x(0) = −1 +with equal probability and s(0) ∼ N(µs, √Ts) is sam- +pled according to the Boltzmann distribution. +As for +the parameter values, we take Tx = 0.15 and vary Ts in +Figs. 2, A3(b). +Driven parabolic potential: We generate 1000 sim- +ulations of the dynamics of Eq. 13 through an Euler- +scheme with a sampling time of ∆t = 10−4 s for Texp = +107 s, τs = 103 × Texp and with initial condition x(0) = 0 +and s(0) ∼ N(0, √Ts) is sampled according to the Boltz- +mann distribution. As for the parameter values, we take +Tx = 0.1 and vary Ts in Figs. 3, A3(c). +Numerical integration +We numerically integrate Eq. A1 with ∆U(λ) = λk and +V (λ) = λn, through a Riemman sum using the midpoint +rule from ωmin = 5×10−10 to ωmax = 1 with ∆ω = 10−9, +yielding the results of Fig. A1. +First passage time distribution estimation +From the simulations of x(t), we first identify all seg- +ments, [t0, tf], in which t0 corresponds to the first time +x returns to x0 for the after reaching xf, and tf is the +time first to reach xf after t0. We then build a normal- +ized histogram of first passages times with logarithmic +bins. We note that while for the slowly driven potential +landscape the position of the minima of the potential are +fixed at x0 = ±1, for the slowly driven harmonic oscilla- +tor and the rugged parabolic potential the minima of the +potential are driven by s. For practical reasons, in the +numerical estimation of the FPTD we take x0 = ⟨s⟩ = 0, +which has no practical impact due to the severe separa- +tion of time scales between the first passage time events +and the relaxation to the well. In practice, even if s ̸= 0, +x is pushed from x to x = s in a negligible time scale +compared to the first passage time to xf. + +10-4 +100 +f(t) 10-8 +Ts +SLZ. +102 +10-12 +Tt +10 +2Ts/4 +100 +102 +104 +106 +(s) 410 +[1] P. H¨anggi, Escape over fluctuating barriers driven by col- +ored noise, Chemical Physics 180, 157 (1994). +[2] N. G. Van Kampen, Stochastic processes in physics and +chemistry (North-Holland, Amsterdam, 1981). +[3] S. Redner, A Guide to First-Passage Processes (Cam- +bridge University Press, 2001). +[4] I. Wijesundera, M. N. Halgamuge, T. Nanayakkara, and +T. Nirmalathas, Natural Disasters, When Will They +Reach Me?, Springer Natural Hazards (Springer Singa- +pore, Singapore, 2016). +[5] A. K. Verma, A. Bhatnagar, D. Mitra, +and R. Pandit, +First-passage-time problem for tracers in turbulent flows +applied to virus spreading, Phys. Rev. Research 2, 033239 +(2020). +[6] A. Szabo, K. Schulten, +and Z. Schulten, First passage +time approach to diffusion controlled reactions, The Jour- +nal of Chemical Physics 72, 4350 (1980). +[7] S. Condamin, V. Tejedor, R. Voituriez, O. B´enichou, +and J. Klafter, Probing microscopic origins of con- +fined subdiffusion by first-passage observables, Proceed- +ings of the National Academy of Sciences 105 (2008), +10.1073/pnas.0712158105. +[8] O. B´enichou and R. Voituriez, From first-passage times +of random walks in confinement to geometry-controlled +kinetics, Physics Reports 539, 225 (2014). +[9] O. B´enichou, T. Gu´erin, +and R. Voituriez, Mean first- +passage times in confined media: from Markovian to non- +Markovian processes, Journal of Physics A: Mathemati- +cal and Theoretical 48, 163001 (2015). +[10] K. R. Ghusinga, +J. J. Dennehy, +and A. Singh, +First-passage time approach to controlling noise in +the +timing +of +intracellular +events, +Proceedings +of +the +National +Academy +of +Sciences +114 +(2017), +10.1073/pnas.1609012114. +[11] R. Chicheportiche and J.-P. Bouchaud, Some applica- +tions of first-passage ideas to finance, in First-Passage +Phenomena and Their Applications (World Scientific, +2014) Chap. 1, pp. 447–476. +[12] X. M. de Wit, A. van Kan, +and A. Alexakis, Bistabil- +ity of the large-scale dynamics in quasi-two-dimensional +turbulence, Journal of Fluid Mechanics 939, R2 (2022). +[13] N. F. Polizzi, M. J. Therien, and D. N. Beratan, Mean +first-passage times in biology, Israel Journal of Chemistry +56, 816 (2016). +[14] B. Meerson and S. Redner, Mortality, redundancy, and +diversity in stochastic search, Phys. Rev. Lett. 114, +198101 (2015). +[15] A. c. v. Godec and R. Metzler, Universal proximity effect +in target search kinetics in the few-encounter limit, Phys. +Rev. X 6, 041037 (2016). +[16] I. Eliazar, T. Koren, and J. Klafter, Searching circular +dna strands, Journal of Physics: Condensed Matter 19, +065140 (2007). +[17] J. Szymanski and M. Weiss, Elucidating the origin of +anomalous diffusion in crowded fluids, Phys. Rev. Lett. +103, 038102 (2009). +[18] Y. Zhang and O. K. Dudko, First-passage processes in +the genome, Annual Review of Biophysics 45, 117 (2016), +pMID: 27391924. +[19] C. +Hyeon +and +D. +Thirumalai, +Can +energy +land- +scape roughness of proteins and rna be measured by +using mechanical unfolding experiments? Proceedings +of +the +National +Academy +of +Sciences +100 +(2003), +10.1073/pnas.1833310100. +[20] R. Nevo, V. Brumfeld, R. Kapon, P. Hinterdorfer, and +Z. Reich, Direct measurement of protein energy landscape +roughness, EMBO reports 6, 482 (2005). +[21] K. Biswas, M. Shreshtha, A. Surendran, and A. Ghosh, +First-passage time statistics of stochastic transcription +process for time-dependent reaction rates, The European +Physical Journal E 42, 24 (2019). +[22] Y. Tu and G. Grinstein, How white noise generates +power-law switching in bacterial flagellar motors, Phys. +Rev. Lett. 94, 208101 (2005). +[23] S. Dev and S. Chatterjee, Optimal search in E. coli +chemotaxis, Phys. Rev. E 91, 042714 (2015). +[24] O. Benichou, M. Coppey, M. Moreau, P.-H. Suet, +and +R. Voituriez, Optimal search strategies for hidden targets, +Physical Review Letters 94, 198101 (2005). +[25] T. Taillefumier and M. O. Magnasco, A phase transi- +tion in the first passage of a Brownian process through a +fluctuating boundary with implications for neural coding, +Proceedings of the National Academy of Sciences 110 +(2013), 10.1073/pnas.1212479110. +[26] K. Pakdaman, S. Tanabe, and T. Shimokawa, Coherence +resonance and discharge time reliability in neurons and +neuronal models, Neural Networks 14, 895 (2001). +[27] T. Spanio, J. Hidalgo, +and M. A. Mu˜noz, Impact of +environmental colored noise in single-species population +dynamics, Phys. Rev. E 96, 042301 (2017). +[28] J. P. Bouchaud, Weak ergodicity breaking and aging +in disordered systems, J. Phys. I France 2 (1992), +10.1051/jp1:1992238. +[29] L. F. Cugliandolo and J. Kurchan, Analytical solution of +the off-equilibrium dynamics of a long-range spin-glass +model, Phys. Rev. Lett. 71, 173 (1993). +[30] K. Binder and A. P. Young, Spin glasses: Experimental +facts, theoretical concepts, and open questions, Reviews +of Modern Physics 58, 801 (1986). +[31] J. Jackle, Models of the glass transition, Reports on +Progress in Physics 49, 171 (1986). +[32] P. G. Debenedetti and F. H. Stillinger, Supercooled liquids +and the glass transition, Nature 410, 259 (2001). +[33] A. P. Young, Spin Glasses and Random Fields, Directions +in condensed matter physics (World Scientific, 1997). +[34] A. X. Brown and B. de Bivort, Ethology as a physical +science, Nature Physics 14, 653 (2018). +[35] G. J. Berman, Measuring behavior across scales, BMC +Biol. 16 (2018), 10.1186/s12915-018-0494-7. +[36] T. D. Pereira, J. W. Shaevitz, +and M. Murthy, Quan- +tifying behavior to understand the brain, Nature Neuro- +science 23, 1537 (2020). +[37] M. W. Mathis and A. Mathis, Deep learning tools for the +measurement of animal behavior in neuroscience, Cur- +rent Opinion in Neurobiology 60, 1 (2020). +[38] E. Yemini, T. Jucikas, L. J. Grundy, A. E. Brown, +and W. R. Schafer, A database of Caenorhabditis el- +egans behavioral phenotypes, Nat Methods 10 (2013), +10.1038/nmeth.2560. +[39] L. Hebert, T. Ahamed, A. C. Costa, L. O’Shaughnessy, +and G. J. Stephens, WormPose: Image synthesis and +convolutional networks for pose estimation in C. elegans, + +11 +PLoS Computational Biology 17, e1008914 (2021). +[40] H. C. Berg, E. coli in motion, Biological and medical +physics series (Springer, New York, 2004). +[41] G. J. Berman, D. M. Choi, W. Bialek, and J. W. Shae- +vitz, Mapping the stereotyped behaviour of freely moving +fruit flies. J. Royal Soc. Interface 11, 1 (2014). +[42] G. J. Stephens, B. Johnson-Kerner, W. Bialek, and W. S. +Ryu, Dimensionality and dynamics in the behavior of C. +elegans. PLoS Comput. Biol. 4, e1000028 (2008). +[43] A. C. Costa, T. Ahamed, and G. J. Stephens, Adaptive, +locally linear models of complex dynamics, Proceedings of +the National Academy of Sciences of the United States +of America 116 (2019), 10.1073/pnas.1813476116. +[44] T. Ahamed, A. C. Costa, and G. J. Stephens, Capturing +the continuous complexity of behaviour in Caenorhabditis +elegans, Nature Physics 17, 275 (2021). +[45] A. B. Wiltschko, M. J. Johnson, G. Iurilli, R. E. Peter- +son, J. M. Katon, S. L. Pashkovski, V. E. Abraira, R. P. +Adams, and S. R. Datta, Mapping Sub-Second Structure +in Mouse Behavior, Neuron 88, 1121 (2015). +[46] R. E. Johnson, S. Linderman, T. Panier, C. L. Wee, +E. Song, K. J. Herrera, A. Miller, and F. Engert, Proba- +bilistic Models of Larval Zebrafish Behavior Reveal Struc- +ture on Many Scales, Current Biology 30, 70 (2020). +[47] V. Alba, G. J. Berman, W. Bialek, +and J. W. Shae- +vitz, Exploring a strongly non-markovian animal behav- +ior, (2020), arXiv:2012.15681 [q-bio.NC]. +[48] A. C. Costa, T. Ahamed, D. Jordan, and G. Stephens, +Maximally predictive ensemble dynamics from data, +(2021), arXiv:2105.12811 [physics.bio-ph]. +[49] J. T. Pierce-Shimomura, T. M. Morse, and S. R. Lock- +ery, The fundamental role of pirouettes in Caenorhabditis +elegans chemotaxis. J. Neurosci. 19, 9557 (1999). +[50] N. Srivastava, D. a. Clark, and a. D. T. Samuel, Temporal +Analysis of Stochastic Turning Behavior of Swimming C. +elegans, Journal of Neurophysiology 102, 1172 (2009). +[51] C. I. Bargmann, Beyond the connectome: +How neu- +romodulators shape neural circuits, BioEssays 34, 458 +(2012). +[52] E. Marder, Neuromodulation of Neuronal Circuits: Back +to the Future, Neuron 76, 1 (2012). +[53] S. W. Flavell, N. Gogolla, M. Lovett-Barron, and M. Ze- +likowsky, The emergence and influence of internal states, +Neuron 110, 2545 (2022). +[54] J. M. Gray, J. J. Hill, +and C. I. Bargmann, A cir- +cuit for navigation in Caenorhabditis elegans, Proceed- +ings of the National Academy of Sciences 102 (2005), +10.1073/pnas.0409009101. +[55] H. Mori, Transport, Collective Motion, and Brownian +Motion, Progress of Theoretical Physics 33 (1965). +[56] R. Zwanzig, Nonlinear generalized Langevin equations, +Journal of Statistical Physics 9, 215 (1973). +[57] D. S. Grebenkov, First exit times of harmonically trapped +particles: a didactic review, Journal of Physics A: Math- +ematical and Theoretical 48, 013001 (2014). +[58] A. J. F. Siegert, On the first passage time probability prob- +lem, Phys. Rev. 81, 617 (1951). +[59] F. R. de Hoog, J. H. Knight, and A. N. Stokes, An im- +proved method for numerical inversion of laplace trans- +forms, SIAM J. Sci. Stat. Comput. 3, 357–366 (1982). +[60] P. Mullowney and S. Iyengar, Parameter estimation for +a leaky integrate-and-fire neuronal model from ISI data, +Journal of Computational Neuroscience 24, 179 (2008). +[61] P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani, and +F. Zamponi, Fractal free energy landscapes in structural +glasses, Nature Communications 5, 3725 (2014). +[62] R. Zwanzig, Diffusion in a rough potential. Proceedings +of the National Academy of Sciences 85, 2029 (1988). +[63] E. Korobkova, T. Emonet, J. M. G. Vilar, T. S. Shimizu, +and P. Cluzel, From molecular noise to behavioural vari- +ability in a single bacterium, Nature 428, 574 (2004). +[64] O. Miramontes, O. DeSouza, L. R. Paiva, A. Marins, and +S. Orozco, L´evy flights and self-similar exploratory be- +haviour of termite workers: Beyond model fitting, PLOS +ONE 9, 1 (2014). +[65] K. Jung, H. Jang, J. D. Kralik, +and J. Jeong, Bursts +and heavy tails in temporal and sequential dynamics of +foraging decisions, PLOS Computational Biology 10, 1 +(2014). +[66] N. E. Humphries, N. Queiroz, J. R. M. Dyer, N. G. +Pade, M. K. Musyl, K. M. Schaefer, D. W. Fuller, J. M. +Brunnschweiler, T. K. Doyle, J. D. R. Houghton, G. C. +Hays, C. S. Jones, L. R. Noble, V. J. Wearmouth, E. J. +Southall, +and D. W. Sims, Environmental context ex- +plains L´evy and Brownian movement patterns of marine +predators, Nature 465, 1066 (2010). +[67] D. W. Sims, E. J. Southall, N. E. Humphries, G. C. Hays, +C. J. A. Bradshaw, J. W. Pitchford, A. James, M. Z. +Ahmed, A. S. Brierley, M. A. Hindell, D. Morritt, M. K. +Musyl, D. Righton, E. L. C. Shepard, V. J. Wearmouth, +R. P. Wilson, M. J. Witt, +and J. D. Metcalfe, Scaling +laws of marine predator search behaviour, Nature 451, +1098 (2008). +[68] D. A. Raichlen, B. M. Wood, A. D. Gordon, A. Z. P. +Mabulla, F. W. Marlowe, +and H. Pontzer, Evidence of +l´evy walk foraging patterns in human hunter–gatherers, +Proceedings of the National Academy of Sciences 111 +(2014), 10.1073/pnas.1318616111. +[69] D. W. Sims, A. M. Reynolds, N. E. Humphries, E. J. +Southall, V. J. Wearmouth, B. Metcalfe, +and R. J. +Twitchett, Hierarchical random walks in trace fossils +and the origin of optimal search behavior, Proceed- +ings of the National Academy of Sciences 111 (2014), +10.1073/pnas.1405966111. +[70] G. M. Viswanathan, S. V. Buldyrev, S. Havlin, M. G. +da Luz, E. P. Raposo, and H. E. Stanley, Optimizing the +success of random searches. Nature 401, 911 (1999). +[71] M. E. Wosniack, M. C. Santos, E. P. Raposo, G. M. +Viswanathan, and M. G. E. da Luz, Robustness of opti- +mal random searches in fragmented environments, Phys. +Rev. E 91, 052119 (2015). +[72] M. E. Wosniack, M. C. Santos, E. P. Raposo, G. M. +Viswanathan, +and M. G. E. da Luz, The evolutionary +origins of l´evy walk foraging, PLOS Computational Biol- +ogy 13, 1 (2017). +[73] B. Guinard and A. Korman, Intermittent inverse-square +levy walks are optimal for finding targets of all sizes, Sci- +ence Advances 7, eabe8211 (2021). +[74] A. Clementi, F. d’Amore, G. Giakkoupis, and E. Natale, +in Proceedings of the 2021 ACM Symposium on Prin- +ciples of Distributed Computing, PODC’21 (Association +for Computing Machinery, New York, NY, USA, 2021) +p. 81–91. +[75] D. W. Sims, N. E. Humphries, N. Hu, V. Medan, +and +J. Berni, Optimal searching behaviour generated intrin- +sically by the central pattern generator for locomotion, +eLife 8, e50316 (2019). + +12 +[76] M. S. Wheatland, P. A. Sturrock, +and J. M. McTier- +nan, The waiting-time distribution of solar flare hard x- +ray bursts, The Astrophysical Journal 509, 448 (1998). +[77] G. Boffetta, V. Carbone, P. Giuliani, P. Veltri, +and +A. Vulpiani, Power laws in solar flares: Self-organized +criticality or turbulence? Phys. Rev. Lett. 83, 4662 +(1999). +[78] J. M. Beggs and D. Plenz, Neuronal avalanches in neocor- +tical circuits, Journal of Neuroscience 23, 11167 (2003). +[79] M. Newman, Power laws, pareto distributions and Zipf’s +law, Contemporary Physics 46, 323 (2005). +[80] T. Mora and W. Bialek, Are Biological Systems Poised +at Criticality? Journal of Statistical Physics 144, 268 +(2011). +[81] J. Touboul and A. Destexhe, Power-law statistics and +universal scaling in the absence of criticality, Phys. Rev. +E 95, 012413 (2017). +[82] V. Priesemann and O. Shriki, Can a time varying exter- +nal drive give rise to apparent criticality in neural sys- +tems? PLOS Computational Biology 14, 1 (2018). +[83] A. Proekt, J. R. Banavar, A. Maritan, and D. W. Pfaff, +Scale invariance in the dynamics of spontaneous behav- +ior, Proceedings of the National Academy of Sciences +109 (2012), 10.1073/pnas.1206894109. +[84] D. J. Schwab, I. Nemenman, +and P. Mehta, Zipf’s law +and criticality in multivariate data without fine-tuning, +Physical Review Letters 113, 1 (2014). +[85] S. Fauve and F. Heslot, Stochastic resonance in a bistable +system, Physics Letters A 97, 5 (1983). +[86] E. Lanzara, R. N. Mantegna, B. Spagnolo, and R. Zan- +gara, Experimental study of a nonlinear system in the +presence of noise: The stochastic resonance, American +Journal of Physics 65, 341 (1997). +[87] M. Vergassola, +V. E. Deneke, +and S. D. Talia, +Mitotic +waves +in +the +early +embryogenesis +of +¡i¿drosophila¡/i¿: +Bistability +traded +for +speed, +Pro- +ceedings of the National Academy of Sciences 115 +(2018), 10.1073/pnas.1714873115. +[88] D. G. Duffy, On the numerical inversion of laplace trans- +forms: comparison of three new methods on character- +istic problems from applications, ACM Transactions on +Mathematical Software 19, 333 (1993). + diff --git a/a9AzT4oBgHgl3EQfLPt-/content/tmp_files/load_file.txt b/a9AzT4oBgHgl3EQfLPt-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..29467b23b0edda53e5bc3000e79805c6f3a8a2e0 --- /dev/null +++ b/a9AzT4oBgHgl3EQfLPt-/content/tmp_files/load_file.txt @@ -0,0 +1,979 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf,len=978 +page_content='Emergent complexity in slowly driven stochastic processes Antonio Carlos Costa∗ and Massimo Vergassola Laboratoire de Physique de l’Ecole normale sup´erieure, ENS, Universit´e PSL, CNRS, Sorbonne Universit´e, Universit´e de Paris, F-75005 Paris, France (Dated: January 4, 2023) We consider the distribution of first passage time events in the presence of non-ergodic modes that drive otherwise ergodic dynamics on a potential landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' We find that in the limit of slow and large enough fluctuations the distribution of first passage time events, f(t), exhibits heavy tails dominated by a power law with exponent f(t) ∼ t−2, and corrections that depend on the strength and the nature of fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' We support our theoretical findings through direct numerical simulations in illustrative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' INTRODUCTION Complex dynamics are ubiquitous in the natural world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Despite their intrinsic irregularity and unpredictabil- ity, they can nonetheless exhibit coherent and univer- sal emergent properties Of particular importance in the study of complex systems is the understanding of the time taken for rare events to occur [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Notable ex- amples include natural disasters [4] or the spreading of a virus [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In fact, first passage times are central to many fields within physics and beyond, with important exam- ples stemming from chemistry, biology and finance (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=', [6–12] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Biology in partic- ular is ripe with examples where time is of the essence [13], such as fertilization [14], intracellular events [15–21], search processes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=', foraging or chemotaxis) [22–24], neural activity [25, 26] or population dynamics [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' We here consider the estimation of first passage time distributions (FPTDs) from finite-time observations in an experimental context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In particular, we are interested in systems with intrinsic time scales comparable to the observation time, for which weak ergodicity breaking be- comes evident [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Such dynamics can be found for instance in glassy systems [30–33], where the time scales of equilibration are so long that one can decompose the dynamics into a stationary component and an “aging” component that breaks time-translation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Our main inspiration comes from the less traditional branch of the physics of animal behavior [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Re- markably, recent advances in machine vision (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=', [36–39]) have resulted in an explosion of high spatio- temporal resolution behavioral data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Analysis of fine- scale posture movements shows that, much like the run- and-tumble behavior of bacteria [40], more complex or- ganisms also exhibit stereotyped behaviors, albeit with a more intricate structure [41–48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The notion of stereo- typy in behavior inherently stems from the time scale sep- aration between variations on what is defined as a behav- ioral state, and the transitions between behavioral states, much like a particle hopping between wells in a poten- tial landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' For example, while foraging for food the ∗ antonio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='costa@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='ens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='fr nematode worm C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' elegans transitions between coarse- grained “runs” and “pirouettes”, which are stereotyped sequences of finer scale movements [48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' However, unlike the particle hopping among potential wells which has a characteristic exponential distribution of transition rates, the time spent in a given behavior can be heavy- tailed (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 4E of [48] or Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 3 of [50]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' We here hypothesize that such heavy-tailed distributions reflect the slow continuous modulation of behavior on longer time scales, resulting from environmental factors or fluc- tuating internal states driven by neuromodulation, such as hunger, stress or arousal (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=', [51–53]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Indeed, it has been shown that C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' elegans continuously modu- lates its rate of reorientation events to explore larger and larger arenas in search for food [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In order to truly capture the multiscale nature of behavior, we therefore need to account for the fact that it can be modulated on time scales comparable to the observation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' We introduce a general model of behavior in which the pose dynamics evolves in potential landscapes that fluctuate over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' We then study how these dynam- ics impact the estimation of the distribution of times spent in a given behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In section II we introduce our phenomenological description of the behavioral dy- namics, decomposing it into ergodic dynamics on a po- tential landscape and the non-ergodic modulation of the landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' We then derive a general result for the dis- tribution of first passage times in section III, which we illustrate through direct numerical simulations of three example systems in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' SLOWLY DRIVEN ERGODIC DYNAMICS Given a set of observations of animal locomotion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' from video imaging), we consider that the dynamics can be decomposed into ergodic and non-ergodic com- ponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The former are the state-space variables that mix sufficiently well and define the potential wells that correspond to the stereotyped behaviors ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' the latter non- ergodic components evolve on time scales comparable to the observation time and slowly modulate the potential landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The full dynamics is thus given by arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='01111v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='stat-mech] 3 Jan 2023 2 � ˙⃗X = F( ⃗X,⃗λ) τλ ˙⃗λ = G( ⃗X,⃗λ) , (1) where ⃗X ∈ RD represents the ergodic components, ⃗λ ∈ RDλ represents the non-ergodic degrees of freedoms, F and G are nonlinear, possibly noisy, functions, and τλ is assumed to be of the order of the measurement time Texp, τλ = O(Texp), such that the ⃗λ dynamics do not mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Given the time scale separation between the dynamics of ⃗X and ⃗λ, we assume that the dynamics of ⃗X is well ap- proximated by quasi-stationary Fokker-Planck dynamics ˙ρ = Lρ, where L represents the Fokker-Planck operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Since we are primarily interested in the long time scale behavior of the system, we consider a projection of the dynamics onto the slowest mode of L, yielding a gener- alized Langevin equation [55, 56] with history-dependent friction and fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Assuming that we can sample the system on a time scale longer than the noise correla- tion time, we obtain an effective overdamped description : ˙⃗X = F( ⃗X,⃗λ) ⇒ ˙x = −∂xU(x, λ) + � 2Txηx(t) , (2) where Tx captures the effective temperature, ηx is Gaus- sian white noise, and λ is a slow control parameter that modulates the effective potential landscape on slow time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Similarly, we consider that λ also obeys an effec- tive overdamped Langevin equation, ˙λ = −τ −1 λ ∂λV (λ) + � 2Tλτ −1 λ ηλ(t), (3) where V is assumed to be uncoupled from the dynamics of x for simplicity, Tλ captures the degree of fluctuations in λ and ηλ is Gaussian white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' FIRST PASSAGE TIME DISTRIBUTIONS We are primarily interested in studying the time spent in a given behavioral state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Within the context of the Langevin dynamics of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 2, this is given by the first pas- sage time to reach an energy barrier xf from the bottom of the potential x0, defined as, τx0,xf (λ) = inf {τ : x(t + τ, λ) = xf|x(t, λ) = x0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' (4) Despite the general interest in this concept, finding an- alytical expressions for the density of first passage time events is generally a formidable task [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Remarkably few closed-form expressions for the FPTD are known, with most results concerning only the mean first pas- sage time (MFPT) which is more tractable (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=', [1, 6, 9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' However, the MFPT provides only limited information on the first passage events, especially when multiple time scales are involved [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Here, we are in- terested in studying the behavior of the full first passage time distribution, with particular focus on its long time behavior in the presence of weakly non-ergodic dynamics for the general dynamics of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' As previously discussed, the measurement time Texp essentially separates ergodic from non-ergodic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In addition, it also sets a lower bound on the slowest observed hopping rates ωmin ∼ T −1 exp, such that when τλ = O(Texp) we can make an adiabatic approximation and assume that transition events occur within a nearly static potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' For a given hopping rate ω, the FPTD is given by f(t, ω) = ωe−ωt , where ω(λ) = 1/τx0,xf (λ) is the dominating slow kinetic transition rate which implicitly depends on the dynamics of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Over multiple realizations of a finite-time experi- ment, we expect that when τλ ∼ Texp the distribution of first passage times, f(t), will be given by the expec- tation value of f(t, ω) over the distribution of ω, p(ω), weighted by the effective number of transition observed within Texp, which is proportional to ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Marginalizing over ω we get, f(t) ∼ � ωmax ωmin p(ω)ω2e−ωtdω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' (5) While the barrier height is going to depend on the dy- namics of a slow control parameter λ, the tail of the dis- tribution is going to be dominated by instances in which the barrier height is the largest, motivating the use of Kramers approximation (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=', [2]), ω(λ) = ω0 exp � −∆U(λ) Tx � , (6) where ∆U(λ) = U(xf, λ)−U(x0, λ) and ω0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Given that λ obeys an overdamped Langevin equation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 3, the distribution of λ is given by the Boltzmann weight p(λ) ∼ exp � −V (λ) 2Tλ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' (7) Leveraging Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 5,6,7 we can obtain an asymptotic ap- proximation of the FPTD in the large t limit (see Ap- pendix A) f(t) ∼ t−2 exp � −V (∆U (−1)(Tx log(ω0t))) 2Tλ � , (8) where ∆U (−1)(·) represents the inverse function of the potential difference defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 6 and we have kept 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Heavy-tailed first passage time distributions for a slowly-driven overdamped harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' (a) We simu- late the dynamics of a particle in a harmonic oscillator while slowly driving the potential landscape, and estimate the dis- tribution of times it takes to reach xf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The gray line rep- resents the minimum of potential, x0 = s, and the color scheme different values of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' See Appendix C for simulation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' (b) FPTDs obtained from direct numerical simula- tions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 9 for different values of the temperature Ts that controls the level of fluctuations for the parameter driving the slow variations of the potential landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' As predicted, the tail of the distribution behaves as a power law with an exponent f(t) ∼ t−2−α, with α = Tx 2Ts .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The color scheme rep- resents different ratios of temperatures, and the black dashed line the Ts → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' only the dominant order of the approximation detailed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' For very general conditions on V (λ) and U(x, λ), we thus get f(t) ∼ t−2 for t → ∞ and Tλ ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In the following section we will demonstrate the validity of this result in three illustrative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' ILLUSTRATIVE EXAMPLES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Slowly driven harmonic oscillator Consider that x evolves in a harmonic potential, U(x, s) = (x − sxf)2, that is driven by a slow parame- ter s that fluctuates within V (s) = s2/2, pushing U(x, s) closer or further from xf in a time scale τs, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The equations of motion are given by a set of Ito stochastic differential equation, corresponding to coupled Ornstein- Uhlenbeck processes, � dxt = −(xt − stxf)dt + √2TxdWt dst = −τ −1 s st + � 2Tsτ −1 s dWt , (9) where Tx and Ts captures the degree of fluctuations, dWt is a Wiener Gaussian white noise process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' We are inter- ested in the density of first passage time events from the minimum of the potential x0 = s to xf = 1, for which it is challenging to find a closed form analytical expres- sion, even when s(t) = s ∈ R [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In Appendix B, we derive the FPTD in Laplace space [58] and leverage it to estimate the FPTD through numerical inversion [59] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Heavy-tailed first passage time distribution of a slowly-driven double-well potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' (a) Schematic of the vari- ation in the double-well potential with s (colored from blue to red;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' the black line represents s = µs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' See Appendix C for simulation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' (b) FPTDs from direct numerical simu- lations of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 11 for different values of Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' As expected, the tail of the distribution behaves as a power law f(t) ∼ t−2−α, where α = Tx 4Ts (colored line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The black dashed line repre- sents the Ts → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' for varying values of τs (as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [60]), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' We find that when s fluctuates fast enough, τs → 0, we can average out s and get the simpler dynamics dxt = − (xt − ⟨s⟩xf) dt + √2TxdWt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In this case, the FPTD is well approximated by f(t) ≈ f(t, ⟨ω⟩) = ⟨ω⟩e−⟨ω⟩t, where ⟨ω⟩ is the average hopping rate which is set by ⟨s⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Even when τs > 0 but short, it is possible to obtain a self-consistent Markovian dynamics for x(t) (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=', [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In this case, the distribution of first passage times is still dominantly exponential, but with a corrected first passage time which depends on the ratio of temperatures Ts/Tx and the slow time scale τs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' However, as we have shown in the previous section and in Appendix A, when τs = O(Texp), the distribution of first passage times be- comes heavy-tailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In this limit, we can leverage Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 8 to derive an asymptotic approximation to the distribu- tion of first passage times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The tail of the distribution will be dominated by low ω values, which correspond to |s| >> 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In this limit, the barrier height primar- ily behaves as ∆U(s) = s2/2 + O(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In addition, since V (s) = s2/2, we see that V (∆U −1(x)) = x and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 8 yields (see Appendix A for details), f(t) ∼ t−2− Tx 2Ts , (10) which matches what we obtain from direct numerical sim- ulations of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 9, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 1(b), A2 and A3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Slowly-driven double-well potential We now consider a symmetric double-well potential in which the barrier height is slowly modulated according to an Ornstein-Uhlenbeck process, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 2(a), � dxt = −4s2 txt(xt − 1)2dt + √2TxdWt dst = −τ −1 s (st − µs)dt + � 2Tsτ −1 s dWt , (11) (a) (b) sf 0 10-1 10-5 10 f (t) Ts 1 10-9 T U(a, s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1 f 100 102 104 106 (s) 4a (b) s 10-1 0 10-5 10 f (t) 2-α 10-9 Ts T U(a, s) 10-13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1 100 102 104 106 a t (s4 where all the parameters are the same as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 9 with an extra µs that represents the expectation value of s, which we set as µs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In this case, we have a quartic potential for x, U(x, s) = s2(x2 − 1)2, which yields ∆U(s) = s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Since V (s) = s2/2, we see that V (∆U −1(x)) = x/2 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 8 yields (see Appendix A for details), f(t) ∼ t−2− Tx 4Ts , (12) matching what we find through direct numerical simula- tions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 11, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 2(b) and A3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Rugged parabolic potential Finally, we consider a rugged parabolic potential as a simple model of the rough energy landscapes found across complex systems, from glasses to proteins (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=', [19, 20, 61]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' We construct a rugged landscape by superimposing a sinusoidal perturbation onto a har- monic potential [62], U(x, s) = U0(x, s) + U1(x), where U0(x, s) = (x − s)2/2 and U1(x) = − cos(2πkx)/(kπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The corresponding dynamics are given by, � dxt = − (xt − st + 2 sin(2πkxt)) dt + √2TxdWt dst = −τ −1 s st + � 2Tsτ −1 s dWt , (13) where k sets the number of smaller barriers between the global minimum of the potential and xf = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' We set k = 10 resulting in a rugged potential as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In this case, since U(x, s) is not as simple as be- fore, it is more challenging to derive the correction terms to the power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' However, it has been shown [62] that by spatial averaging of U1(x) = − cos(2πkx)/(kπ) over one period, the resulting hopping rate is simply corrected by a constant prefactor ω = I−2 0 (k−1π−1T −1 x )ω0, where I0 is the modified Bessel function and ω0 is the hopping rate in the absence of the sinusoidal perturbation (from U0(x, s) = (x − s)2/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' As such, we expect the asymp- totic behavior of f(t) to be the same as for the slowly driven harmonic potential, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Indeed, this is what we observe in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 3(b), A3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' DISCUSSION Inspired by quantitative analysis of animal behavior, we here examined how the existence of slow non-ergodic modes impacts the statistics collected experimentally, fo- cusing on the distribution of first passage time events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Our results show the emergence of heavy-tailed distribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In particular, we find that the distribution asymp- totes to a power law with an exponent f(t) ∼ t−2 in the limit of large fluctuations, regardless of the details of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' As remarked in the Introduction, our results have important implications to a wide variety of fields, and we here discuss some of these in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Heavy-tailed first passage time distribution in slowly driven rugged parabolic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' (a) We estimate the first passage time to reach xf from the global minimum of a rugged parabolic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' See Appendix C for simulation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' (b) FPTDs from direct numerical simulations of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 13 for different values of Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' As expected, the tail of the distribution behaves as a power law f(t) ∼ t−2−α (colored lines) with α = Tx 2Ts .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The black dashed line corresponds to the Ts → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In the context of animal behavior, heavy-tailed first passage times with an exponent f(t) ≈ t−2 have been found extensively across multiple species, from bacteria [63], termites [64] and rats [65] to marine animals [66, 67], humans [68] and even fossil records [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In the context of search behaviors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=', when foraging for food), such ob- servations have led researchers to hypothesize that L´evy- flights (power law distributed run lengths) are efficient search strategies and thus evolutionarily favorable [70– 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' However, we here show that such fat tails may emerge when the animal is continuously adapting its be- havior (slowly modulating the potential landscape), even in the absence of external drives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In fact, even in the ab- sence of active control by the nervous system, morpholog- ical changes resulting from maturation and development may be enough to give rise to the emergent power law behavior, as was observed in fly larvae with a disrupted nervous system [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Importantly, power laws have been observed in a wide variety of systems, from solar flares [76, 77] to the brain [78] and different hypotheses have been put forward to ex- plain their emergence (for a review see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [79]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Among these, work inspired by phase transitions in statistical mechanics associates power laws to “criticality”, mostly due to the fact that models inferred from the data appear to require fine-tuning of the model parameters to a spe- cial regime between two qualitatively different “phases” (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=', [80]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' However, as we have shown here, power laws can emerge without fine tuning and far from “crit- icality”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Indeed, slow modes that evolve on time scales comparable to the observation time are challenging to in- fer from data, and can give rise to best-fit models that appear “critical”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' While some of the arguments we have put forward have also been proposed in other contexts [22, 81–83], we here place them into the framework of out- of-equilibrium statistical mechanics, explicitly connect- a (b) 10-1 10-5 10 f (t) S<0 Ts 10-9 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1 100 102 ¥104106 t(s)5 ing the long time scale emergent behavior with the un- derlying effective fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In addition, unlike other approaches [82, 84], our framework does not require ex- plicit external drives, but simply collective modes that evolve in a weakly non-ergodic fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Our starting point is an effective description of the long time scale dynamics, and further work will be required to fully bridge between microscopic collective dynamics, and the emergent long time behavior of the first passage time distribution that we uncovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' For example, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A2 shows that for intermediate values of 1 ≪ τλ ≪ Texp the FPTD behaves as a truncated power law with an effective exponent that is slightly smaller that -2, which goes be- yond arguments presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' What are the minimum τλ and Tλ for power laws to be measurable, and how do simple exponentials (τλ ≪ Texp) transition to power law behavior?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' These are important questions if one hopes to test our predictions in an experimental context (using for example a set-up akin to the ones used to test stochas- tic resonance [85, 86]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Additionally, we note that when τλ ≫ Texp, the distribution of initial conditions is impor- tant to define the emergent behavior, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Inspired by experiments in animal behavior, which are typically done with multiple animals, we here assume that the ini- tial condition for the slow mode is sampled according to its Boltzmann distribution λ(t = 0) ∼ e− V (λ) 2Tλ , re- flecting the variability across individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In this case, the emergent behavior we have derived holds true from τλ ∼ Texp to τλ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' However, if the variability across experiments is smaller than that of the Boltzmann dis- tribution, the τλ → ∞ limit will differ from the behavior at τλ ∼ Texp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Indeed, in the limit τλ → ∞ the observed distribution of λ is directly set by the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' As such, if the variance of the initial distribution of λ is smaller than that of the Boltzmann distribution, the temperature Tλ in our derivation should be changed to a new effective temperature T 0 λ < Tλ reflecting the lower variance of the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Making this transfor- mation we still get a power law distribution of first pas- sage times, but with a modified exponent that reflects the lower variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Naturally, for τλ ∼ Texp our result is independent of the initial conditions, since λ will be distributed according to the Boltzmann distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' To conclude, we have considered the effect of slow non- ergodic modulations and theoretically captured their ef- fects on the distribution of first passage times, a result that we believe is widely relevant to a range of natural systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank Adrian van Kan, St´ephan Fauve, Feder- ica Ferretti, Tosif Ahamed and Arghyadip Mukherjee for comments and lively discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' This work was supported by the LabEx ENS-ICFP: ANR-10-LABX- 0010/ANR-10-IDEX-0001-02 PSL*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' AC also acknowl- edges useful discussions at the Aspen Center for Physics, which is supported by National Science Foundation Grant PHY-1607611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Appendix A: General derivation We here derive the expression for the first passage time distribution in a fluctuating potential landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Since ω = ω0 exp {−∆U(λ)/Tx}, we have λ(ω) = ∆U (−1)(−Tx log (ω/ω0)), where ω0 is a typical (fast) fre- quency of the hopping dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The distribution p(ω) obeys p(ω)dω = p(λ)dλ, and it is thus given by p(ω) ∼ exp � −V (λ(ω)) 2Tλ � Tx/ω ∂λ∆U(λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' By using the adiabatic limit, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 5, we get f(t) ∝ � ωmax ωmin exp � −V (λ(ω)) 2Tλ � Tx ∂λ∆U(λ)ωe−ωtdω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' (A1) The exponential factor e−ωt restricts the contributions to ω ∼ 1/t, which motivates the change of variable ω = θ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The above integral is then recast in the form f(t) ∝ t−2 � θmax(t) θmin(t) exp � −θ − V (λ(θ)) 2Tλ + log (θ) � ∂λ∆U(λ(θ)) dθ, (A2) where λ(θ) = ∆U (−1) � −Tx log � θ ω0t �� , θmin(t) = ωmint and θmax(t) = ωmaxt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' To grasp the structure of the integral, it is convenient to consider first the special case where V and ∆U can be written as a power series expansion V (λ) ∼ aλn and ∆U(λ) ∼ bλn, a, b ∈ R with an equal dominant (at large values of the argument, see below) exponent n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The in- tegral reduces then to the form f(t) ∝ t−2− aTx 2bTλ � θmax θmin θ1+ aTx 2bTλ e−θ � − log � θ ωot ��1− 1 n dθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The power law factor in t yields the scaling result an- nounced in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' It remains to verify that the time dependencies at the denominator of the inte- grand and the limits of integration do not spoil the be- havior at large times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' This is verified by noting that the numerator of the integrand has the structure of an Euler-Γ function of order 2 + aTx 2bTλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The integrand has its maximum at θ∗ = 1 + aTx 2bTλ , decays over a range of values of order unity and vanishes at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In that range, the argument of the power at the denominator log (ω0t) − log (θ) ≃ log (ω0t), which yields the final scal- ing with subdominant logarithmic corrections f(t) ∼ t−2− aTx 2bTλ × log(ω0t) 1 n −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' (A3) 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Numerical integration of f(t) for different choices of V (λ) and ∆U(λ), compared to the asymptotic approximation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A5 (black dashed line) with Tx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1 and Tλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' To complete the argument, we note that the time depen- dency of θmin is not an issue as long as values θ ∼ O(1) are in the integration range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In practice, this means that the minimum hopping rate ωmin should be comparable to (or larger than) the measurement time, ω−1 min ∼ O(Texp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Before moving to the general case, two remarks are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' First, for ω0t ≫ 1 the functions V and ∂λ∆U that appear in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A2 have their argument λ ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The dominant behavior of the two functions should then be understood for large values of their arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Second, the denominator ∂λ∆U could a priori be included in the exponential at the numerator but this does not modify our conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' It is indeed easy to verify that the maxi- mum θ∗ and the decay range would not be not shifted at the dominant order (and this holds also for the general case considered hereafter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' We can now consider the general case with different dominant exponents V (λ) ∼ aλn and ∆U(λ) ∼ bλk, a, b ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The argument of the exponential in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A2 L (θ) = −θ − V (λ(θ)) 2Tλ + log (θ) , (A4) has its maximum at θ∗, defined by the implicit equation θ∗ = 1 + Tx 2Tλ ∂λV (λ (θ∗)) ∂λ∆U(λ (θ∗)) = 1 + Tx 2Tλ an bk λn−k , where we have used ∂θV (λ) = ∂λV (λ) × dλ(θ) dθ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' dλ(θ) dθ = − Tx/θ ∂λ∆U(λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' For n < k, the maximum θ∗ ≃ 1 and the integrand decays in a range of order unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Indeed, it is checked that the dominant order of the derivatives ∂pL (p ≥ 2) at θ = θ∗ coincide with those of log (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' It follows that L(θ) − L(θ∗) ≃ log (θ/θ∗)−(θ − θ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The resulting integral over θ is an Euler Γ-function of order two, which indeed forms at values O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In that range, λ ∼ � Tx b log (ω0t) �1/k and the integral is then approximated by exp{L (θ∗)} and the function f(t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A2 by f(t) ∼ t−2 exp � −a � Tx b log(ω0t) �n/k 2Tλ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The factor at the denominator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A2 is O [exp {(1/k − 1) log[log(ω0t)]}] and thus of the same or- der as terms that we have discarded in our approximation so we neglect it as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Since the integral over θ forms for values O(1), the constraint on the minimum hopping rate is the same as for the n = k case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=', ω−1 min ∼ O(Texp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' For n > k, the maximum θ∗ ∼ (log ω0t)n/k−1, which is now large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The dominant order of the derivatives ∂pL (p ≥ 2) at θ = θ∗ is given by (−1)p−1 (p − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' (θ∗)−p−1, that is they coincide with those of θ∗ log (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' It follows that L(θ) − L(θ∗) ≃ θ∗ [log (θ/θ∗) − (θ − θ∗) /θ∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The resulting integral over θ is an Euler Γ-function of (large) argument θ∗ + 1 : its value is approximated by Stirling formula, which yields � (θ/θ∗)θ∗ e−(θ−θ∗) dθ ≃ √ θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The √ θ∗ reflects the fact that the integral forms around the maximum at θ∗ of the integrand over a range √ θ∗, which implies that the approximation − log (θ/ω0t) ≃ log (ωot) still holds, as in the previous cases n ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The √ θ∗, as well as the log (ω0t)1/k−1 coming from the denominator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A2, is subdominant with respect to terms that we have neglected in the expansion of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' We therefore discard them from our final approximation for n > k : f(t) ∼ t−2 exp � −a � Tx b log(ω0t) �n/k 2Tλ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Since the integral over θ forms for values O � (log ω0t)n/k−1� ≫ 1, the condition ω−1 min ∼ O(Texp) ensures a fortiori that the finite value of ωmin does not affect the above result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Discarding subdominant terms, in all three cases we V() = >2 V() = ^4 V() = >2 10-5 10-5 10-5 △U(V) = >2 AU() = >2 △U() = 4 10-10 10-10 10-10 f(t) 10-15 10-15 10-15 10-20 10-20 10-20 10-25 10-10 10-5 10-20 10-15 10-10 10-5 100 10-20 10-15 10-20 10-15 10-10 10-5 (Tα log(t)n /k (Tα log(t)n /k (Ta log(t)n /k exp exp exp 2Tx 2Tx 2Tx7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Emergence of heavy-tails when the driving is sufficiently slow (a) First passage time distribution for different values of τs (colored from purple to yellow) show the emergence of heavy tails as τs → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' We also plot the full FPTDs fL(t, s), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' (A4), for different fixed values of s (color coded from red to blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' From these estimates, we highlight fL(t, ⟨s⟩ = 0) (gray dashed line), which corresponds to the mean first passage time in the limit τs → 0 (purple line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' We also plot fL(t) (black dashed line), which corresponds to the adiabatic approximation of the FPTD obtained through a weighted average of the FPTDs obtained for fixed s, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' (A5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' and f(t), which corresponds to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' (b) First passage times for 3 qualitatively different regimes and the corresponding distribution of s at the first passage event (inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' When τs → 0 (left) the distribution of s at the first passage time events corresponds to the Boltzmann distribution of s, p(s) ∼ e−V (s)/Ts (gray dashed line in the inset), and the distribution of first passage times f(t) matches the one obtained from a fixed s = ⟨s⟩ = 0 (dashed line);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' when τs matches the time scale of x (middle), we observe that the mean first passage time is reduced, boosted by events in which the harmonic potential moves closer to the boundary as shown in the inset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' finally, when τs ≫ 1, we observe the emergence of a heavy tailed first passage time accurately predicted by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' (10) (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' thus get the general expression, f(t) ∼ t−2 exp � −a � Tx b log(ω0t) �n/k 2Tλ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' (A5) To verify the validity of the above arguments, we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='A1 how, to dominant order, asymptotic pre- dictions agree with a detailed numerical integration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A1 for ∆U(λ) = λk and V (λ) = λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Appendix B: First-passage time through an absorbing boundary in a harmonic oscillator We here derive the first passage distribution for the harmonic oscillator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 9 but for fixed s, so for dxt = −(xt − sxf)dt + � 2TxdWt, (A1) where s ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The corresponding Fokker-Planck equation is given by ∂tρ =Lρ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' L =∂x((x − sxf)•) + Tx∂2 x(•) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' We are interested in the distribution of first passage times from x0 to the energy barrier located at xf, derived in the Supplementary Information of [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' We here reformulate this derivation and tune it to our particular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The survival probability can be written as S(xf, t|x0) = � xf −∞ Pxf (x, t|x0)dx , where Pxf (x, t|x0) is the propagator from t = 0 to t with the constraint that x < xf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In other words, we have an absorbing boundary condition at xf, P(xf, t|x0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' From the survival probability, the first passage density can be obtained as fxf (t|x0) = −∂tSxf (t|x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The backward Kolmogorov equation for the propagator Pxf (x, t|x0) is ∂tPxf (x, t|x0) = L† x0Pxf (x, t|x0), where L† x0 is the adjoint of the generator of the stochastic pro- cess, L† x0 = −(x0 − sxf)∂x0 • +Tx∂2 x0•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Integrating the backward Kolmogorov equation and then taking the time derivative we get ∂tSxf (t|x0) = L† x0Sxf (t|x0) ∂tfxf (t|x0) = L† x0fxf (t|x0), which vanishes for x0 > xf, fxf (0|x0) = 0, fxf (t|xf) = δ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' To solve for f we take the Laplace transform, Lfxf (t|x) = ˆfxf (p|x) � ∞ 0 e−pt∂tfxf (t|x0)dt =L† x0 ˆfxf (p|x0) e−ptfxf (t|x0)|∞ 0 + p � ∞ 0 e−ptfxf (t|x0)dt =L† x0 ˆfxf (p|x0) fxt(0|x0) + p ˆfxf (p|x0) =L† x0 ˆfxf (p|x0) (L† x0 − p) ˆfxf (p|x0) =0 where we performed integration by parts and made use of the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' The unique solution to the above (a) (b) 10-2s 10ls 106s 1 0 1 Ts --- fr(t, (s)) 10-1, 10-2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='0 --- fL(t) T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='5 pB(s) 一 f(t) PDF 106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='0 f(t) 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='5 f(t) 10-5 102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 1 0 1 1 0 1 1 0 1 10-2 10-6 s s s 2 10-9 4 6 8 2 4 6 8 2 4 6 8 10-1 101 103 105 t(×103s) t(×103s) t(×103s) (s) ↑8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Details of the accuracy of the asymptotic prediction for the behavior of the tail of f(t) across temperatures Ts for the slowly driven harmonic oscillator (a), the slowly driven double well potential (b) and the slowly driven rugged parabolic potential (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' problem is given by ˆfxf (p|x0) = vp(x0)/vp(xf), where vp(•) is the unique increasing positive solution of the equation (L† x0−p)vp = 0 [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' For the harmonic oscillator we get −(x − sxf)∂xvp(x) + Tx∂2 xvp(x) − pvp(x) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' which we can solve by rewriting vp(x) as vp(x) = exp �x2 − 2sxfx 4D � Zp(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' yielding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Tx∂2 xZp(x) + Zp(x) � −p + 1 2 − (x − sxf)2 4Tx � = 0 Rescaling x = √Txy + sxf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' and using the chain rule,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' ∂2 xZp(x) = T −1 x ∂2 yZp(y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' we get,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' d2Zp(y) dy2 + Zp(y) � −y2 4 + 1 2 − p � = 0 which is Weber’s parabolic cylinder differential equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' with solution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Zp(x) = D−p � − � T −1 x (x − sxf)2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' where Dα is the parabolic cylinder function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Thus, vp(x) = exp �x2 − 2sxfx 4Tx � D−p � − � T −1 x (x − sxf)2 � (A2) From this we can write an expression for the Laplace transform of the distribution of times for a particle to go from x0 = 0 to xf = 1, a 100 10-1 10-1 10-1 10-4 10-5, 10-5, 10-5 f(t) 10-8 10-9 10-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 10-9, 100 102 104 100 102 104 100 102 104 100 102 104 106 (b) 100 100 10-1] 10-1 10-4 10-4 10-5 10-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' f(t) 10-8 10-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 10-9 10-9, 10-12 10-12 100 102 104 106 100 102 104 106 100 102 104 106 100 102 104 106 (c) 100 10-1 10-1 10-1 10-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 10-5 10-5 10-5 f(t) 10-8 10-9 10-9/ 10-9, 100 102 104 100 102 104 106 100 102 104 106 100 102 104 106 t(s) t(s) t(s) t(s) Ts 10-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='2 10-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='8 10-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='4 1009 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' First passage time distribution for different values of τs (colored from purple to yellow) when the initial condition is a narrower Boltzmann distribution with T 0 s = Ts/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' As discussed in the main text, when τs → ∞ the FPTD exhibits a deeper power law exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' When τs ∼ Texp we recover the asymptotic behavior derived in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' ˆfL(p, s) = ˆfxf =1(p|x0 = 0) = vp(x0 = 0) vp(xf = 1) = e− 1−2s 4Tx D−p � − � s2/Tx � D−p � − � (1 − s)2/Tx �, (A3) which we can invert numerically to evaluate the first pas- sage time distribution (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [88]), fL(t, s) = L(−1) � ˆfL(p, s) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' (A4) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A2(a) we used the method of de Hoog et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [59] to numerically invert the Laplace transform and obtain the FPTDs for each fixed s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In addition, we estimate the adiabatic approximation of the full FPTD through, fL(t) = � p(s)fL(t, s)ds, (A5) where p(s) is the distribution of the values of s during first passage time events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Appendix C: Numerics Simulations Driven harmonic oscillator : We generate 1000 simu- lations of the dynamics of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 9 through an Euler-scheme with a sampling time of ∆t = 10−4 s for Texp = 107 s and with initial condition x(0) = 0 and s(0) ∼ N(0, √Ts) is sampled according to the Boltzmann distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' As for the parameter values, we take Tx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1, Ts = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1 and vary τs in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A2, A4, and fix τs = 103 × Texp and vary Ts in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 1, A3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Driven double well potential: We generate 1000 sim- ulations of the dynamics of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 11 through an Euler- scheme with a sampling time of ∆t = 10−4 s for Texp = 107 s, τs = 103 × Texp and with initial condition x(0) which is randomly chosen as x(0) = 1 and x(0) = −1 with equal probability and s(0) ∼ N(µs, √Ts) is sam- pled according to the Boltzmann distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' As for the parameter values, we take Tx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='15 and vary Ts in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 2, A3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Driven parabolic potential: We generate 1000 sim- ulations of the dynamics of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 13 through an Euler- scheme with a sampling time of ∆t = 10−4 s for Texp = 107 s, τs = 103 × Texp and with initial condition x(0) = 0 and s(0) ∼ N(0, √Ts) is sampled according to the Boltz- mann distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' As for the parameter values, we take Tx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1 and vary Ts in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 3, A3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Numerical integration We numerically integrate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A1 with ∆U(λ) = λk and V (λ) = λn, through a Riemman sum using the midpoint rule from ωmin = 5×10−10 to ωmax = 1 with ∆ω = 10−9, yielding the results of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' First passage time distribution estimation From the simulations of x(t), we first identify all seg- ments, [t0, tf], in which t0 corresponds to the first time x returns to x0 for the after reaching xf, and tf is the time first to reach xf after t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' We then build a normal- ized histogram of first passages times with logarithmic bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' We note that while for the slowly driven potential landscape the position of the minima of the potential are fixed at x0 = ±1, for the slowly driven harmonic oscilla- tor and the rugged parabolic potential the minima of the potential are driven by s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' For practical reasons, in the numerical estimation of the FPTD we take x0 = ⟨s⟩ = 0, which has no practical impact due to the severe separa- tion of time scales between the first passage time events and the relaxation to the well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' In practice, even if s ̸= 0, x is pushed from x to x = s in a negligible time scale compared to the first passage time to xf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 10-4 100 f(t) 10-8 Ts SLZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 102 10-12 Tt 10 2Ts/4 100 102 104 106 (s) 410 [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' H¨anggi, Escape over fluctuating barriers driven by col- ored noise, Chemical Physics 180, 157 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [2] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Van Kampen, Stochastic processes in physics and chemistry (North-Holland, Amsterdam, 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Redner, A Guide to First-Passage Processes (Cam- bridge University Press, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [4] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Wijesundera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Halgamuge, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Nanayakkara, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Nirmalathas, Natural Disasters, When Will They Reach Me?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=', Springer Natural Hazards (Springer Singa- pore, Singapore, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Verma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Bhatnagar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Mitra, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Pandit, First-passage-time problem for tracers in turbulent flows applied to virus spreading, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Research 2, 033239 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Szabo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Schulten, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Schulten, First passage time approach to diffusion controlled reactions, The Jour- nal of Chemical Physics 72, 4350 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Condamin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Tejedor, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Voituriez, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' B´enichou, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Klafter, Probing microscopic origins of con- fined subdiffusion by first-passage observables, Proceed- ings of the National Academy of Sciences 105 (2008), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='0712158105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [8] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' B´enichou and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Voituriez, From first-passage times of random walks in confinement to geometry-controlled kinetics, Physics Reports 539, 225 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [9] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' B´enichou, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Gu´erin, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Voituriez, Mean first- passage times in confined media: from Markovian to non- Markovian processes, Journal of Physics A: Mathemati- cal and Theoretical 48, 163001 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [10] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Ghusinga, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Dennehy, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Singh, First-passage time approach to controlling noise in the timing of intracellular events, Proceedings of the National Academy of Sciences 114 (2017), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1609012114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [11] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Chicheportiche and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Bouchaud, Some applica- tions of first-passage ideas to finance, in First-Passage Phenomena and Their Applications (World Scientific, 2014) Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 447–476.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [12] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' de Wit, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' van Kan, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Alexakis, Bistabil- ity of the large-scale dynamics in quasi-two-dimensional turbulence, Journal of Fluid Mechanics 939, R2 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [13] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Polizzi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Therien, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Beratan, Mean first-passage times in biology, Israel Journal of Chemistry 56, 816 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [14] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Meerson and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Redner, Mortality, redundancy, and diversity in stochastic search, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 114, 198101 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Godec and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Metzler, Universal proximity effect in target search kinetics in the few-encounter limit, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' X 6, 041037 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [16] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Eliazar, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Koren, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Klafter, Searching circular dna strands, Journal of Physics: Condensed Matter 19, 065140 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Szymanski and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Weiss, Elucidating the origin of anomalous diffusion in crowded fluids, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 103, 038102 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [18] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Zhang and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Dudko, First-passage processes in the genome, Annual Review of Biophysics 45, 117 (2016), pMID: 27391924.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [19] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Hyeon and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Thirumalai, Can energy land- scape roughness of proteins and rna be measured by using mechanical unfolding experiments?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences 100 (2003), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1833310100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [20] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Nevo, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Brumfeld, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Kapon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Hinterdorfer, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Reich, Direct measurement of protein energy landscape roughness, EMBO reports 6, 482 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [21] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Biswas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Shreshtha, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Surendran, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Ghosh, First-passage time statistics of stochastic transcription process for time-dependent reaction rates, The European Physical Journal E 42, 24 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [22] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Tu and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Grinstein, How white noise generates power-law switching in bacterial flagellar motors, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 94, 208101 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [23] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Dev and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Chatterjee, Optimal search in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' coli chemotaxis, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' E 91, 042714 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [24] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Benichou, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Coppey, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Moreau, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Suet, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Voituriez, Optimal search strategies for hidden targets, Physical Review Letters 94, 198101 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [25] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Taillefumier and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Magnasco, A phase transi- tion in the first passage of a Brownian process through a fluctuating boundary with implications for neural coding, Proceedings of the National Academy of Sciences 110 (2013), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1212479110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [26] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Pakdaman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Tanabe, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Shimokawa, Coherence resonance and discharge time reliability in neurons and neuronal models, Neural Networks 14, 895 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [27] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Spanio, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Hidalgo, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Mu˜noz, Impact of environmental colored noise in single-species population dynamics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' E 96, 042301 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [28] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Bouchaud, Weak ergodicity breaking and aging in disordered systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' I France 2 (1992), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1051/jp1:1992238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [29] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Cugliandolo and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Kurchan, Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 71, 173 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [30] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Binder and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Young, Spin glasses: Experimental facts, theoretical concepts, and open questions, Reviews of Modern Physics 58, 801 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [31] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Jackle, Models of the glass transition, Reports on Progress in Physics 49, 171 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [32] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Debenedetti and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Stillinger, Supercooled liquids and the glass transition, Nature 410, 259 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [33] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Young, Spin Glasses and Random Fields, Directions in condensed matter physics (World Scientific, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [34] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Brown and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' de Bivort, Ethology as a physical science, Nature Physics 14, 653 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [35] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Berman, Measuring behavior across scales, BMC Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 16 (2018), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1186/s12915-018-0494-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [36] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Pereira, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Shaevitz, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Murthy, Quan- tifying behavior to understand the brain, Nature Neuro- science 23, 1537 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [37] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Mathis and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Mathis, Deep learning tools for the measurement of animal behavior in neuroscience, Cur- rent Opinion in Neurobiology 60, 1 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [38] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Yemini, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Jucikas, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Grundy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Brown, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Schafer, A database of Caenorhabditis el- egans behavioral phenotypes, Nat Methods 10 (2013), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1038/nmeth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='2560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [39] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Hebert, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Ahamed, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Costa, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' O’Shaughnessy, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Stephens, WormPose: Image synthesis and convolutional networks for pose estimation in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' elegans, 11 PLoS Computational Biology 17, e1008914 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [40] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Berg, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' coli in motion, Biological and medical physics series (Springer, New York, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [41] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Berman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Choi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Bialek, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Shae- vitz, Mapping the stereotyped behaviour of freely moving fruit flies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Royal Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Interface 11, 1 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [42] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Stephens, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Johnson-Kerner, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Bialek, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Ryu, Dimensionality and dynamics in the behavior of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' elegans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' PLoS Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 4, e1000028 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Costa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Ahamed, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Stephens, Adaptive, locally linear models of complex dynamics, Proceedings of the National Academy of Sciences of the United States of America 116 (2019), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1813476116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [44] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Ahamed, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Costa, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Stephens, Capturing the continuous complexity of behaviour in Caenorhabditis elegans, Nature Physics 17, 275 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [45] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Wiltschko, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Johnson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Iurilli, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Peter- son, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Katon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Pashkovski, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Abraira, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Adams, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Datta, Mapping Sub-Second Structure in Mouse Behavior, Neuron 88, 1121 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [46] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Johnson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Linderman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Panier, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Wee, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Song, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Herrera, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Miller, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Engert, Proba- bilistic Models of Larval Zebrafish Behavior Reveal Struc- ture on Many Scales, Current Biology 30, 70 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [47] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Alba, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Berman, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Bialek, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Shae- vitz, Exploring a strongly non-markovian animal behav- ior, (2020), arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='15681 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='NC].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [48] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Costa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Ahamed, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Jordan, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Stephens, Maximally predictive ensemble dynamics from data, (2021), arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='12811 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='bio-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [49] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Pierce-Shimomura, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Morse, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Lock- ery, The fundamental role of pirouettes in Caenorhabditis elegans chemotaxis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Neurosci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 19, 9557 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [50] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Srivastava, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Clark, and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Samuel, Temporal Analysis of Stochastic Turning Behavior of Swimming C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' elegans, Journal of Neurophysiology 102, 1172 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [51] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Bargmann, Beyond the connectome: How neu- romodulators shape neural circuits, BioEssays 34, 458 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [52] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Marder, Neuromodulation of Neuronal Circuits: Back to the Future, Neuron 76, 1 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [53] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Flavell, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Gogolla, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Lovett-Barron, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Ze- likowsky, The emergence and influence of internal states, Neuron 110, 2545 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [54] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Gray, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Hill, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Bargmann, A cir- cuit for navigation in Caenorhabditis elegans, Proceed- ings of the National Academy of Sciences 102 (2005), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='0409009101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [55] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Mori, Transport, Collective Motion, and Brownian Motion, Progress of Theoretical Physics 33 (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [56] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Zwanzig, Nonlinear generalized Langevin equations, Journal of Statistical Physics 9, 215 (1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [57] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Grebenkov, First exit times of harmonically trapped particles: a didactic review, Journal of Physics A: Math- ematical and Theoretical 48, 013001 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [58] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Siegert, On the first passage time probability prob- lem, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 81, 617 (1951).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [59] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' de Hoog, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Knight, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Stokes, An im- proved method for numerical inversion of laplace trans- forms, SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 3, 357–366 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [60] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Mullowney and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Iyengar, Parameter estimation for a leaky integrate-and-fire neuronal model from ISI data, Journal of Computational Neuroscience 24, 179 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [61] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Charbonneau, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Kurchan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Parisi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Urbani, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Zamponi, Fractal free energy landscapes in structural glasses, Nature Communications 5, 3725 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [62] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Zwanzig, Diffusion in a rough potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences 85, 2029 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [63] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Korobkova, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Emonet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Vilar, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Shimizu, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Cluzel, From molecular noise to behavioural vari- ability in a single bacterium, Nature 428, 574 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [64] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Miramontes, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' DeSouza, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Paiva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Marins, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Orozco, L´evy flights and self-similar exploratory be- haviour of termite workers: Beyond model fitting, PLOS ONE 9, 1 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [65] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Jung, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Jang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Kralik, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Jeong, Bursts and heavy tails in temporal and sequential dynamics of foraging decisions, PLOS Computational Biology 10, 1 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [66] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Humphries, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Queiroz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Dyer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Pade, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Musyl, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Schaefer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Fuller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Brunnschweiler, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Doyle, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Houghton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Hays, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Jones, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Noble, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Wearmouth, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Southall, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Sims, Environmental context ex- plains L´evy and Brownian movement patterns of marine predators, Nature 465, 1066 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [67] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Sims, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Southall, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Humphries, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Hays, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Bradshaw, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Pitchford, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' James, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Ahmed, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Brierley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Hindell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Morritt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Musyl, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Righton, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Shepard, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Wearmouth, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Wilson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Witt, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Metcalfe, Scaling laws of marine predator search behaviour, Nature 451, 1098 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [68] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Raichlen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Wood, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Gordon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Mabulla, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Marlowe, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Pontzer, Evidence of l´evy walk foraging patterns in human hunter–gatherers, Proceedings of the National Academy of Sciences 111 (2014), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1318616111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [69] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Sims, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Reynolds, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Humphries, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Southall, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Wearmouth, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Metcalfe, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Twitchett, Hierarchical random walks in trace fossils and the origin of optimal search behavior, Proceed- ings of the National Academy of Sciences 111 (2014), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1405966111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [70] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Viswanathan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Buldyrev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Havlin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' da Luz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Raposo, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Stanley, Optimizing the success of random searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Nature 401, 911 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [71] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Wosniack, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Santos, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Raposo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Viswanathan, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' da Luz, Robustness of opti- mal random searches in fragmented environments, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' E 91, 052119 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [72] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Wosniack, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Santos, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Raposo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Viswanathan, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' da Luz, The evolutionary origins of l´evy walk foraging, PLOS Computational Biol- ogy 13, 1 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [73] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Guinard and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Korman, Intermittent inverse-square levy walks are optimal for finding targets of all sizes, Sci- ence Advances 7, eabe8211 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [74] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Clementi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' d’Amore, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Giakkoupis, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Natale, in Proceedings of the 2021 ACM Symposium on Prin- ciples of Distributed Computing, PODC’21 (Association for Computing Machinery, New York, NY, USA, 2021) p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 81–91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [75] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Sims, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Humphries, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Hu, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Medan, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Berni, Optimal searching behaviour generated intrin- sically by the central pattern generator for locomotion, eLife 8, e50316 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 12 [76] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Wheatland, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Sturrock, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' McTier- nan, The waiting-time distribution of solar flare hard x- ray bursts, The Astrophysical Journal 509, 448 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [77] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Boffetta, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Carbone, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Giuliani, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Veltri, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Vulpiani, Power laws in solar flares: Self-organized criticality or turbulence?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' 83, 4662 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [78] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Beggs and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Plenz, Neuronal avalanches in neocor- tical circuits, Journal of Neuroscience 23, 11167 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [79] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Newman, Power laws, pareto distributions and Zipf’s law, Contemporary Physics 46, 323 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [80] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Mora and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Bialek, Are Biological Systems Poised at Criticality?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Journal of Statistical Physics 144, 268 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [81] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Touboul and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Destexhe, Power-law statistics and universal scaling in the absence of criticality, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' E 95, 012413 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [82] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Priesemann and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Shriki, Can a time varying exter- nal drive give rise to apparent criticality in neural sys- tems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' PLOS Computational Biology 14, 1 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [83] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Proekt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Banavar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Maritan, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Pfaff, Scale invariance in the dynamics of spontaneous behav- ior, Proceedings of the National Academy of Sciences 109 (2012), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1206894109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [84] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Schwab, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Nemenman, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Mehta, Zipf’s law and criticality in multivariate data without fine-tuning, Physical Review Letters 113, 1 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [85] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Fauve and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Heslot, Stochastic resonance in a bistable system, Physics Letters A 97, 5 (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [86] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Lanzara, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Mantegna, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Spagnolo, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Zan- gara, Experimental study of a nonlinear system in the presence of noise: The stochastic resonance, American Journal of Physics 65, 341 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [87] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Vergassola, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Deneke, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Talia, Mitotic waves in the early embryogenesis of ¡i¿drosophila¡/i¿: Bistability traded for speed, Pro- ceedings of the National Academy of Sciences 115 (2018), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content='1714873115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' [88] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} +page_content=' Duffy, On the numerical inversion of laplace trans- forms: comparison of three new methods on character- istic problems from applications, ACM Transactions on Mathematical Software 19, 333 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfLPt-/content/2301.01111v1.pdf'} diff --git a/adFQT4oBgHgl3EQffzab/content/tmp_files/load_file.txt b/adFQT4oBgHgl3EQffzab/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bf66f4ff4d5a07819ede87a286c8ac7a71fdc9bf --- /dev/null +++ b/adFQT4oBgHgl3EQffzab/content/tmp_files/load_file.txt @@ -0,0 +1,1551 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf,len=1550 +page_content='1 Neural Target Speech Extraction: An Overview Katerina Zmolikova, Marc Delcroix, Tsubasa Ochiai, Keisuke Kinoshita, Jan ˇCernock´y, Dong Yu Abstract—Humans can listen to a target speaker even in challenging acoustic conditions that have noise, reverberation, and interfering speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This phenomenon is known as the cocktail-party effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For decades, researchers have focused on approaching the listening ability of humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' One critical issue is handling interfering speakers because the target and non-target speech signals share similar characteristics, complicating their discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Target speech/speaker extraction (TSE) isolates the speech signal of a target speaker from a mixture of several speakers with or without noises and reverberations using clues that identify the speaker in the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Such clues might be a spatial clue indicating the direction of the target speaker, a video of the speaker’s lips, or a pre-recorded enrollment utterance from which their voice characteristics can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' TSE is an emerging field of research that has received increased attention in recent years because it offers a practical approach to the cocktail- party problem and involves such aspects of signal processing as audio, visual, array processing, and deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This paper focuses on recent neural-based approaches and presents an in-depth overview of TSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We guide readers through the different major approaches, emphasizing the similarities among frameworks and discussing potential future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Index Terms—Speech processing, target speech extraction, speech enhancement, multi-modal, deep learning I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' INTRODUCTION In everyday life, we are constantly immersed in complex acoustic scenes consisting of multiple sounds, such as a mix- ture of speech signals from multiple speakers and background noise from air-conditioners or music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Humans naturally extract relevant information from such noisy signals as they enter our ears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The cocktail-party problem is a typical example [1], where we can follow the conversation of a speaker of interest (target speaker) in a noisy room with multiple interfering speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Humans can manage this complex task due to selective attention or a selective hearing mechanism that allows us to focus our attention on a target speaker’s voice and ignore others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Although the mechanisms of human selective hearing are not fully understood yet, many studies have identified essential cues exploited by humans to attend to a target speaker in a speech mixture: spatial, spectral (audio), visual, or semantic cues [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' One long-lasting goal of speech processing research is designing machines that can achieve similar listening abilities as humans, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', selectively extracting the speech of a desired speaker based on auxiliary cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In this paper, we present an overview of recent devel- opments in target speech/speaker extraction (TSE), which estimates the speech signal of a target speaker in a mixture Katerina Zmolikova and Jan ˇCernock´y are with Brno University of Tech- nology, Speech@FIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Marc Delcroix, Tsubasa Ochiai and Keisuke Kinoshita are with NTT Corporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Dong Yu is with Tencent, AI Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Target speech extraction Spatial Target speaker clues Visualal ua Audio Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' TSE problem and examples of clues of several speakers, given auxiliary cues to identify the tar- get1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In the following, we call auxiliary cues, clues, since they represent hints for identifying the target speaker in the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1 illustrates the TSE problem and shows that by exploiting the clues, TSE can focus on the voice of the target speaker while ignoring other speakers or noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Inspired by psychoacoustic studies [1], several clues have been explored to tackle the TSE problem, such as spatial clues that provide the direction of the target speaker [2], [3], visual clues from video of their face [4]–[9], or audio clues extracted from pre- recorded enrollment recording of their voice [10]–[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The TSE problem is directly related to human selective hearing, although we approach it from an engineering point of view and do not try to precisely mimic human mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' TSE is related to other speech and audio-processing tasks such as noise reduction and blind source separation (BSS) that do not use clues about the target speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Although noise reduction does suppress the background noise, it cannot handle well interfering speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' BSS estimates each speech source signal in a mixture, which usually requires estimating the number of sources, a step that is often challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Moreover, it estimates the source signals without identifying them, which leads to global permutation ambiguity at its output;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' it remains ambiguous which of the estimated source signals corresponds to the target speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In contrast, TSE focuses on the target speaker’s speech by exploiting clues without assuming knowledge of the number of speakers in the mixture and avoids global permutation ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' It thus offers a practical alternative to noise reduction or BSS when the use case requires focusing on a desired speaker’s voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Solving the TSE problem promises real implications for the development of many applications: (1) robust voice user interfaces or voice-controlled smart devices that only respond to a specific user;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' (2) teleconferencing systems that can remove 1Alternative terms in the literature for TSE include informed source sepa- ration, personalized speech enhancement, or audio-visual speech separation, depending on the context and the modalities involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Copyright ©2023 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Personal use of this material is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='13341v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='AS] 31 Jan 2023 12 interfering speakers close by;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' (3) hearing aids/hearables that can emphasize the voice of a desired interlocutor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' TSE ideas can be traced back to early works on beam- formers [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Several works also extended BSS approaches to exploit clues about the target speaker [4], [5], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Most of these approaches required a microphone array [5] or models trained on a relatively large amount of speech data from the target speaker [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The introduction of neural networks (NNs) enabled the building of powerful models that learn to perform complex conditioning on various clues by leveraging large amounts of speech data of various speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This evolution re- sulted in impressive extraction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Moreover, neural TSE systems can operate with a single microphone and with speakers unseen during the training of the models, allowing more flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This overview paper covers recent TSE development and focuses on neural approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Its remaining sections are organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In Section II, we formalize the TSE problem and its relation to noise reduction and BSS and introduce its historical context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We then present a taxonomy of TSE approaches and motivate the focus of this overview paper in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We describe a general neural TSE framework in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The later sections (V, VI, and VII) introduce implementations of TSE with different clues, such as audio, visual, and spatial clues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We discuss extensions to other tasks in Section VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Finally, we conclude by describing the outlook on remaining issues in Section IX and provide pointers to available resources for experimenting with TSE in Section X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' PROBLEM DEFINITION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Speech recorded with a distant microphone Imagine recording a target speaker’s voice in a living room using a microphone placed on a table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This scenario represents a typical use case of a voice-controlled smart device or a video-conferencing device in a remote-work situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Many sounds may co-occur while the speaker is speaking, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', a vacuum cleaner, music, children screaming, voices from another conversation, or from a TV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The speech signal captured at a microphone thus consists of a mixture of the target speaker’s speech and interference from the speech of other speakers and background noise2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We can express the mixture signal recorded at a microphone as ym = xm s + � k̸=s xm k + vm � �� � ≜im , (1) where ym = [ym[0], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' , ym[T]] ∈ RT , xm s ∈ RT , xm k ∈ RT , and vm ∈ RT are the time-domain signal of the mixture, the target speech, the interference speech, and noise signals, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Variable T represents the duration (number of samples) of the signals, m is the index of the microphone in an array of microphones, s represents the index of the target speaker and k is the index for the other speech sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 2In this paper, we do not explicitly consider the effect of reverberation caused by the reflection of sounds on the walls and surfaces in a room, which also corrupt the recorded signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Some of the approaches we discussed implicitly handle reverberation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We drop microphone index m whenever we deal with single microphone approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In the TSE problem, we are interested in only recovering the target speech of speaker s, xm s , and view all the other sources as undesired signals to be suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We can thus define the interference signal as im ∈ RT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Note that we make no explicit hypotheses about the number of interfering speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' TSE problem and its relation to BSS and noise reduction The TSE problem is to estimate the target speech, given a clue, Cs, as ˆxs = TSE(y, Cs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' θTSE), (2) where ˆxs is the estimate of the target speech, TSE(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' θTSE) represents a TSE system with parameters θTSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The clue, Cs, allows identifying the target speaker in the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' It can be of various types, such as a pre-recorded enrollment utterance, C(a) s , a video signal capturing the face or lips movements of the target speaker, C(v) s , or such spatial information as the direction of arrival (DOA) of the speech of the target speaker, C(d) s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In the later sections, we expand on how to design TSE systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Here, we first emphasize the key difference between TSE and BSS and noise reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 2 compares these three problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' BSS [13], [14] estimates all the source signals in a mixture without requiring clues: {ˆx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' , ˆxK} = BSS(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' θBSS), (3) where BSS(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' θBSS) represents a separation system with pa- rameters θBSS, ˆxk are the estimates of the speech sources, and K is the number of sources in the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' As seen in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' (3), BSS does not and cannot differentiate the target speech from other speech sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Therefore, we cannot know in advance which output corresponds to the target speech, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', there is a global permutation ambiguity problem between the outputs and the speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Besides, since the number of outputs is given by the number of sources, the number of sources K must be known or estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Comparing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' (2) and (3) emphasizes the fundamental difference between TSE and BSS: (1) TSE estimates only the target speech signal, while BSS estimates all the signals, and (2) TSE is conditioned on speaker clue Cs, while BSS only relies on the observed mixture3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Typical use cases for BSS include applications that require estimating speech signals of every speaker, such as automatic meeting transcription systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Noise reduction is another related problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' It assumes that the interference only consists of background noise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', i = v, and can thus enhance the target speech without requiring clues: ˆxs = Denoise(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' θDenoise), (4) where Denoise(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' θDenoise) represents a noise reduction sys- tem with parameters θDenoise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Unlike BSS, a noise reduction system’s output only consists of target speech ˆxs, and there 3Another setup sitting between TSE and BSS is a task that extracts multiple target speakers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', extracting the speech of all the meeting attendees given such information about them as enrollment or videos of all the speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Comparison of TSE with BSS and noise reduction is thus no global permutation ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This is possible if the background noise and speech have distinct characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For example, we can assume that ambient noise and speech signals exhibit different spectro-temporal characteristics that enable their discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, noise reduction cannot suppress interfering speakers because it cannot discriminate among different speakers in a mixture without clues4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Noise reduction is often used, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', in video-conferencing systems or hearing aids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' TSE is an alternative to BSS and noise reduction, which uses a clue to simplify the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Like BSS, it can handle speech mixtures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Like noise reduction, it only estimates the target speaker, thus avoiding global permutation ambiguity and the need to estimate the number of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, TSE requires access to clues, unlike BSS and noise reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Moreover, it must internally perform two sub-tasks: (1) iden- tifying the target speaker and (2) estimating the speech of that speaker in the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' TSE is thus a challenging problem that introduces specific issues and requires dedicated solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A straightforward way to achieve TSE using BSS methods is to first apply BSS and next select the target speaker among the estimated sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Such a cascade system allows the separate development of BSS and speaker identification modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' How- ever, this scheme is usually computationally more expensive and imports some disadvantages of BSS, such as the need to estimate the number of speakers in the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Therefore, we focus on approaches that directly exploit the clues in the extraction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Nevertheless, most TSE research is rooted in BSS, as argued in the following discussion on the historical context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Historical context The first studies related to TSE were performed in the 1980s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Flanagan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [2] explored enhancing a target speaker’s voice in a speech mixture, assuming that the target speech originated from a fixed and known direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' They employed a microphone array to record speech and designed a fixed beamformer that enhanced the signals from the target direction [2], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We consider that this work represents an early TSE system that relies on spatial clues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In the mid-1990s, the BSS problem gained attention with pioneering works on independent component analysis (ICA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 4Some works propose to exploit clues for noise reduction and apply similar ideas of TSE to reduce background noise (and sometimes interfering speakers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In the literature, this is called personalized speech enhancement, which in this paper, we view as a special case of the TSE problem, where only the target speaker is actively speaking [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' ICA estimates spatial filters that separate the sources by relying on the assumption of the independence of the sources in the mixture and the fact that speech signals are non- Gaussian [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A frequency-domain ICA suffers from a fre- quency permutation problem because it treats each frequency independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In the mid-2000s, independent vector analysis (IVA) addressed the frequency-permutation problem by work- ing on vectors spanning all frequency bins, which allowed modeling dependency among frequencies [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Several works have extended ICA and IVA to perform TSE, which simplifies inference by focusing on a single target source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For example, in the late 2000s, TSE systems were designed by incorporating the voice activity information of the target speaker derived from video signals to the ICA criterion, allowing identification and extraction of only the target source [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In the late 2010s, independent vector extraction (IVE) extended IVA to extract a single source out of the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In particular, IVE exploits clues to guide the extraction process, such as the enrollment of the target speaker to achieve TSE [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' All these approaches require a microphone array to capture speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In the first decade of the 2000s, single-channel approaches for BSS emerged, such as factorial hidden Markov model (F-HMM) [17] and non-negative matrix factorization (NMF) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' These approaches relied on pre-trained spectral models of speech signals learned on clean speech data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' An F-HMM is a model of speech mixtures, where the speech of each speaker in the mixture is explicitly modeled using a separate hidden Markov model (HMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The parameters of each speaker-HMM are learned on the clean speech data of that speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The sep- aration process involves inferring the most likely HMM state sequence associated with each speaker-HMM, which requires approximations to make inference tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This approach was the first to achieve super-human performance using only single-channel speech [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In the early 2000s, F-HMM was also among the first approaches to exploit visual clues [4]5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In NMF, the spectrogram of each source is modeled as a multi- plication of pre-learned bases, representing the basic spectral patterns and their time-varying activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' NMF methods have also been extended to multi-channel signals [13] and used to extract a target speaker [19] with a flexible multi-source model of the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The main shortcoming of the F-HMM and NMF methods is that they require pre-trained source models and thus struggle with unseen speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Furthermore, 5This framework needs having clues for all of the speakers, a requirement that negates some of the advantages of TSE, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', the number of speakers must be known beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Despite that, the method does not suffer from global permutation ambiguity, since visual clues identify the target speaker, and we thus include this work in the broader view of TSE methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 4 the inference employs a computationally expensive iterative optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In the mid-2010s, deep NNs (DNNs) were first introduced to address the BSS problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' These approaches rapidly gained attention with the success of deep-clustering and permutation invariant training (PIT) [20], [21], which showed that single- channel speaker-open6 BSS was possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In particular, the introduction of DNNs enabled more accurate and flexible spec- trum modeling and computationally efficient inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' These advances were facilitated by supervised training methods that can exploit a large amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Neural BSS rapidly influenced TSE research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For example, Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [22] trained a speaker-close NN to extract the speech of a target speaker using training data with mixed various interfering speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This work is an initial neural TSE system using audio clues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, using speaker-close models requires a significant amount of data from the target speaker and cannot be extended to speakers unseen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Subsequently, the introduction of TSE systems con- ditioned on speaker characteristics derived from an enrollment utterance significantly mitigated this requirement [10], [11], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Enrollment consists of a recording of a target speaker’s voice, which amounts to a few seconds of speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' With these approaches, audio clue-based TSE became possible for speakers unseen during training as long as an enrollment utterance was available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Furthermore, the flexibility of NNs to integrate different modalities combined with the high modeling capability of face recognition or lip-reading systems offered new possibilities for speaker-open visual clue-based TSE [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' More recently, neural approaches have also been intro- duced for spatial-clue-based TSE [3], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' TSE has gained increased attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For example, dedi- cated tasks were part of such recent evaluation campaigns as the deep noise suppression (DNS)7 and Clarity8 challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Many works have extended TSE to other tasks, such as a direct automatic speech recognition (ASR) of a target speaker from a mixture, which is called target speaker ASR (TS- ASR) [25], [26], or personalized voice activity detection (VAD)/diarization [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Notably, target speaker VAD (TS-VAD)-based diarization [28] has been very successful in such evaluation campaigns as CHiME-69 or DIHARD- 310, outperforming state-of-the-art diarization approaches in challenging conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' TSE TAXONOMY TSE is a vast research area spanning a multitude of ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This section organizes them to emphasize their relations and differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We categorized the techniques using four criteria: 1) type of clues, 2) number of channels, 3) speaker-close vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' open, and 4) generative vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' discriminative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Table I summarizes the taxonomy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' the works in the scope of this overview paper are emphasized in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 6BSS is possible for speakers unseen during training, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', not present in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 7https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='com/en-us/research/academic-program/ deep-noise-suppression-challenge-icassp-2022/ 8https://claritychallenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='io/clarity CC doc 9https://chimechallenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='io/chime6/results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='html 10https://dihardchallenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='io/dihard3/results A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Type of clue The type of clue used to determine the target speaker is an important factor in distinguishing among TSE approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The most prominent types are audio, visual, and spatial clues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This classification also defines the main organization of this article, which covers such approaches in Sections V, VI, and VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Other types have and could be proposed, as we briefly discuss in Section IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' An audio clue consists of a recording of a speech signal of the target speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Such a clue can be helpful, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', in the use case of personal devices, where the user can pre-record an example of their voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Alternatively, for long recordings, such as meetings, clues can be obtained directly from part of the recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The interest in audio clues sharply increased recently with the usage of neural models for TSE [10]–[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Audio clues are perhaps the most universal, because they do not require using any additional devices, such as multiple microphones or a camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, the performance may be limited compared to other clues, since discriminating speakers based only on their voice characteristics is prone to errors due to inter- and intra-speaker variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For example, the voice characteristics of different speakers, such as family members, often closely resemble each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' On the other hand, the voice characteristics of one speaker may change depending on such factors as emotions, health, or age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A visual clue consists of a video of the target speaker talking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This type is often constrained to the speaker’s face, sometimes just to the lip area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Unlike audio clues, visual clues are typically synchronized with audio signals that are processed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', not pre-recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A few works also explored just using a photo of the speaker [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Visual clues have been employed to infer the activity pattern and location of the target speaker [5] or to jointly model audio and visual signals [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Recent works usually use visual clues to guide discriminative models toward extracting the target speaker [7]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Visual clues are especially useful when speakers in the recording have similar voices [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, they might be sensitive to physical obstructions of the speaker in the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A spatial clue refers to the target speaker’s location, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', the angle from the recording devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The location can be inferred in practice from a video of the room or a recording of a speaker in the same position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Extracting the speaker based on their location has been researched from mid 1980’s, with beamforming techniques that pioneered this topic [2], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' More recent IVE models use location for initialization [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Finally, several works have shown that NNs informed by location can also achieve promising performance [3], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Spatial clues are inherently applicable only when a recording from multiple microphones is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, they can identify the target speaker in the mixture rather reliably, especially when the speakers are stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Different clues may work better in different situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For example, the performance with audio clues might depend on the similarity of voices of the present speakers, and obstruc- tions in the video may influence visual clues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' As such, it is advantageous to use multiple clues simultaneously to combine their strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Many works have combined audio and visual 5 TABLE I TAXONOMY OF TSE WORKS: APPROACHES WITHIN SCOPE OF THIS OVERVIEW PAPER ARE EMPHASIZED IN RED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Representative approaches References Year Type of clues Number of mic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Speaker-close/open Audio Visual Spatial Single Multi Close Open Fixed beamforming [2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [16]11 1985 ✓ ✓ ✓ Generative Audio-visual F-HMM [4] 2001 ✓12 ✓ ✓ ✓ ICA with visual voice activity [5] 2007 ✓ ✓ ✓ Multi-channel NMF [19] 2011 ✓12 ✓ ✓ IVE with x-vectors [12] 2020 ✓ ✓ ✓ Audio-visual VAE [29] 2020 ✓ ✓ ✓ Discriminative Speaker-specific network [22] 2014 ✓12 ✓ ✓ Multi-channel SpeakerBeam [10],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [30] 2017 ✓ ✓ ✓ SpeakerBeam [10] 2019 ✓ ✓ ✓ VoiceFilter [11] 2019 ✓ ✓ ✓ SpEx [31] 2020 ✓ ✓ ✓ The conversation [7] 2018 ✓ ✓ ✓ Looking-to-listen [8] 2018 ✓ ✓ ✓ On/off-screen audio-visual separation [9] 2018 ✓ ✓ ✓ Landmark-based AV speech enh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [32] 2019 ✓ ✓ ✓ Multi-modal SpeakerBeam [33], [34] 2019 ✓ ✓ ✓ ✓ AV speech enh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' through obstructions [35] 2019 ✓ ✓ ✓ ✓ Neural spatial filter [3] 2019 ✓ ✓ ✓ ✓ Spatial speaker extractor [24] 2019 ✓ ✓ ✓ ✓ Multi-channel multi-modal TSE [36] 2020 ✓ ✓ ✓ ✓ ✓ clues [4], [33], and some have even added spatial clues [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Number of microphones Another way to categorize the TSE approaches is based on the number of microphones (channels) they use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Mul- tiple channels allow the spatial diversity of the sources to be exploited to help discriminate the target speaker from interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Such an approach also closely follows human audition, where binaural signals are crucial for solving the cocktail-party problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' All approaches with spatial clues require using a micro- phone array to capture the direction information of the sources in the mixture [2], [3], [16], [24], [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Some TSE approaches that exploit audio or visual clues also assume multi-channel recordings, such as the extensions of ICA/IVA approaches [5], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Multi-channel approaches generally generate extracted sig- nals with better quality and are thus preferable when record- ings from a microphone array are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, some- times they might fail when the sources are located in the same direction from the viewpoint of the recording device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Moreover, adopting a microphone array is not always an option when developing applications due to cost restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In such cases, single-channel approaches are requested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' They rely on spectral models of speech mixture using either F-HMM or recently NNs and exploit audio [10], [11] or visual clues [7], [8] to identify the target speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Recent single-channel neural TSE systems have achieved re- markable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Interestingly, such approaches can also be easily extended to multi-channel processing by augmenting the input with spatial features [3] or combining the processing 11Since the first works that proposed beamforming were not model-based, we consider them neither generative nor discriminative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 12In speaker-close cases, the models are trained on target speaker’s audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We consider this an audio clue in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' with beamforming [24], [30], as discussed in Section IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For example, using a beamformer usually extracts a higher quality signal due to employing a spatial linear filter to perform extraction, which can benefit ASR applications [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Speaker-open vs speaker-close methods We usually understand the clues used by TSE as short evidence about the target speaker obtained at the time of executing the method, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', one utterance spoken by the target speaker, a video of him/her speaking, or their current location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' There are, however, also methods that use a more significant amount of data from the target speaker (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', several hours of their speech) to build a model specific to that person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' These methods can also be seen as TSE except that the clues involve much more data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We refer to these two categories as the speaker-open and speaker-close methods13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In speaker-open methods, the data of the target speaker are available only during the test time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', the model is trained on the data of different speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In contrast, the target speaker is part of the training data in speaker-close methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Many methods in the past were speaker-close, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', [4] or [19], where the models were trained on the clean utterances of the target speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Also, the first neural models for TSE used a speaker-specific network [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Most recent works on neural methods, which use a clue as an additional input, are speaker-open methods [3], [7], [8], [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Recent IVE methods [12] are also speaker-open, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', they guide the inference of IVE using the embedding of a previously unseen speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 13Speaker-open and speaker-close categories are sometimes referred to as speaker-independent and speaker-dependent, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We avoid this terminology, as in TSE, all systems are informed about the target speaker, and therefore the term speaker-independent might be misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 6 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Generative vs discriminative We can classify TSE into approaches using generative or discriminative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Generative approaches model the joint distribution of the observations, target signals, and clues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The estimated target speech is obtained by maximizing the likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In contrast, discriminative approaches directly estimate the target speech signal given observations and clues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In the TSE literature, generative models were the dominant choice in the pioneering works, including one [4] that used HMMs to jointly model audio and visual modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' IVE [12] is also based on a generative model of the mixtures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The popularity of discriminative models, in particular NNs, has increased since mid-2010’s, and such models today are the choice for many problems, including TSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' With dis- criminative models, TSE is treated as a supervised problem, where the parameters of a TSE model are learned using artificially generated training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The modeling power of NNs enables us to exploit large amounts of such data to build strong speech models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Moreover, the versatility of NNs enables complex dependencies to be learned between different types of observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', speech mixture and video/speaker embeddings), which allows the successful conditioning of the extraction process on various clues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, NNs also bring new challenges, such as generalization to unseen conditions or high computational requirements [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Some recent works have also explored using generative NNs, such as variational autoencoders (VAEs) [29], which might represent a middle-ground between the traditional gen- erative approaches and those using discriminative NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Scope of overview paper In the remainder of our paper, we focus on the neu- ral methods for TSE emphasized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Recent neural TSE approaches opened the possibility of achieving high- performance extraction with various clues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' They can be op- erated with a single microphone and applied for speaker- open conditions, which are very challenging constraints for other schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Consequently, these approaches have received increased attention from both academia and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In the next section, we introduce a general framework to provide a uniformized view of the various NN-based TSE approaches, for both single- and multi-channel approaches, and independently of the type of clues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We then respectively review the approaches relying on audio, visual, and spatial clues in Sections V, VI, and VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' GENERAL FRAMEWORK FOR NEURAL TSE In the previous section, we introduced a taxonomy that de- scribed the diversity of approaches to tackle the TSE problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, recent neural TSE systems have much in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In this section, we introduce a general framework that provides a unified view of a neural TSE system, which shares the same processing flow independently of the type of clue used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' By organizing the existing approaches into a common framework, we hope to illuminate their similarities and differences and establish a firm foundation for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A neural TSE system consists of an NN that estimates the target speech conditioned on a clue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 3 is a schematic diagram of a generic neural TSE system that consists of two main modules: a clue encoder and a speech extraction module, described in more detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Clue encoder The clue encoder pulls out (from the clue, Cs) information that allows the speech extraction module to identify and extract the target speech in the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We can express the processing as Es = ClueEncoder(Cs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' θClue), (5) where ClueEncoder(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' θClue) represents the clue encoder, which can be an NN with learnable parameters θClue, and Es are the clue embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Naturally, the specific implementa- tion of the clue encoder and the information carried within Es largely depend on the type of clues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For example, when the clue is an enrollment utterance, Es = E(a) s ∈ RDEmb will be a speaker embedding vector of dimension DEmb that represents the voice characteristics of the target speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' When dealing with visual clues, Es = E(v) s ∈ RDEmb×N can be a sequence of the embeddings of length N, representing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', the lip movements of the target speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Here N represents the number of time frames of the mixture signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Interestingly, the implementation of the speech extraction module does not depend on the type of clues used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' To provide a description that is independent of the type of clues, hereafter, we consider that Es ∈ RDEmb×N consists of a sequence of embedding vectors of dimension DEmb of length N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Note that we can generate a sequence of embedding vectors for audio clue-based TSE systems by repeating the speaker embedding vector for each time frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Speech extraction module The speech extraction module estimates the target speech from the mixture, given the target speaker embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We can use the same configuration independently of the type of clue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Its process can be decomposed into three main parts: a mixture encoder, a fusion layer, and a target extractor: Zy = MixEncoder(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' θMix), (6) Zs = Fusion(Zy, Es;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' θFusion), (7) ˆxs = TgtExtractor(Zs, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' θTgtExtractor), (8) where MixEncoder(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' θMix), Fusion(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' θFusion), and TgtExtractor(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' θTgtExtractor) respectively represent the mixture encoder, the fusion layer, and the target extractor with parameters θMix, θFusion, and θTgtExtractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Zy ∈ RDy×N and Zs ∈ RDs×N are the internal representations of the mixture before and after conditioning on embedding Es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The mixture encoder performs the following: Y = FE(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' θFE), (9) Zy = MixNet(Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' θMixNet), (10) where FE(·) and MixNet(·) respectively represent the feature extraction process and an NN with parameters θFE and θMixNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 7 Mixture encoder Clue encoder ������������ ������������������������ Mixture Clue Target extractor Fusion layer ������������������������� Target speech ������������������������ ������������������������ ������������������������ Feature extraction NN (MixNet) Extraction process (Mask/Beamformer) NN (MaskNet) Signal reconstruction Speech extraction module Mixture encoder Target extractor Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' General framework for neural TSE TABLE II TYPE OF FUSION LAYERS: L, L1, AND L2 ARE LINEAR TRANSFORMATIONS FOR MAPPING THE DIMENSION OF THE CLUE EMBEDDINGS, DEMB, TO THE DIMENSION OF Zy, DZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' ⊙ REPRESENTS THE ELEMENT-WISE HADAMARD MULTIPLICATION OPERATION OF MATRICES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' ei IS A VECTOR CONTAINING THE ELEMENTS OF THE i-TH ROW OF Es AND diag(·) IS AN OPERATOR THAT CONVERTS A VECTOR INTO A DIAGONAL MATRIX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Fusion type Equation Parameters (θFusion) Concatenation Zs = [Zy, Es] Addition Zs = Zy + LEs L ∈ RDZ×DEmb Multiplication Zs = Zy ⊙ (LEs) L ∈ RDZ×DEmb Feature-wise Linear Modulation (FiLM) Zs = Zy ⊙ (L1Es) + L2Es, L1 ∈ RDZ×DEmb, L2 ∈ RDZ×DEmb Factorized layer Zs = �DEmb i=1 LiZy diag(ei), Li ∈ RDZ×DZ The feature extractor computes the features from the observed mixture signal, Y ∈ RD×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' These can be such spectral features as magnitude spectrum coefficients derived from the short-time Fourier transform (STFT) of the input mixture [7], [8], [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' When using a microphone array, spatial features like interaural phase difference (IPD) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' (21) in Section VII can also be appended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Alternatively, the feature extraction process can be implemented by an NN such as a 1-D convolutional layer that operates directly on the raw input waveform of the microphone signal [23], [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This enables learning of a feature representation optimized for TSE tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The features are then processed with an NN, MixNet(·), which performs a non-linear transformation and captures the time context, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', several past and future frames of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The resulting representation, Zy, of the mixture is (at this point) agnostic of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The fusion layer, sometimes denoted as an adaptation layer, is a key component of a TSE system and allows conditioning of the process on the clue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' It combines Zy with the clue embeddings, Es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Conditioning an NN on auxiliary information is a general problem that has been studied for multi-modal processing or the speaker adaptation of ASR systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' TSE systems have borrowed fusion layers from these fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Table II lists several options for the fusion layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Some widely used fusion layers include: (1) the concatenation of Zy with the clue embeddings Es [7], [8];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' (2) addition14 after transforming the embeddings with linear transformation L to match the dimension of Zy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' (3) multiplication [10];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' (4) a combination of 14Concatenation is similar to addition if a linear transformation follows it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' addition and multiplication denoted as FiLM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' (5) a factorized layer [10], [30], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', the combination of different transfor- mations of the mixture representation weighted by the clue embedding values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Other alternatives have also been proposed, including attention-based fusion [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Note that the fusion operations described here assume just one clue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' It is also possible to use multiple clues, as discussed in Section VI-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Some works also employ the fusion repeatedly at multiple positions in the model [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The last part of the speech extraction module is the target extractor, which estimates the target signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We explain below the time-frequency masking-based extractor, which has been widely used [3], [7], [8], [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Recent approaches also perform a similar masking operation in the learned feature domain [23], [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The time-frequency masking approach was inspired by early BSS studies that relied on the sparseness assumption of speech signals, an idea based on the observation that the energy of a speech signal is concentrated in a few time-frequency bins of a speech spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Accordingly, the speech signals of different speakers rarely overlap in the time-frequency domain in a speech mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We can thus extract the target speech by applying a time-frequency mask on the observed speech mixture, where the mask indicates the time-frequency bins where the target speech is dominant over other signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 4 shows an example of an ideal binary mask for extracting a target speech in a mixture of two speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Such an ideal binary mask assumes that all the energy in each TF bin belongs to one speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In recent mask-based approaches that use real- 8 Mixture TF-mask Extracted signal Time Frequency Time Frequency Time Frequency Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Example of time-frequency mask for speech extraction: Time-frequency mask shows spectrogram regions where target source is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' By applying this mask to the mixture, we obtain an extracted speech signal that estimates the target speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' valued (or complex) masks, this assumption or observation is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The processing of the masking-based extractor can be summarized as Ms = MaskNet(Zs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' θMask), (11) ˆXs = Ms ⊙ Y, (12) ˆxs = Reconstruct( ˆXs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' θReconst), (13) where MaskNet(·) is an NN that estimates the time-frequency mask for the target speech, Ms ∈ RD×N, θMask are the net- work parameters, and ⊙ denotes the element-wise Hadamard multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Y and ˆXs are the mixture and the estimated target speech signals in the feature domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' (12) shows the actual extraction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Reconstruct(·) is an operation to reconstruct the time-domain signal by performing the inverse operation of the feature extraction of the mixture encoder, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', either inverse STFT (iSTFT) or a transpose convolution if using a learnable feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In the latter case, the reconstruction layer has learnable parameters, θReconst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' There are other possibilities to perform the extraction pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For example, we can modify the MaskNet(·) NN to directly infer the target speech signal in the feature domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Alternatively, as discussed in Section IV-C, we can replace the mask-based extraction process with beamforming when a microphone array is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Integration with microphone array processing If we have access to a microphone array to record the speech mixture, we can exploit the spatial information to extract the target speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' One approach is to use spatial clues to identify the speaker in the mixture by informing the system about the target speaker’s direction, as discussed in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Another approach combines TSE with beamforming and uses the latter to perform the extraction process instead of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For example, we can use the output of a TSE system to estimate the spatial statistics needed to compute the coefficients of a beamformer steering in the direction of the target speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This approach can also be used with audio or visual clue-based TSE systems and requires no explicit use of spatial clues to identify the target speaker in the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We briefly review the mask-based beamforming approach, which was introduced initially for noise reduction and BSS [42], [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A beamformer performs the linear spatial filtering of the observed microphone signals: ˆXs[n, f] =WH[f]Y[n, f], (14) where ˆXs[n, f] ∈ C is the STFT coefficient of the esti- mated target signal at time frame n and frequency bin f, W[f] ∈ CM is a vector of the beamformer coefficients, Y[n, f] = � Y 1[n, f], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' , Y M[n, f] � T ∈ CM is a vector of the STFT coefficients of the microphone signals, M is the number of microphones, and H is the conjugate transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We can derive the beamformer coefficients from the spatial correlation matrices of the target speech and the interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' These correlation matrices can be computed from the observed signal and the time-frequency mask estimated by the TSE system [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This way of combining a TSE system with beamforming replaces the time-frequency masking operation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' (12) with the spatial linear filtering operation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' It allows distortionless extraction, which is often advantageous when using TSE as a front-end for ASR [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Training a TSE system Before using a TSE model, we first need to learn its param- eters: θTSE = {θMix, θClue, θFusion, θTgtExtractor}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Most existing studies use fully supervised training, which requires a large amount of training data consisting of the triplets of speech mixture y, target speech signal xs, and corresponding clue Cs to learn parameters θTSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Since this requires access to a clean target speech signal, such training data are usually simulated by artificially mixing clean speech signals and noise following the signal model of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Figure 5 illustrates the data generation process using a multi-speaker audio-visual speech corpus containing multiple videos for each speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' First, we generate a mixture using randomly selected speech signals from the target speaker, the interference speaker, and the background noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We obtain an audio clue by selecting another speech signal from the target speaker as well as a visual clue from the video signal associated with the target speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The training of a neural TSE framework follows the training scheme of NNs with error back-propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The parameters are estimated by minimizing a training loss function: θTSE = arg min θ L (xs, ˆxs) , (15) where L(·) is a training loss, which measures how close estimated target speech ˆxs = TSE (y, Cs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' θ) is to the target source signal xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We can use a similar loss as that employed for training noise reduction or BSS systems [14], [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 9 Audio clue (Enrollment) Visual clue (Video) Speech/Video (target speaker) Speech (interfering speaker) Audio clue: Speech sample from the target speaker different from the target speech Visual clue: Video signal associated with the target speech Noise samples Mixture Target speech Interfering speech Noise (1) Sample source signals (2) Get the clues (1) Sample target speech,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' interfering speech and noise signal from the dataset (2) Get the clues as: Audio clue: Sample from the dataset an utterance from the target speaker different from the target speech Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Example of generating simulation data for training or testing: This example assumes videos are available so that audio and visual clues can be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' No video is needed for audio clue-based TSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For visual clue-based TSE, we do not necessarily need multiple videos from the same speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Several variants of the losses operating on different domains exist, such as the cross-entropy between the oracle and the estimated time-frequency masks and the mean squared error (MSE) loss between the magnitude spectra of the source and the estimated target speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Recently, a negative signal-to- noise ratio (SNR) measured in the time-domain has been widely used [6], [23], [39]: LSNR(xs, ˆxs) = −10 log10 � ∥xs∥2 ∥xs − ˆxs∥2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' (16) The SNR loss is computed directly in the time-domain, which forces the TSE system to learn to correctly estimate the magnitude and the phase of the target speech signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This loss has improved extraction performance [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Many works also employ versions of the loss which are invariant to arbitrary scaling, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', scale-invariant SNR (SI-SNR) [39] or linear filtering of the estimated signal, often calledsignal-to-distortion ratio (SDR) [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Besides training losses operating on the signal or mask levels, it is also possible to train a TSE system end-to-end with a loss defined on the output of an ASR system [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Such a loss can be particularly effective when targeting ASR applications, as discussed in Section VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The clue encoder can be an NN trained jointly with a speech extraction module [10] or pre-trained on a different task, such as speaker identification for audio clue-based TSE [11] or lip- reading for visual clue-based TSE [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Using a pre-trained clue encoder enables the leveraging of large amounts of data to learn robust and highly discriminative embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' On the other hand, jointly optimizing the clue encoder allows learning embeddings to be optimized directly for TSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' These two trends can also be combined by fine-tuning the pre-trained encoder or using multi-task training schemes, which add a loss to the output of the clue embeddings [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Considerations when designing a TSE system We conclude this section with some considerations about the different options for designing a TSE system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In the above description, we intentionally ignored the details of the NN architecture used in the speech extraction module, such as the type of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Indeed, novel architectures have been and will probably continue to be proposed regularly, leading to gradual performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For concrete examples, we refer to some public implementations of TSE frameworks presented in Section X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Most TSE approaches borrow a network configuration from architectures proven effective for BSS or noise reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' One important aspect is that an NN must be able to see enough context in the mixture to identify the target speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This has been achieved using such recurrent neural network (RNN)- based architectures as a stack of bidirectional long short-term memory (BLSTM) layers [10], convolutional neural network (CNN)-based architectures with a stack of convolutional layers that gradually increases the receptive field over the time axis to cover a large context [7], [23] or attention-based architectures [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The networks in the mixture encoder and the extraction pro- cess generally use a similar architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The best performance was reported when using a shallow mixture encoder (typically a single layer/block) and a much deeper extraction network, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', where a fusion layer is placed on the lower part of the extraction module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Furthermore, we found in our experiments that the multiplication or FiLM layers usually perform well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Illustration of i-vector, NN-based vector, and jointly-trained embeddings: Orange parts are included only in training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, the impact of the choice of the fusion layer seems rather insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For the feature extraction, early studies used spectral fea- tures computed with STFT [7], [8], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, most recent approaches employ a learned feature extraction module following its success for separation [23], [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This approach allows direct optimization of the features for the given task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, the choice of input features may depend on the acoustic conditions, and some have reported superior perfor- mance using STFT under challenging reverberant conditions [48] or using handcrafted filterbanks [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Except for such general considerations, it is difficult to make solid arguments for a specific network configuration since performance may depend on many factors, such as the task, the type of clue, the training data generation, and the network and training hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' AUDIO-BASED TSE In this section, we explain how the general framework introduced in Section IV can be applied in the case of audio clues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In particular, we discuss different options to implement the clue encoder, summarize the development of the audio- based TSE, and present some representative experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Audio clue encoder An audio clue is an utterance spoken by the target speaker from which we derive the characteristics of their voice, al- lowing identification in a mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This enrollment utterance can be obtained by pre-recording the user of a personal device or with a part of a recording where a wake-up keyword was uttered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The clue encoder is usually used to extract a single vector that summarizes the entire enrollment utterance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Since the clue encoder’s goal is to extract information that defines the voice characteristics of the target speaker, embeddings from the speaker verification field are often used, such as i-vectors or NN-based embeddings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', d-vectors or x-vectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Clue encoders trained directly for TSE tasks are also used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 6 describes these three options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1) I-vectors: From their introduction around 2010, i-vectors [50] were the ruling speaker verification paradigm until the rise of NN speaker embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The main idea behind i-vectors is modeling the features of an utterance using a Gaussian mixture model (GMM), whose means are constrained to a subspace and depend on the speaker and the channel effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The subspace is defined by the Universal Background model (UBM), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', GMM trained on a large amount of data from many speakers, and a total variability subspace matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The super-vector of the means of utterance GMM µ is decom- posed: µ = m + Tw, (17) where m is a super-vector of the means of the UBM, T is a low-rank rectangular matrix representing the bases spanning the subspace, and w is a random variable with standard normal prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Since an i-vector is the maximum a posteriori estimate of w, it thus consists of values that enable the adaptation of the parameters of the generic UBM speaker model (m) to a specific recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' As a result, it captures the speaker’s voice characteristics in the recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' An important characteristic of i-vectors is that they capture both the speaker and channel variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This case may be desired in some TSE applications, where we obtain enrollment utterances in identical conditions as the mixed speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In such a situation, the channel information might also help distinguish the speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' I-vectors have also been used in several TSE works [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 2) Neural network-based embeddings: The state-of-the- art speaker verification systems predominantly use NN-based speaker embeddings, which were adopted later for TSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The common idea is to train an NN for the task of speaker classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Such an NN contains a “pooling layer” which converts a sequence of features into one vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The pooling layer computes the mean and optionally the standard deviation of the sequence of features over the time dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The pooled vector is then classified into speaker classes or used oro11 in other loss functions that encourage speaker discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For TSE, the speaker embedding is then the vector of the activation coefficients of one of the last network layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The most common of such NN-based speaker embeddings are d- vectors and x-vectors [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Many TSE works employ d-vectors [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Since NNs are trained for speaker classification or a related task, embeddings are usually highly speaker-discriminative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Most other sources of variability are discarded, such as the channel or content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Another advantage of this class of em- beddings is that they are usually trained on large corpora with many speakers, noises, and other variations, resulting in very robust embedding extractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Trained models are often publicly available, and the embeddings can be readily used for TSE tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 3) Jointly-learned embeddings: NN-based embeddings, such as x-vectors, are designed and trained for the task of speaker classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Although this causes them to contain speaker information, it is questionable whether the same representation is optimal for TSE tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' An alternative is to train the neural embedding extractor jointly with a speech extraction module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The resulting embeddings are thus directly optimized for TSE tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This approach has been used for TSE in several works [10], [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The NN performing the speaker embedding extraction takes an enrollment utterance C(a) s as input and generally contains a pooling layer converting the frame-level features into one vector, similar to the embedding extractors discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This NN is trained with the main NN using a common ob- jective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A second objective function can also be used on the embeddings to improve their speaker discriminability [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' As mentioned above, the advantage of such embeddings is that they are trained directly for TSE and thus collect essential information for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' On the other hand, the pre-trained embedding extractors are often trained on larger corpora and may be more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A possible middle ground might take a pre-trained embedding extractor and fine-tune it jointly with the TSE task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, this has, to the best of our knowledge, not been done yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Existing approaches The first neural TSE methods were developed around 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' One of the first published works, SpeakerBeam [10], ex- plored both the single-channel approach, where the target extractor was implemented by time-frequency masking, and the multi-channel approach using beamforming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This work also compared different variants of fusion layers and clue encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This was followed by VoiceFilter [11], which put more emphasis on ASR applications using TSE as a front-end and also investigated streaming variants with minimal latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A slightly modified variant of the task was presented in works on speaker inventory [40], where not one but multiple speakers can be enrolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Such a setting might be suitable for meeting scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Recently, many works, such as SpEx [31], have started to use time-domain approaches, following their success in BSS [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 7 9 11 13 15 17 WSJ0-2mix WHAM WHAMr SI-SNR improvement [dB] BSS (oracle) BSS (x-vector) TSE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Comparison of TSE and cascade BSS systems when using an audio clue in terms of SI-SNR improvement (higher is better) [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Experiments An audio clue is a simple way to condition the system for extracting the target speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Many works have shown that the speaker information extracted from audio clues is suffi- cient for satisfactory performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Demonstrations of many works are available online15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We present here some results to demonstrate the potential of audio clue-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The experiments were done with time-domain SpeakerBeam16, which uses a convolutional architecture, a multiplicative fusion layer, and a jointly-learned clue encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The experiments were done on three different datasets (WSJ0-2mix, WHAM!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', and WHAMR!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=') to show the perfor- mance in different conditions (clean, noisy, and reverberant, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We describe these datasets in more detail in Section X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' All the experiments were evaluated with the SI- SNR metric and measured the improvements over the SI-SNR of the observed mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' More details about the experiments can be found in [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Figure 7 compares the TSE results with a cascade system, first doing BSS and then independent speaker identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Speaker identification is done either in an oracle way (selecting the output closest to the reference) or with x-vectors (ex- tracting the x-vectors from all the outputs and the enrollment utterances and selecting the output with the smallest cosine distance as the target).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The BSS system uses the same con- volutional architecture as TSE, differing only in that it does not have a clue encoder and the output layer is twice larger as it outputs two separated speech signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The direct TSE scheme outperformed the cascade system, especially in more difficult conditions such as WHAMR!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='. This difference reflects a couple of causes: 1) the TSE model is directly optimized for the TSE task and does not spend any capacity on extracting other speakers or 2) the TSE model has additional speaker information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Figure 8 shows an example of spectrograms obtained using TSE on a recording of two speakers from the WHAMR!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' database, including noise and reverberation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' TSE correctly 15Demonstrations of audio clues approaches: VoiceFilter [11] https://google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='io/speaker-id/publications/VoiceFilter/, SpeakerBeam [10] https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='v=7FSHgKip6vI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 16https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='com/butspeechfit/speakerbeam 12 2 4 6 8 10 0 2000 4000 Frequency [Hz] Mixture 2 4 6 8 10 0 2000 4000 Frequency [Hz] Reference 2 4 6 8 10 Time [s] 0 2000 4000 Frequency [Hz] Extracted (SI-SNR 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='56 dB) 100 50 100 50 100 50 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Example of spectrograms of mixed, reference, and extracted speech: Example is taken from WHAMR!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' identifies the target speaker and removes all the interference, including the second speaker, noise, and reverberation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Limitations and outlook Using TSE systems conditioned on audio clues is particu- larly practical due to the simplicity of obtaining the clues, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', no additional hardware is needed, such as cameras or multi- ple microphones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Considering the good performance demon- strated in the literature, these systems are widely applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Nowadays, the methods are rapidly evolving and achieving increasingly higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The main challenge in audio-clue-based systems is correct identification of the target speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The speech signal of the same speaker might have highly different characteristics in different conditions due to such factors as emotional state, channel effects, or the Lombard effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' TSE systems must be robust enough to such intra-speaker variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' On the other hand, different speakers might have very similar voices, leading to erroneous identification if the TSE system lacks sufficient accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Resolving both issues requires precise speaker modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In this regard, the TSE methods may draw inspiration from the latest advances in the speaker verification field, including advanced model architectures, realistic datasets with a huge number of speakers for training, or using pre-trained features from self-supervised models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' VISUAL/MULTI-MODAL CLUE-BASED TSE Visual clue-based TSE assumes that a video camera captures the face of the target speaker who is talking in the mixture [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Using visual clues is motivated by psycho-acoustic studies (see the references in a previous work [6]) that revealed that humans look at lip movements to understand speech better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Similarly, the visual clues of TSE systems derive hints about the state of the target speech from the lip movements, such as whether the target speaker is speaking or silent or more refined information about the phoneme being uttered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A visual clue, which presents different characteristics than audio clues because it captures information from another modality, is time-synchronized with the target speech in the mixture without being corrupted by the interference speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Therefore, a visual clue-based TSE can better handle mixtures of speakers with similar voices, such as same-gender mixtures, than audio clue-based systems because the extraction process is not based on the speaker’s voice characteristics17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Another potential advantage is that the users may not need to pre-enroll their voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Video signals are also readily available for many applications such as video-conferencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Figure 9 shows a diagram of a visual TSE system that follows the same structure as the general TSE framework introduced in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Only the visual clue encoder part is specific to the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We describe it in more detail below and then introduce a multi-modal clue extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We conclude this section with some experimental results and discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Visual clue encoder The visual clue encoder computes from the video signal a representation that allows the speech extraction module to identify and extract the target speech in the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This processing involves the steps described below: E(v) s = Upsample(NN(VFE(C(v) s ), θv-clue)), (18) where E(v) s ∈ RDEmb×N represents the sequence of the visual embedding vectors, C(v) s is the video signal obtained after pre-processing, VFE(·) is the visual feature extraction module, NN(·, θv-clue) is an NN with parameters θv-clue, and Upsample(·) represents the up-sampling operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The latter up-sampling step is required because the sampling rates of the audio and video devices are usually different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Up-sampling 17Some works can even perform extraction from a mixture of the same speaker’s speech [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 13 Pre- processing Feature extraction Up- sampling Speech extraction module Visual clue encoder NN Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Visual clue-based TSE system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' matches the number of frames of the mixture and visual clue encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1) Pre-processing: First, the video signal captured by the camera requires pre-processing to isolate the face of the target speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Depending on the application, this may require detecting and tracking the target speaker’s face and cropping the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' These pre-processing steps can be performed using previously well-established video processing algorithms [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 2) Visual feature extraction: Similar to an audio-clue-based TSE, the visual clue encoder can directly extract embeddings from raw video data or visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' With the first option, the raw video is processed with a CNN whose parameters are jointly-learned with the speech extraction module to enable direct optimization of the features for the extraction task without any loss of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, since the video signals are high-dimensional data, achieving joint optimization can be complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This approach has been used successfully with speaker-close conditions [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Extending it to speaker- open conditions might require a considerable amount of data or careful design of the training loss using, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', multi-task training to help the visual encoder capture relevant informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Most visual TSE works use instead a visual feature extractor pre-trained on another task to reduce the dimensionality of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Such feature extractors can leverage a large amount of image or video data (that do not need to be speech mixtures) to learn representation robust to variations, such as resolution, luminosity, or head orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The first option is to use facial landmark points as features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Facial landmarks are the key points on a face that indicate the mouth, eyes, or nose positions and offer a very low-dimension representation of a face, which is interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Moreover, face landmarks can be easily computed with efficient off-the-shelf algorithms [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The other option is to use neural embeddings derived from an image/video processing NN trained on a different task, which was proposed in three concurrent works [7]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ephrat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [8] used visual embeddings obtained from an intermediate layer of a face recognition system called FaceNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This face recognition system is trained so that embeddings derived from photographs of the same person are close and embeddings from different persons are far from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' It thus requires only a corpus of still images with person identity labels for training the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, the embeddings do not capture the lip movement dynamics and are not explicitly related to the acoustic content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Alternatively, Afouras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [7] proposed using embeddings obtained from a network trained to perform lip-reading, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', where a network is trained to estimate the phoneme or word uttered from the video of the speaker’s lips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The resulting embeddings are thus directly related to the acoustic content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, the training requires video with the associated phoneme or word transcriptions, which are more demanding and costly to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The third option introduced by Owens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [9] exploits embeddings derived from an NN trained to predict whether the audio and visual tracks of a video are synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This approach enables self-supervised training, where the training data are simply created by randomly shifting the audio track by a few seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The embeddings capture information on the association between the lip motions and the timing of the sounds in the audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' All three options [7]–[9] can successfully perform a visual TSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 3) Transformation and up-sampling: Except with joint- training approaches, the visual features are (pre-)trained on different tasks and thus do not provide a representation optimal for TSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Besides, since some of the visual features are ex- tracted from the individual frames of a video, the dynamics of lip movements are not captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Therefore, the visual features are further transformed with an NN, which is jointly trained with the speech extraction module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The NN, which allows learning a representation optimal for TSE, can be implemented with long short-term memory (LSTM) or convolutional layers across the time dimension to model the time series of the visual features, enabling the lip movement dynamics to be captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Finally, the visual embeddings are up-sampled to match the sampling rate of audio features Zy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Audio-visual clue-based TSE Audio and visual clue-based TSE systems have complemen- tary properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' An audio clue-based TSE is not affected by speaker movements and visual occlusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In contrast, a visual clue-based TSE is less affected by the voice characteristics of the speakers in the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' By combining these approaches, we can build TSE systems that exploit the strengths of both clues for improving the robustness to various conditions [33], [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Figure 10 shows a diagram of an audio-visual TSE system, which assumes access to the pre-recorded enrollment of the target speaker to provide an audio clue and a video camera for a visual clue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The system uses the audio and visual clue encoders described in Sections V-A and VI-A and combines these clues into an audio-visual embedding, which is given to the speech extraction module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Audio-visual embeddings can be simply the concatenation [35] or the summation of the audio and visual embeddings, or obtained as a weighted sum [33], 14 Speech extraction module Enrollment Video Mixture Visual clue encoder Audio clue encoder Multi-modal clue fusion Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Audio-visual clue-based TSE system [34], where the weights can vary depending on the reliability of each clue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The weighted sum approach can be implemented with an attention layer widely used in machine learning, which enables dynamic weighting of the contribution of each clue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Experimental results and discussion Several visual TSE systems have been proposed, which dif- fer mostly by the type of visual features used and the network configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' These systems have demonstrated astonishing results, which can be attested by the demonstrations available online18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Here we briefly describe experiments using the audio, visual, and audio-visual time-domain SpeakerBeam systems [34], which use a similar configuration as the system in Section V-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The speech extraction module employs a stack of time- convolutional blocks and a multiplicative fusion layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The audio clue encoder consists of the jointly-learned embeddings described in Section V-A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The visual clue encoder uses visual features derived from face recognition like a previous work [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The audio-visual system combines the visual and audio clues with an attention layer [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The experiments used mixtures of utterances from the LRS3-TED corpus19, which consists of single speaker ut- terances with associated videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We analyzed the behavior under various conditions by looking at results from same and different gender mixtures and two examples of clue corruptions (enrollment corrupted with white noise at SNR of 0 dB and video with a mask on the speaker’s mouth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The details of the experimental setup are available in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Figure 11 compares the extraction performance measured in terms of SDR improvement for audio, visual, and audio- visual TSE under various mixture and clue conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We confirmed that a visual clue-based TSE is less sensitive to the characteristics of the speakers in the mixture since the performance gap between different- and same-gender mixtures is smaller than with an audio clue-based TSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' When using a single clue, performance can be degraded when this clue is corrupted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, the audio-visual system that exploits both clues can achieve superior extraction performance and is more robust to clue corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 18Demo samples for several approaches are available, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', for [9]: https://andrewowens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='com/multisensory, for [8]: https://looking-to-listen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='io, for [7]: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='uk/∼vgg/demo/theconversation, and for [34]: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='kecl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='ntt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='jp/icl/signal/member/demo/audio visual speakerbeam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='html 19https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='uk/∼vgg/data/lip reading/lrs3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='html D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Discussions and outlook Visual clue-based TSE approaches offer an alternative to audio-clue-based ones when a camera is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The idea of using visual clues for TSE is not new [4], [5], although recent neural systems have achieved an impressive level of performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This is probably because NNs can effectively model the relationship between the different modalities learned from a large amount of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Issues and research opportunities remain with the current visual clue-based TSE systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' First, most approaches do not consider the speaker tracking problem and assume that the audio and video signals are synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' These aspects must be considered when designing and evaluating future TSE sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Second, video processing involves high computational costs, and more research is needed to develop efficient online systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' SPATIAL CLUE-BASED TSE When using a microphone array to record a signal, spatial information can be used to discriminate among sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In particular, access to multi-channel recordings opens the way to extract target speakers based on their location, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', using spatial clues (as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This section explains how such spatial clues can be obtained and used in TSE systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' While enhancing speakers from a given direction has a long research history [2], we focus here on neural methods that follow the scope of our overview paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Note that multi-channel signals can also be utilized in the extraction process using beamforming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Such an extraction process can be used in systems with any type of clue, only requiring that the mixed speech be recorded with multiple microphones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This beamforming process was reviewed in Section IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In this section, we focus specifically on the processing of spatial clues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Obtaining spatial clues In some situations, the target speaker’s location is approxi- mately known in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For example, for an in-car ASR, the driver’s position is limited to a certain region in a car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In other scenarios, we might have access to a multi-channel enrollment utterance of the speaker recorded in the same position as the final mixed speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In such a case, audio source localization methods can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Conventionally, this can be done by methods based on generalized cross-correlation or steered- response power, but recently, deep learning methods have also shown success in this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' An alternative is to skip the explicit estimation of the location and directly extract features in which the location is encoded when a multi-channel enrollment is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We will detail this approach further in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Spatial clues can also be obtained from a video using face detection and tracking systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A previous work [36] demonstrated this possibility with a 180-degree wide-angle camera positioned parallel to a linear microphone array20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' By identifying the target speaker in the video, the azimuth with 20https://yongxuustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='io/grnnbf 15 12 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='5 13 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='5 14 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='5 15 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='5 16 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='5 Different gender Same gender Corrupted audio clue Corrupted visual clue SDR improvement (dB) Audio Visual Audio-visual Clues: Visual clean no mask Audio Effect of gender in mixtures Effect of clue corruption no mask 0dB Visual Audio mask clean Visual Audio Visual clean no mask Audio Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' SDR Improvement of TSE with audio, visual, and audio-visual clues for mixtures of same/different gender and for corruptions of audio and visual clues: Audio clues were corrupted by adding white noise at SNR of 0 dB to enrollment utterance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Video clues were corrupted by masking mouth region in video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Illustration of usage of spatial clue encoder and directional features respect to the microphone array was roughly approximated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Depth cameras can also be used to estimate not only the azimuth but also the elevation and distance of the speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Spatial clue encoder The left part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 12 shows the overall structure and the usage of a spatial clue encoder, which usually consists of two parts: the extraction of directional features and an NN post- processing of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Two possible forms of spatial clues are dominant in the literature: the angle of the target speaker with respect to the microphone array or a multi-channel enrollment utterance recorded in the target location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Both can be encoded into directional features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' When the spatial clue is DOA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' the most commonly used directional features are the angle features,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' which are computed as the cosine of the difference between the IPD and the target phase difference (TPD): AF[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' f] = � m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='m2∈M cos � TPD (m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' φs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' f) − IPD (m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' f) � (19) TPD(m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' φs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' f) = 2πfFs F cos φs ∆m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='m2 c (20) IPD(m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' f) = ∠Y m2[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' f] − ∠Y m1[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' f],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' (21) where M is a set of pairs of microphones used to compute the feature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Fs is the sampling frequency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' φs is the target direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' c is the sound’s velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' and ∆m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='m2 is the distance from microphone m1 to microphone m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' An example of angle features is shown on the right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For time-frequency bins dominated by the source from direction φs, the value of the angle feature should be close to 1 or -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Other directional features have been proposed that exploit a grid of fixed beamformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A directional power ratio measures the ratio between the power of the response of a beamformer steered into the target direction and the power of the beamformer responses steered into all the directions in the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In a similar fashion, a directional signal-to-noise ratio can also be computed, which compares the response of a beamformer in the target direction with the response of a beamformer in the direction with the strongest interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' If the spatial clue consists of a multi-channel enrollment utterance, the directional feature can be formed as a vector of IPDs computed from the enrollment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Alternatively, the DOA can be estimated from the enrollment, and the spatial features derived from it can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Note that when using a spatial clue to determine the target speaker, the multi-channel input of the speech extraction module must also be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This enables the identification of the speaker coming from the target location in the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Furthermore, a target extractor is often implemented as beam- forming, as explained in Section IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 5 10-10 15-5 10 1516 0 2 4 6 8 10 12 <15° 15-45° 45-90° >90° SI-SNR improvement [dB] Audio Visual Spatial Combined Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' SI-SNR improvement of TSE with audio, visual, and spatial clues in four conditions based on angle separation between speakers [36] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Combination with other clues Although a spatial clue is very informative and generally can lead the TSE system to a correct extraction of the target, it does fail in some instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Estimation errors of DOA are harmful to proper extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Furthermore, if the spatial separation of the speakers with respect to the microphone array is not significant enough, the spatial clue may not discriminate between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Combining a spatial clue with audio or visual clues is an option to combat such failure cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Experimental results We next report the results from an experiment with spatial clues [36] that compared the effectiveness of using audio, visual, and spatial clues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The audio-clue encoder was trained jointly with the extraction module, and the visual encoder was a pre-trained lip-reading network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The target speaker’s direction was encoded in the angle feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The spatial and visual embeddings were fused with the extraction network by concatenation and the audio embedding with a factorized layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The extraction module employed a neural network consisting of temporal convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The experiments were performed on a Mandarin audio- visual dataset containing mixtures of two and three speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 13 were divided into several conditions, based on the angle separation between the closest speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The spatial clue is very effective, although the performance declines when speakers are near each other (< 15°).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A combi- nation with other modalities outperformed any individual type of clue in all the conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Discussion Using spatial clues is a powerful way of conditioning a TSE system to extract the target speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' It relies on the availability of signals from a microphone array and a way to determine the location of the target speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Unfortunately, these restrictions limit the applications to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Neural TSE methods with spatial clues follow a long history of research on the topic, such as beamforming techniques, and extend them with non-linear processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This approach unifies the methods with those using other clues and allows a straightforward combi- nation of different clues into one system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Such combinations can alleviate the shortcomings of spatial clues, including the failures when the speakers are located in the same direction from the point of view of the microphones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In most current neural TSE works, the target speaker’s location is assumed to be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Although the methods should be easily extended to a dynamic case, investigations of such settings remain relatively rare [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' EXTENSION TO OTHER TASKS The ideas of TSE can be applied to other speech processing tasks, such as ASR and diarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Target-speaker ASR An important application of TSE is TS-ASR, where the goal is to transcribe the target speaker’s speech and ignore all the interference speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The TSE approaches we described can be naturally used as a front-end to an ASR system to achieve TS-ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Such a cascade combination allows for a modular system, which offers ease of development and interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, the TSE system is often optimized with a signal loss, as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Such a TSE system inevitably introduces artifacts caused by the remaining interferences, over-suppression, and other non-linear processing distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' These artifacts limit the expected performance improvement from a TSE front-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' One approach to mitigate the effect of such artifacts is to optimize the TSE front-end with an ASR criterion [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The TSE front-end and the ASR back-end are NNs and can be interconnected with differentiable operations, such as beam- forming and feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Therefore, a cascade system can be represented with a single computational graph, allowing all parameters to be jointly trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Such joint-training can significantly improve the TS-ASR performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Another approach inserts a fusion layer into an ASR system [26], [45] to directly perform clue conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' These inte- grated TS-ASR systems avoid any explicit signal extraction step, a decision that reduces the computational cost, although such systems may be less interpretable than cascade systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' TS-ASR can use the audio clues provided by pre-recorded enrollment utterances [10], [26], [45] or from a keyword (anchor) for a smart-device scenario [54], for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Some works have also exploited visual clues, which can be used for the extraction process and to implement an audio-visual ASR back-end, since lip-reading also improves ASR performance [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Target-speaker VAD and diarization The problem of speech diarization consists of detecting who spoke when in a multi-speaker recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This technology is essential for achieving, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', meeting recognition and analysis systems that can transcribe a discussion between multiple participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Several works have explored using speaker clues to perform this task [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For example, a personalized VAD [27] exploits a speaker embedding vector derived from an enrollment utterance of the target speaker to predict its activity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', whether they are speaking at a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In principle, this can be done with a system like that presented in Section IV, where the output layer performs the binary classification of the speaker activity 17 instead of estimating the target speech signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Similar systems have also been proposed using visual clues, called audio-visual VAD [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Predicting the target speaker’s activity is arguably a more straightforward task than estimating its speech signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Consequently, TS-VAD can use simpler network architectures, leading to more lightweight processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The above TS-VAD systems, which estimate the speech activity of a single target speaker, have been extended to simul- taneously output the activity of multiple target speakers [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The resulting system achieved the top diarization performance in the CHiME 6 evaluation campaign21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' REMAINING ISSUES AND OUTLOOK Research toward computational selective hearing has been a long endeavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Recent developments in TSE have enabled identifying and extracting a target speaker’s voice in a mixture by exploiting audio, visual, or spatial clues, which is one step closer to solving the cocktail-party problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Progress in speech processing (speech enhancement, speaker recognition) and image processing (face recognition, lip-reading), com- bined with deep learning technologies to learn models that can effectively condition processing on auxiliary clues, triggered the progress in the TSE field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Some of the works we presented have achieved levels of performance that seemed out-of-reach just a few years ago and are already being deployed in products22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Despite substantial achievements, many opportunities re- main for further research, some of which we list below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Deployment of TSE systems Most of the systems we described operate offline and are computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' They are also evaluated under controlled (mostly simulated mixture) settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Deploying such systems introduces engineering and research challenges to re- duce computational costs while maintaining high performance under less controlled recording conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We next discuss some of these aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1) Inactive target speaker: Most TSE systems have been evaluated assuming that the target speaker is actively speaking in the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In practice, we may not know beforehand whether the target speaker will be active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We expect that a TSE system can output no signal when the target speaker is inactive, which may not actually be the case with most current systems that are not explicitly trained to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The inactive target speaker problem is specific to TSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The type of clue used may also greatly impact the difficulty of tackling this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For instance, visual voice activity detection [5] might alleviate this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, it is more challenging with audio clues [57], and further research may be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 21The results of the CHiME 6 challenge can be found at: https:// chimechallenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='io/chime6/results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The top system used TS-VAD among other technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' DiHARD 3 performed a diarization evaluation on the CHiME 6 challenge data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Here the top system also used TS-VAD: https://dihardchallenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='io/dihard3/results 22The following blog details the effort for deploying a visual clue- based TSE system for on-device processing: https://ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='googleblog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='com/2020/ 10/audiovisual-speech-enhancement-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 2) Training and evaluation criteria: Most TSE systems are trained and evaluated using such signal level metrics as SNR or SDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Although these metrics are indicative of the extraction performance, their use presents two issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' First, they may not always be correlated with human per- ception and intelligibility or with ASR performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This issue is not specific to TSE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' it is common to BSS and noise reduction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For ASR we can train a system end-to-end, as discussed in Section VIII-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' When targeting applications for human listeners, the problem can be partly addressed using other metrics for training or evaluation that correlate better with human perception, such as short-time objective intelligibility (STOI) or perceptual evaluation of speech quality (PESQ) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, controlled listening tests must be conducted to confirm the impact of a TSE on human listeners [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Second, unlike BSS and noise reduction, a TSE system needs to identify the target speech, implying other sources of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Indeed, failing to identify the target may lead to incorrectly estimating an interference speaker or inaccurately outputting the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Although these errors directly impact the SDR scores, it would be fruitful to agree on the evaluation metrics that separate extraction and identification performance to better reveal the behavior of TSE systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Signal level metrics might not satisfactorily represent the extraction perfor- mance for inactive speaker cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A better understanding of the failures might help develop TSE systems that can recognize when they cannot identify the target speech, which is appealing for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Consequently, developing better training and evaluation criteria are critical research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 3) Robustness to recording conditions: Training neural TSE systems requires simulated mixtures, as discussed in Section IV-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Applying these systems to real conditions (multi-speaker mixtures recorded directly with a microphone) requires that the training data match the application scenario relatively well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For example, the type of noise and reverberation may vary significantly depending on where a system is deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This raises questions about the robustness of TSE systems to various recording conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Neural TSE systems trained with a large amount of simu- lated data have been shown to generalize to real recording conditions [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, exploiting real recordings where no reference target speech signal is available could further improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Real recordings might augment the training data or be used to adapt a TSE system to a new environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The issue is defining unsupervised training losses correlated with the extraction performance of the target speech without requiring access to the reference target signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Another interesting research direction is combining neural TSE systems, which are powerful under matched conditions, with such generative-based approaches as IVE [12], which are adaptive to recording conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 4) Lightweight and low-latency systems: Research on lightweight and low-latency TSE systems is gaining mo- mentum as the use of teleconferencing systems in noisy environments has risen in response to the Covid pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Other important use cases for TSE are hearing aids and 18 hearables, both of which impose very severe constraints in terms of computation costs and latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The recent DNS23 and Clarity24 challenges that target teleconferencing and hearing aid application scenarios include tracks where target speaker clues (enrollment data) can be exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This demonstrates the growing interest in practical solutions for TSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Since TSE is related to BSS and noise reduction, the development of online and low-latency TSE systems can be inspired from the progress of BSS/noise reduction in that direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, TSE must also identify the target speech, which may need specific solutions that exploit the long-context of the mixture to reliably and efficiently capture a speaker’s identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 5) Spatial rendering: For applications of TSE to hearing aids or hearables, sounds must be localized in space after the TSE processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Therefore, a TSE system must not only extract the target speech but also estimate its direction to allow rendering it so that a listener feels the correct direction of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Self-supervised and cross-modal learning A TSE system identifies the target speech in a mixture based on the intermediate representation of the mixture and the clue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Naturally, TSE benefits from better intermediate representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For example, speech models learned with self- supervised learning criteria have gained attention as a way to obtain robust speech representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' They have shown potential for pre-training many speech processing downstream tasks, such as ASR, speaker identification, and BSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Such self-supervised models could also reveal advantages for TSE since they could improve robustness by allowing efficient pre- training on various acoustic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Moreover, for audio- based TSE, using the same self-supervised pre-trained model for the audio clue encoder and the speech extraction module will help to learn the common embedding space between the enrollment and speech signals in the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Similarly, the progress in cross-modal learning, which aims to learn the joint representation of data across modalities, could benefit such multi-modal approaches as visual clue-based TSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Exploring other clues We presented three types of clues that have been widely used for TSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, other clues can also be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For example, recent works have explored other types of spatial clues such as the distance [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Moreover, humans do not only rely on physical clues to perform selective hearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We also use more abstract clues, such as semantic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Indeed, we can rapidly focus our attention on a speaker when we hear our name or a topic we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Reproducing a similar mechanism would require TSE systems that operate with semantic clues, which introduces novel challenges concerning how to represent semantic information and exploit it within a TSE system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Some works have started to explore this direction, 23https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='com/en-us/research/academic-program/ deep-noise-suppression-challenge-icassp-2022/ 24https://claritychallenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='io/clarity CC doc/ such as conditioning on languages [59] or more abstract concepts [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Other interesting clues consist of signals that measure a listener’s brain activity to guide the extraction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Indeed, the electroencephalogram (EEG) signal of a listener focusing on a speaker correlates with the envelope of that speaker’s speech signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ceolini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' identified the possibility of using EEG as clues for TSE with a system similar to the one described in Section IV [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' An EEG-guided TSE might open the door for futuristic hearing aids controlled by the user’s brain activity, which might automatically emphasize the speaker a user wants to hear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' However, research is still needed because developing a system that requires marginal tuning to the listener is especially challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Moreover, collecting a large amount of training data is very complicated since it is more difficult to control the quality of such clues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Compared to audio and visual TSE clues, EEG signals are very noisy and affected by changes in the attention of the listener, body movements, and other factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Beyond speech Human selective listening abilities go beyond speech sig- nals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For example, we can focus on listening to the part of an instrument in an orchestra or switch our attention to a siren or a barking dog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' In this paper, we focused on TSE, but similar extraction problems have also been explored for other audio-processing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For example, much research has been performed on extracting the track of an instrument in a piece of music conditioned on, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', the type of instrument [62], video of the musician playing [63], or EEG signal of the listener [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' These approaches may be important to realize, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', audio-visual music analysis [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Recently, the problem was extended to the extraction of arbitrary sounds from a mixture [66], [67], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', extracting the sound of a siren or a klaxon from a recording of a mixture of street sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We can use such systems as that introduced in Section IV to tackle these problems, where the clue can be a class label indicating the type of target sound [66], the enrollment audio of a similar target sound [67], a video of the sound source [9] or a text description of the target sound [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Target sound extraction may become an important technology to design, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', hearables or hearing aids that could filter out nuisances and emphasize important sounds in our surroundings, or audio visual scene analysis [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Psycho-acoustic studies suggest that humans process speech and music partly using shared auditory mechanisms and that exposure to music can lead to better discrimination of speech sounds [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' It would be interesting to explore whether, similarly to humans, TSE systems could benefit from exposure to other acoustic signals by training a system to extract target speech, music, or arbitrary sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' RESOURCES We conclude by providing pointers to selected datasets and toolkits available for those motivated to experiment with TSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' TSE works mostly use datasets designed for BSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' These datasets consist generally of artificial mixtures generated from 19 TABLE III SOME DATASETS AND TOOLKITS Name Description Link Dataset WSJ0-mix Mixtures of two or three speakers www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='merl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='com/demos/deep-clustering WHAM(R)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Noisy and reverberant versions of WSJ0-mix wham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='whisper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='ai Librimix Larger dataset of mixtures of two or three speakers github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='com/JorisCos/LibriMix LibriCSS Meeting-like mixtures recorded in a room github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='com/chenzhuo1011/libri css MC-WSJ0-mix Spatialized version of WSJ0-2mix www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='merl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='com/demos/deep-clustering SMS-WSJ Multi-channel corpus based on WSJ github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='com/fgnt/sms wsj LRS Audio-visual corpus from TED or BBC videos www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='uk/∼vgg/data/lip reading AVSpeech Very large audio-visual corpus from YouTube videos looking-to-listen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='io/avspeech Tools SpeakerBeam Time-domain audio-based TSE system github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='com/butspeechfit/speakerbeam SpEx+ Time-domain audio-based TSE system [31] github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='com/xuchenglin28/speaker extraction SpEx VoiceFilter Time-domain audio-based TSE system (Unofficial) [11] github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='com/mindslab-ai/voicefilter Multisensory Visual clue-based TSE [9] github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='com/andrewowens/multisensory AV Speech enh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Face landmark-based visual clue-based TSE [32] github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='com/dr-pato/audio visual speech enhancement FaceNet Visual feature extractor used in [8], [33], [34] github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='com/davidsandberg/facenet the isolated signals of the individual speakers and background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' This allows evaluation of the performance by comparing the estimated signals to the original references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Additionally, TSE methods also require a clue, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', an enrollment utterance for the target speaker or video signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' We can obtain enrollment utterances by choosing a random utterance of the target speaker from the same database, provided that the utterance is different from the one in the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' For a video clue, it requires using an audio-visual dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The top of Table III lists some of the most commonly used datasets for audio and visual TSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Several implementations of TSE systems are openly avail- able and listed in the lower part of Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Although there are no public implementations for some of the visual TSE systems, they can be re-implemented following the audio TSE toolkits and using openly available visual feature extractors such as FaceNet, which was used in some previous works [8], [33], [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was partly supported by the Czech Ministry of Education, Youth and Sports from project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' LTAIN19087 ”Multi-linguality in speech technologies.” Computing on IT4I supercomputer was supported by the Czech Ministry of Ed- ucation, Youth and Sports from the Large Infrastructures for Research, Experimental Development and Innovations project ”e-Infrastructure CZ – LM2018140”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' The figures contain elements designed by pikisuperstar/Freepik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Bronkhorst, “The cocktail-party problem revisited: early pro- cessing and selection of multi-talker speech,” Attention, Perception, & Psychophysics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 77, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1465–1487, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Flanagan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Johnston, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Zahn, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Elko, “Computer- steered microphone arrays for sound transduction in large rooms,” The Journal of the Acoustical Society of America, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 78, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1508– 1518, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Gu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Zheng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Xu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Yu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Su, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Zou, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Yu, “Neural spatial filter: Target speaker speech separation assisted with directional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=',” in Interspeech, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 4290–4294, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Hershey and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Casey, “Audio-visual sound separation via Hidden Markov Models,” Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 14, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [5] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Rivet, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Naqvi, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Chambers, “Audiovisual speech source separation: An overview of key methodologies,” IEEE Signal Processing Magazine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 31, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 125–134, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Michelsanti, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Tan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Xu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Yu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Yu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Jensen, “An overview of deep-learning-based audio-visual speech en- hancement and separation,” IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 29, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1368–1396, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [7] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Afouras, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Chung, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Zisserman, “The Conversation: Deep Audio-Visual Speech Enhancement,” in Interspeech, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 3244–3248, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ephrat, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Mosseri, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Lang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Dekel, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Wilson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Hassidim, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Freeman, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Rubinstein, “Looking to listen at the cocktail party: A speaker-independent audio-visual model for speech separation,” ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 37, jul 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Owens and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Efros, “Audio-visual scene analysis with self- supervised multisensory features,” in Proceedings of the European Conference on Computer Vision (ECCV), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 631–648, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [10] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' ˇZmol´ıkov´a, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Delcroix, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Kinoshita, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ochiai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Nakatani, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Burget, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' ˇCernock`y, “SpeakerBeam: Speaker aware neural network for target speaker extraction in speech mixtures,” IEEE Journal of Selected Topics in Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 800–814, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [11] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Muckenhirn, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Wilson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Sridhar, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Hershey, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Saurous, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Weiss, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Jia, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Moreno, “VoiceFilter: Tar- geted Voice Separation by Speaker-Conditioned Spectrogram Masking,” in Interspeech, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 2728–2732, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Jansk´y, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' M´alek, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' ˇCmejla, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Kounovsk´y, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Koldovsk´y, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' ˇZd’´ansk´y, “Adaptive blind audio source extraction supervised by dominant speaker identification using x-vectors,” in ICASSP, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 676– 680, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [13] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Sawada, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ono, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Kameoka, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Kitamura, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Saruwatari, “A review of blind source separation methods: two converging routes to ILRMA originating from ICA and NMF,” APSIPA Transactions on Signal and Information Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' e12, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [14] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Wang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Chen, “Supervised speech separation based on deep learning: An overview,” IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 26, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1702–1726, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Eskimez, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Yoshioka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Chen, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Huang, “Personalized speech enhancement: new models and comprehensive evaluation,” in ICASSP 2022 - 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 356–360, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Gannot, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Vincent, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Markovich-Golan, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ozerov, “A consol- idated perspective on multimicrophone speech enhancement and source separation,” IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 692–730, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Hershey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Rennie, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Olsen, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Kristjansson, “Super- human multi-talker speech recognition: A graphical modeling approach,” Computer Speech and Language, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 24, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 45 – 66, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [18] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Virtanen, “Monaural sound source separation by nonnegative ma- trix factorization with temporal continuity and sparseness criteria,” IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 15, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1066–1074, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ozerov and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Vincent, “Using the fasst source separation toolbox for 20 noise robust speech recognition,” in International Workshop on Machine Listening in Multisource Environments (CHiME 2011), 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Hershey, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Le Roux, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Watanabe, “Deep clus- tering: Discriminative embeddings for segmentation and separation,” in ICASSP, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 31–35, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [21] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Yu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Kolbaek, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Tan, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Jensen, “Permutation invariant training of deep models for speaker-independent multi-talker speech separation,” in ICASSP, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 241–245, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [22] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Du, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Tu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Xu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Dai, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Lee, “Speech separation of a target speaker based on deep neural networks,” in 2014 12th International Conference on Signal Processing (ICSP), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 473–477, IEEE, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [23] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Xu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Rao, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Chng, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Li, “Time-domain speaker extraction network,” in ASRU, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 327–334, IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [24] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Heitkaemper, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Feh´er, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Freitag, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Haeb-Umbach, “A study on online source extraction in the presence of changing speaker posi- tions,” in International Conference on Statistical Language and Speech Processing, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 198–209, Springer, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Delcroix, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Zmolikova, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Kinoshita, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ogawa, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Nakatani, “Single channel target speaker extraction and recognition with Speaker- Beam,” in ICASSP, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 5554–5558, IEEE, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [26] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Denisov and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Vu, “End-to-End Multi-Speaker Speech Recogni- tion Using Speaker Embeddings and Transfer Learning,” in Interspeech, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 4425–4429, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [27] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ding, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Chang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Wan, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Lopez Moreno, “Per- sonal VAD: Speaker-Conditioned Voice Activity Detection,” in Odyssey, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 433–439, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [28] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Medennikov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Korenevsky, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Prisyach, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Khokhlov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ko- renevskaya, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Sorokin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Timofeeva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Mitrofanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Andrusenko, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Podluzhny, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Laptev, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Romanenko, “Target-Speaker Voice Activity Detection: A Novel Approach for Multi-Speaker Diarization in a Dinner Party Scenario,” in Interspeech, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 274–278, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [29] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Sadeghi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Leglaive, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Alameda-Pineda, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Girin, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Horaud, “Audio-visual speech enhancement using conditional variational auto- encoders,” IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 28, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1788–1800, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [30] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' ˇZmol´ıkov´a, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Delcroix, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Kinoshita, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Higuchi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ogawa, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Nakatani, “Speaker-aware neural network based beamformer for speaker extraction in speech mixtures,” in Interspeech, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 2655–2659, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [31] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ge, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Xu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Wang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Chng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Dang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Li, “SpEx+: A Complete Time Domain Speaker Extraction Network,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Interspeech 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1406–1410, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [32] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Morrone, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Bergamaschi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Pasa, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Fadiga, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Tikhanoff, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Badino, “Face landmark-based speaker-independent audio- visual speech enhancement in multi-talker environments,” in ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 6900–6904, IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [33] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ochiai, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Delcroix, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Kinoshita, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ogawa, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Nakatani, “Multimodal SpeakerBeam: Single channel target speech extraction with audio-visual speaker clues,” in Interspeech, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 2718–2722, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [34] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Sato, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ochiai, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Kinoshita, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Delcroix, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Nakatani, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Araki, “Multimodal attention fusion for target speaker extraction,” in 2021 IEEE Spoken Language Technology Workshop (SLT), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 778–784, IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [35] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Afouras, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Chung, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Zisserman, “My lips are concealed: Audio-visual speech enhancement through obstructions,” arXiv preprint arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='04975, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [36] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Gu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Xu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Zou, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Yu, “Multi-modal multi-channel target speech separation,” IEEE Journal of Selected Topics in Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 530–541, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [37] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Chung, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Choe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Chung, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Kang, “FaceFilter: Audio- Visual Speech Separation Using Still Images,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Interspeech 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 3481–3485, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [38] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Maciejewski, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Sell, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Fujita, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Garcia-Perera, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Watanabe, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Khudanpur, “Analysis of robustness of deep single-channel speech separation using corpora constructed from multiple domains,” in WASPAA, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 165–169, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [39] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Luo and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Mesgarani, “TasNet: Surpassing ideal time-frequency masking for speech separation,” in ICASSP, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [40] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Xiao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Chen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Yoshioka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Erdogan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Liu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Dimitriadis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Droppo, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Gong, “Single-channel speech extraction using speaker inventory and attention network,” in ICASSP, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 86–90, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [41] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' ˇZmol´ıkov´a, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Delcroix, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Kinoshita, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Higuchi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ogawa, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Nakatani, “Learning speaker representation for neural network based multichannel speaker extraction,” in ASRU, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 8–15, IEEE, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [42] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Heymann, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Drude, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Haeb-Umbach, “Neural network based spectral mask estimation for acoustic beamforming,” in 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 196–200, IEEE, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [43] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Erdogan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Hershey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Watanabe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Mandel, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Le Roux, “Improved MVDR beamforming using single-channel mask prediction networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=',” in Interspeech, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1981–1985, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [44] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Vincent, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Gribonval, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' F´evotte, “Performance measurement in blind audio source separation,” IEEE transactions on audio, speech, and language processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1462–1469, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [45] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Delcroix, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ochiai, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Kinoshita, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Karita, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ogawa, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Nakatani, “End-to-end speakerbeam for single channel target speech recognition,” in Interspeech, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [46] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Mun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Choe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Huh, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Chung, “The sound of my voice: Speaker representation loss for target voice separation,” in ICASSP, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 7289–7293, IEEE, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [47] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Xu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Yu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Xu, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Yu, “MIMO Self-Attentive RNN Beamformer for Multi-Speaker Speech Separation,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Interspeech 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1119–1123, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [48] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Cord-Landwehr, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Boeddeker, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Von Neumann, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Zoril˘a, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Dod- dipatla, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Haeb-Umbach, “Monaural source separation: From anechoic to reverberant environments,” in IWAENC, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1–5, IEEE, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [49] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ditter and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Gerkmann, “A multi-phase gammatone filterbank for speech separation via Tasnet,” in ICASSP, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 36–40, IEEE, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [50] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Dehak, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Kenny, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Dehak, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Dumouchel, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ouellet, “Front- end factor analysis for speaker verification,” IEEE Transactions on Audio, Speech, and Language Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 19, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 788–798, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [51] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Snyder, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Garcia-Romero, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Sell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Povey, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Khudan- pur, “X-vectors: Robust DNN embeddings for speaker recognition,” in 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 5329–5333, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [52] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' ˇZmol´ıkov´a, Neural target speech extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' thesis, Brno University of Technology, Faculty of Information Technology, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [53] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Gabbay, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Shamir, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Peleg, “Visual Speech Enhancement,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Interspeech 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1170–1174, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [54] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' King, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Vaizman, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Liu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Maas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Parthasarathi, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Hoffmeister, “Robust speech recognition via anchor word repre- sentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=',” in Interspeech, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 2471–2475, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [55] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Yu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Hu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Geng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Meng, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Yu, “Audio-visual multi-channel integration and recognition of overlapped speech,” IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 29, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 2067–2082, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [56] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Sodoyer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Rivet, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Girin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Schwartz, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Jutten, “An analysis of visual speech information applied to voice activity detection,” in 2006 IEEE International Conference on Acoustics Speech and Signal Processing Proceedings, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' I–I, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [57] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Yu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Weng, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Yu, “Towards robust speaker verification with target speaker enhancement,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' of ICASSP’21, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 6693–6697, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [58] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Tzinis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Wichern, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Subramanian, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Smaragdis, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Le Roux, “Heterogeneous Target Speech Separation,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Interspeech 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1796–1800, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [59] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Borsdorf, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Li, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Schultz, “Target language extraction at mul- tilingual cocktail parties,” in 2021 IEEE Automatic Speech Recognition and Understanding Workshop (ASRU), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 717–724, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [60] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ohishi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Delcroix, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ochiai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Araki, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Takeuchi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ni- izumi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Kimura, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Harada, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Kashino, “Conceptbeam: Concept driven target speech extraction,” MM ’22, (New York, NY, USA), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 4252–4260, Association for Computing Machinery, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [61] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ceolini, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Hjortkjær, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Wong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' O’Sullivan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Raghavan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Herrero, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Mehta, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Liu, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Mesgarani, “Brain-informed speech separation (BISS) for enhancement of target speaker in mul- titalker speech perception,” NeuroImage, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 223, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 117282, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [62] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Seetharaman, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Wichern, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Venkataramani, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Roux, “Class- conditional embeddings for music source separation,” in ICASSP 2019 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 301–305, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [63] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Zhao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Gan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Rouditchenko, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Vondrick, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' McDermott, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Torralba, “The sound of pixels,” in The European Conference on Computer Vision (ECCV), September 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [64] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Cantisani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Essid, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Richard, “Neuro-steered music source separation with eeg-based auditory attention decoding and contrastive- nmf,” in ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 36–40, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 21 [65] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Duan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Essid, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Liem, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Richard, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Sharma, “Audiovisual analysis of music performances: Overview of an emerging field,” IEEE Signal Processing Magazine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 36, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 63–73, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [66] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ochiai, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Delcroix, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Koizumi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Ito, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Kinoshita, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Araki, “Listen to What You Want: Neural Network-Based Universal Sound Selector,” in Interspeech, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1441–1445, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [67] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Gfeller, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Roblek, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Tagliasacchi, “One-shot conditional audio filtering of arbitrary sounds,” in ICASSP, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 501–505, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [68] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Liu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Kong, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Mei, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Zhao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Huang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Plumbley, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Wang, “Separate What You Describe: Language-Queried Audio Source Separation,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Interspeech 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 1801–1805, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' [69] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' Asaridou and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' McQueen, “Speech and music shape the listen- ing brain: evidence for shared domain-general mechanisms,” Frontiers in psychology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} +page_content=' 321, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFQT4oBgHgl3EQffzab/content/2301.13341v1.pdf'} diff --git a/bNFQT4oBgHgl3EQfhDYG/content/tmp_files/2301.13344v1.pdf.txt b/bNFQT4oBgHgl3EQfhDYG/content/tmp_files/2301.13344v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4bb3e8b7e212e140ccec6bb082a8a5d91dfeb34c --- /dev/null +++ b/bNFQT4oBgHgl3EQfhDYG/content/tmp_files/2301.13344v1.pdf.txt @@ -0,0 +1,2238 @@ +MNRAS 000, 1–18 (2022) +Preprint 1 February 2023 +Compiled using MNRAS LATEX style file v3.0 +Anomalous compressible mode generation by global frame +projections of pure Alfven mode +Ka Ho Yuen,1,2⋆ Huirong Yan,3,4† Alex Lazarian1‡ +1Department of Astronomy, University of Wisconsin-Madison, USA, 53715 +2Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA +3Deutsches Elektronen-Synchrotron DESY +4Institut f¨ur Physik & Astronomie, Universit¨at Potsdam, Germany +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +Alfven wave is the single most important physical phenomenon of magneto-hydrodynamic turbulence and has far- +reaching impact to almost all studies related to astrophysical magnetic field. Yet the restoration of the Alfven wave +fluctuations from a given magnetic field, aka the local Alfven wave problem, is never properly addressed in literature +albeit its importance. Previous works model the Alfven wave fluctuation as the perturbation along a straight-line, +constant magnetic field. However, Lazarian & Pogosyan (2012) suggested that the decomposition of Alfven wave +along a straight line, aka. the global frame decomposition, has a factor of discrepancy to the true local Alfven wave +fluctuation. Here we provide a geometric interpretation on how the local Alfven wave is related to the global frame +through the use of vector frame formulation. We prove both analytically and numerically that the local frame Alfven +wave is an orthogonal transformation of that of the global frame and related by the local Alfvenic Mach number. In +other words, when we observe Alfven wave in the global frame of reference, some of the Alfven wave will be mistaken +as compressible waves. The importance of frame choices have a far-reaching impact to the analytical studies of MHD +turbulence. Combining the frame formalism and the new techniques we can have accurate measurement to some of +the fundamental turbulence properties like the inclination angle of mean magnetic field relative to the line of sight. +Key words: turbulence – ISM: magnetic fields – ISM: structure — galaxies: ISM +1 INTRODUCTION +Turbulence is ubiquitous in astrophysical environment and +the interstellar gases are permeated by turbulent magnetic +fields. Magneto-hydrodynamic (MHD) turbulence plays a +very important role in various astrophysical phenomena (see +Armstrong et al. (1995); Chepurnov et al. (2010); Biskamp +(2003); Elmegreen & Scalo (2004); McKee & Ostriker (2007)), +including star formation (see McKee & Ostriker (2007); Fis- +sel et al. (2016)), propagation and acceleration of cosmic rays +(see Chandran (2000); Yan & Lazarian (2002); Farmer & Gol- +dreich (2004); Lazarian (2016)), as well as regulating heat and +mass transport between different ISM phases (Green (1993); +Deshpande et al. (2000); Dickey et al. (2001); Lazarian & +Pogosyan (2004, 2006); Khalil et al. (2006); Begum et al. +(2006); Padoan et al. (2006) see Draine (2009, 2011) for the +list of the phases). +MHD turbulence is usually highly compressible, and has +been thoughtfully studied by a number of authors in the +community (e.g. Kowal et al. (2007)). However, the com- +⋆ kyuen@lanl.gov (Oppenheimer Fellow), ORCID: 0000-0003- +1683-9153 +† huirong.yan@desy.de, ORCID: 0000-0003-2560-8066 +‡ alazarian@facstaff.astro.wisc.edu +pressibility of the turbulence adds additional difficulty in the +understanding of how the three fundamental MHD modes +(namely Alfven, slow and fast modes) would behave in vari- +ous astrophysical phenomena, each carrying different spectra +and anisotropies. For instance, it is believed that the Alfven +mode plays a central role in making the cold neutral media +aligned with the magnetic field (Lazarian et al. 2018) and +controls the transport of heat and particles across magnetic +fields (Narayan & Medvedev 2001; Lazarian 2006; Yan & +Lazarian 2008; Maiti et al. 2021). In comparison, fast modes +play an important role in the scattering and acceleration of +cosmic rays (Yan & Lazarian 2002, 2004; Cho & Lazarian +2005; Lazarian & Pogosyan 2008; Brunetti & Lazarian 2007). +The modes composition strongly depends on the way of driv- +ing Makwana & Yan (2020). It is therefore essential to have +a handy way in decomposing the three fundamental MHD +modes in numerical analysis. +A notable development is the statistical mode decomposi- +tion developed by Cho & Lazarian (2002; 2003, latter here- +after CL03), which allows one to obtain the realization of the +three fundamental MHD modes in numerical simulations by +considering a perturbation along a locally strong magnetic +field direction. The realization of MHD modes allowed the +community to validate the theory of MHD turbulence (Gol- +© 2022 The Authors +arXiv:2301.13344v1 [astro-ph.GA] 31 Jan 2023 + +2 +Yuen et.al +dreich & Sridhar (1995) hereafter GS95, see also Lazarian +& Vishniac (1999); Cho & Vishniac (2000); Maron & Gol- +dreich (2001); Lithwick & Goldreich (2001); Cho & Lazarian +(2002, 2003)) through numerical simulations. In particular, +the scaling relation of compressible modes were first verified +through the realization of MHD modes using the mode de- +composition algorithm developed by CL03. The realization of +MHD modes also excites the development of different tech- +niques in studying MHD turbulence in observations, includ- +ing the Velocity Gradient Technique (VGT, Yuen & Lazarian +(2017b,a)) which uses the anisotropy of different modes in re- +trieving the magnetic field directions in spectroscopic data, +and also the Synchrotron Polarization Analysis (SPA, Zhang +et al. (2020)) which utilizes the properties of the projected +statistics in predicting the dominance of Alfven or compress- +ible modes in observational synchrotron data, as well as de- +tailed analysis of solar wind turbulence (e.g. Zhao et al. 2021, +2022). +However, Goldreich & Sridhar (1995) model of MHD tur- +bulence is of centre importance in the modern theory of MHD +turbulence. The latter is employs the concept of ”local frame +of reference” that was added to the theory later (Lazarian +& Vishniac 1999; Cho & Vishniac 2000). This means that +the eddies, which are usually elliptical in shape, are aligned +to the local magnetic field rather than the mean magnetic +field. As pointed out by Kowal & Lazarian (2010), the de- +composition of CL03 is a global frame decomposition, as op- +posed to the local frame MHD theory stressed in the works +that followed the original GS95 study (Lazarian & Vishniac +1999). As described in Fig.1, when one considers a different +volume, the realization of the three fundamental modes will +be different due to the change of the mean magnetic field +directions under the CL03 decomposition algorithm. The dif- +ficulty of obtaining the statistics of three modes in a local- +ized manner has been attempted, including abandoning the +realization of modes but focusing on the structure functions +Beresnyak et al. (2005), decomposing the MHD quantities +into linear combination of fundamental localized patches be- +fore performing the CL03 decomposition (Kowal & Lazarian +2010), or the introduction of the frame changing parame- +ters in the framework of correlation functions (Lazarian & +Pogosyan 2012). Yet, how the three fundamental modes are +realized in the local system of reference is still an unsolved +problem for numerical community. +In this paper, we explore how the Alfven and compressible +modes in the local system of reference are expressed globally. +In §2 we review the CL03 method and its possible improve- +ments. In particular, in §3 we discuss about the generation of +”compressible waves signature” due to the wrong choice of lo- +cal frame of reference. From §4 to §5, we describe a few appli- +cations that utilize the concept of Alfven leakage, namely the +applications of the Synchrotron Polarization Analysis Tech- +nique to regimes with strong Faraday rotation (§4) and the +determination of the line of sight angle γ (§5). In §6 we discuss +about the possible impacts of our method and the caveats of +our work. In §7 we conclude our paper. +2 MODE DECOMPOSITION +2.1 Review of the MHD mode decomposition +methods +In this section we review the underlying assumptions of the +mode decomposition method as introduced by CL03 and the +development since then. In CL03 they consider a volume dΩ +in which the perturbation of magnetic field is small compared +to the mean field δB(dΩ) < ⟨B⟩, so does the density fluctua- +tions δρ/⟨ρ⟩ < 1. Fig.1 shows how the volume dΩ is selected. +Readers should be careful that once the volume is selected the +mean magnetic field direction ˆλ is also defined respectively. +In this scenario, the small perturbation in the presence of +a strong mean magnetic field will provide a linearized set of +MHD equations in which three non-trivial eigenvectors would +be found. In this localized box, the Alfven, slow and fast mode +eigenvectors are1: +ζA(ˆk, ˆλ) ∝ ˆk × ˆλ +ζS(ˆk, ˆλ) ∝ (−1 + α − +√ +D)(k · ˆλ)ˆλ ++ (1 + α − +√ +D)(ˆλ × (k × ˆλ)) +ζF (ˆk, ˆλ) ∝ (−1 + α + +√ +D)(k · ˆλ)ˆλ ++ (1 + α + +√ +D)(ˆλ × (k × ˆλ)) +(1) +where α = βγ/2, D = (1 + α)2 − 4α cos2 θλ, cos θλ = ˆk · ˆλ, +plasma β ≡ Pgas/Pmag measures the compressibility and +γ = ∂P/∂ρ is the polytropic index of the adiabatic equa- +tion of state. The presence of ˆk suggests that the direction +of the three mode vectors are changing as k changes. In this +scenario, the perturbed quantities, say for the velocity fluc- +tuations v1 = v − ⟨v⟩ can be written as: +v1(r) = +� +d3keik·r +� +X∈A,S,F +F0,X(k)F1,X(k, ˆλ)CXζX(ˆk, ˆλ) +(2) +for some power spectrum Ev(k) = F 2 +0 = k−n/2 (Yuen et al. +2022), some anisotropy weighting function F1 and the mode +constants CX denoting the relative weight of the three modes. +Notice that the decomposition (Eq.(1)) is a global decom- +position method since the magnetic field fluctuations within +the volume dΩ is not considered when computing the eigen- +vectors of the three modes. One of the most important conse- +quences of performing global decomposition is the loss of the +GS95 scaling for small k. In fact, Cho & Lazarian (2002) (see +also Lazarian & Pogosyan (2012)) pointed out that in the +global system of reference the anisotropic scaling is scale in- +dependent, meaning that the elongation of turbulence eddies +is fixed and does not change as the eddies cascade. Another +important consequence of the concept of dΩ is that, when +one changes the sampling volume, e.g. from the volume in +blue to that of yellow or orange in Fig.1, the weighting of the +1 However, recent publication from Gan et al. (2022) suggests that +a significant portion of the projected spectral powers are not in +the form of propagating waves, but fluctuations with miniature +frequencies. The nature of the non-wave fluctuations as dubbed +in Gan et al. (2022) requires further clarifications from the theory +of MHD turbulence. See Beresnyak & Lazarian (2019); Fu et al. +(2022); Schekochihin (2022). +MNRAS 000, 1–18 (2022) + +Alfven leakage +3 +three modes will also change due to the change of the mean +direction of magnetic field from volume to volume. +Notice that the selection of the volume dΩ has to fulfill the +conditions as assumed in CL03: Both the sonic and Alfven +Mach number within the volume must be smaller than unity. +Notice that if the volume is smaller than the volume defined +by the correlation length of the turbulence, the dispersion +of the turbulence observables will be scaled as a function of +distance according to their respective turbulence statistics, +meaning that if ρ, v follows GS95, +δρ2(r) ∝ r2/3 +δv2(r) ∝ r2/3 +(3) +To address the issue of the locality, the community has ex- +plored a number of ways to include the local fluctuations of +magnetic field during the calculation of statistics of MHD +modes. For instance, one of the most notable ways of ob- +taining the statistics of MHD modes is to compute the local +structure functions Beresnyak et al. (2005). The mathemati- +cal expression of the 3D structure function of the turbulence +variable v in the local frame of reference is given by: +SF{v}(r) = +� � +(v(r′ + r) − v(r′)) · ˆλ(r, r′) +�2 � +r′ +(4) +where +ˆλ(r, r′) = B(r′ + r) + B(r′) +|B(r′ + r) + B(r′)| +(5) +The anisotropy computed throughout this manner is scale +dependent and exhibit the GS95 scaling r∥ ∝ r2/3 +⊥ . The use +of the local structure function correctly recovers the GS95 +statistics, yet it is not possible to obtain the realization of +the modes in this manner, meaning that further study of the +features of the modes, e.g. how does the mode look like when +projected, are prohibited when using the structure functions. +Another important way of improving the CL03 is to rewrite +the turbulence variables into the linear sum of small local- +ized patches through the wavelet transform (Kowal & Lazar- +ian 2010). Physically, they are looking for specific functional +forms obeying the orthogonality requirement and represent +the volumes as shown in Fig.1.By considering the set of or- +thogonal wavelets φ, one can write the velocity field v(r) as: +˜v(w; a) = a−3/2 +� +d3xφ(r − w +a +)v(r) +(6) +where φ is the set of wavelet functions. For (Kowal & Lazarian +2010) they select the 12-tap Daubechies wavelet. To perform +mode decomposition, they first convert the velocity field into +the linear combinations of wavelets, and then proceed with +the procedures of CL03 for the wavelet transformed variable. +The contributions for all wavelets for a specific mode are +added before the inverse Fourier transform takes place. No- +tice that the wavelet functions is simply a mathematical con- +struction that may contain spatially dispatched regions (e.g. +D4 and D12 of the Daubechies wavelet), for which taking +the statistical calculations within the wavelet might not be +physically justifiable. Nevertheless, the improved mode de- +composition method proposed by Kowal & Lazarian (2010) +still retrieve seemingly the correct GS95 statistics from the +simulations. +2.2 The locality of the mode decomposition +problem due to magnetic field wandering effect +In realistic MHD simulations the magnetic field lines are fluc- +tuating within any selected volume, which is named wander- +ing effect as in CL03. However the mode decomposition algo- +rithm available in the community had not considered any of +these wandering effect, which makes the estimation of modes +be rather unrealistic for larger MA. To model the additional +effect when decomposing the modes in the global frame of +reference, Lazarian & Pogosyan (2012) model the magnetic +field correlation function as the linear combination of the +isotropic tensor ˆTE and the axis-symmetric tensor ˆTF (See +Yan & Lazarian 2004 also our Appendix for a detailed math- +ematical construction). In general the direct tensor of the +magnetic field in the Fourier space at a given wavevector k +in the local frame of reference can be written as: +˜Hi ˜Hj ∝ E(k) ˆTE,ij + F(k) ˆTF,ij +(7) +The transformation from the local to global frame as mod- +elled by LP12 is: +ˆTE,local → ˆTE,global +ˆTF,local → WI ˆTE,global + WL ˆTF,global +(8) +where WI,L are two modelling constants that are functions +of MA and also k. +3 ALFVEN LEAKAGE +3.1 Description of the problem +Since ˆTE = ˆTC + ˆTA and ˆTF = ˆTC (See the Appendix A2) +one can actually write the magnetic field in the Fourier space +in the global frame of reference using vector notations due to +the orthogonality of the base vectors: +˜H(k) = CCζC + CAζA +(9) +where +ζC = (ˆk × (ˆk × ˆλ))i +|ˆk × ˆλ| +ζA = (ˆk × ˆλ)i +|ˆk × ˆλ| +(10) +(See Appendix). The modelling of LP12 simply means that, +assuming if in the local frame of reference the magnetic field +only has the Alfven component ˜H(k) = +� +E(k)C0ζA, then +the transformation from local to global reference frame sim- +ply means a vector projection: +˜H(k) = +� +E(k)C0ζA +global +−−−−→ +� +E(k)(CCζC + CAζA) +(11) +where CC = C0 sin(θf), CA = C0 cos(θf) for some θf that +we will explore in the coming subsection. For Alfven waves, +E(k) = −F(k), and the relation between CC,A and WI,L can +be easily derived: +C2 +A = 1 − WI +C2 +C = 1 − WI − WL +(12) +Notice that when MA ≪ 1, there is no magnetic field wander- +ing effect. In this limit, CA(MA → 0) = 1, CC(MA → 0) = 0. +The vector notation (cf. Eq.(9)) allows us to think of this +MNRAS 000, 1–18 (2022) + +4 +Yuen et.al +Figure 1. The concept of mode decomposition in CL03: (1) By selecting a volume dΩ, a local mean magnetic field direction ˆλ would +then be defined for later decomposition. (2) All the wavevector k that is contained in this volume dΩ are used for the decomposition. (3) +For each k there is a local reference frame (See Fig.A1) that decomposes the magnetic field into the three eigenmodes. The change of the +selected volume dΩ will lead to different mean field vector ˆλ. As a result, the local decomposition result would be functions of both λ and +the wavevector k . +Figure 2. An illustrative figure showing how the Alfven Leakage phenomenon happens during the mode decomposition process. Here the +red line represents the B-field line, point X is the origin of the volume dΩ, that is represented by the dash orange circle. The vector ˆλ +represents the mean field averaged over dΩ. For a given point within dΩ, assuming the k-vector points inside the plane, the local Alfven +wave unit vector (green) makes an angle to that of the Alfven wave unit vector defined by ˆλ. This effect will be stronger if the magnetic +field fluctuations within dΩ is larger, and vice versa. +problem geometrically and relates to a very important phys- +ical effect that mentioned in both Yuen et al. (2018) and +more recently Yuen & Lazarian (2020): The Alfven leak- +age effect. Alfven leakage describes the effect that the locally +Alfven component of the turbulent variables are projected as +linear combinations of Alfven and non-Alfven components in +the presence of a curved magnetic field +averaged over se- +lected volume (local mean magnetic field). In Yuen & +Lazarian (2020) we consider the effect that the gravitational +forces creates extra compression to the Alfven waves and thus +some of the Alfven waves from the self-gravitating systems +are transferred into compressible components. In fact, in the +MNRAS 000, 1–18 (2022) + +Local Volume dΩ +B-field line +Wavevector k +Wavevector k +Local +Local Mean Field +Local +Mean +αr +Mean +Field +d3r B(r) +d2 JrEd2 +Field +Wavevector kGreen: Real, local {A = k X +H; = CA,oA,local +Blue: From the +decomposition algorithm +what A = k × looks like +k +> +Cc,global = CA,local sin SQdn +Point X +Local volume d2 +CA,global = CA,local cos SQdn +Ssα = Dispersion of B +within the local yolume +B-field lineAlfven leakage +5 +presence of non-trivial magnetic field structures, there exist +non-zero (Yuen & Lazarian 2020) field lines in any volume +due to magnetic field wandering. Similar effect has been con- +sidered in Maron & Goldreich (2001) but being corrected by +Cho & Vishniac (2000) as the ”rotation effect” by recogniz- +ing that the global frame anisotropy axis ratio is a function +of MA. However the actual relation between the local and +global frame of reference has yet to be explored. +Fig.2 gives a pictorial illustration on how the Alfven leakage +happens during mode decomposition. For a given magnetic +field Hi(r), its Fourier transform ˜Hi(k) can be written as the +linear sum of Alfven and compressible components. To sim- +plify our argument, we consider a pure Alfven wave B-field in +the local frame of reference, which ˜Hi is simply proportional +to the local Alfven vector ζA,local, which is represented by +the green vector. However if the volume dΩ is selected (the +orange dash circle), the mean field is defined in dΩ. In this +case, if we consider a pure Alfven mode magnetic field in the +local frame of reference as we show in Fig.2, the projection +of the Alfven mode in the global frame of reference will be +an linear combination of Alfven and compressible modes as +written in Fig.2. By selecting a k-vector that points inside +the plane, the local Alfven wave unit vector (green) , which +is defined by the cross product between the k-vector and the +local magnetic field direction, makes an angle to that of the +Alfven wave unit vector defined by ˆk × ˆλ which we will name +that angle δθdΩ. The projection effect causes artificial com- +pressible modes to be detected within dΩ. It is very apparent +that if the magnetic field line is aligned with the mean field +vector ˆλ in dΩ, then there is no artificial compressible modes +due to the projection effect. We call this effect Alfven Leak- +age. This effect is artificial and does not involve the change +of the cascade laws or anisotropies. +3.2 Modelling of the Alfven leakage problem +We can further model the leakage phenomenon through the +use of δθdΩ. From Fig.2 we see that δθdΩ is a measure of +the average magnetic field angle dispersion in this selected +volume. We can postulate that the dispersion of angles are +related to the Alfven Mach number measured within dΩ: +δθdΩ ∼ MA,dΩ +(13) +Notice that the dispersion of magnetic field angle can scale +up to MA ≈ 2 as shown in the appendix of Lazarian et al. +(2022b). Notice that according to the definition of the Alfven +Mach number and the self-similarity of turbulence cascade, +any localized calculation of statistical observables in a tur- +bulence system can be approximated by the scale relation +through the observable’s structure function. For the case of +Alfven Mach number, the corresponding observable is δB/B, +then +MA,dΩ ≈ +� 1 +2B2 ⟨δB(r + r′) − δB(r′)⟩r′ +�1/2 +∼ MA,global +� +r +Linj +�1/3 +(14) +where the last equality comes from the fact that the magnetic +field fluctuation is small and Kolmogorov: ⟨δB2⟩ ∼ r2/3. We +can then relate the CC,A with Eq.(13) and (14). +3.3 Numerical tests +We can perform a very simple numerical test to illustrate the +behavior of the Alfven leakage, which can be done by the +following steps: +(i) We select the local frame vectors ζA,C, where they can be +approximated by selecting a very small volume in the numer- +ical simulations and use Eq.9. We shall call this volume 1. +(ii) We then perform Alfven wave decomposition using ζA from +simulations so that we can put CA,local = 1 in all our case +(See Fig.1). +(iii) We then select a larger volume with size σ. According to +§2, if there is indeed the effect of Alfven leakage happening, +then CC = sin MA ∼ sin(Cσ1/3), where C = MA,globalL−1/3. +In this new volume, there will be another pair of ζA,C being +defined. We shall call this volume 2. +(iv) We plot the quantity C2 +C = 1 − (⟨ ˆBAlf,1 · ˆBAlf,2)2⟩. Notice +we only compare the directions of B within volume 1, since +this calculation only makes sense there. +(v) There could be three outcome from this test +(a) If there is no such leakage effect, CC should be a constant +zero as we already removed all non-compressible compo- +nents in the previous step. +(b) If there is indeed CC but our model in the previous section +is incorrect, then there should not be a dependence of CC ∼ +sin(Cσ1/3). +(c) If the Alfven leakage effect indeed exists, we expect CC ∼ +sin(Cσ1/3). +Fig.3 shows how the CC behave as a function of the size of +the volume σ, where we plot the regime when σ is not in the +dissipation range. Notice that here we intentionally pick simu- +lation cubes from various numerical codes and with different +conditions to show that the leakage effect is universal and +rather independent to what choices of MHD code one works +with. For each figure, we draw the predicted proportional- +ity CC ∼ sin2(Cσ1/3) as the red dash curve. As we can see +from these three subplots, our prediction CC ∼ sin2(Cσ1/3), +which came analytically from previous sections and not from +a fitting algorithm, follows the trend reasonably well. +This indicates that (1) the Alfven leakage effect actually ex- +ists even in incompressible mode turbulence (2) the leakage +is smaller when one goes to smaller scales (3) our postulate +that δθdΩ ∼ MA,local is a good approximation. These three +consequences indicate that the mode decomposition as pro- +posed by CL03 requires an additional update in addressing +the contributions of large scale magnetic field wandering to +the relative composition of modes within the volume. +As a direct consequence of this section, the WI,L constants +originated from Lazarian & Pogosyan (2012) as a function of +MA,local (≪ 1) are given by Eq.(8) are : +WI = 1 − C2 +A ∝ M 2 +A +WL = C2 +A − C2 +C ∝ 1 − 2M 2 +A +(15) +MNRAS 000, 1–18 (2022) + +6 +Yuen et.al +Figure 3. Three figures showing the ratio of compressible modes CC as a function of the size of the volume selected σ for three simulations. +From the left: ma20, ma59, huge-4. (See Table 1). +Model +MS +MA +β = 2M2 +A/M2 +S +Resolution +ZEUS-MP Simulations +Ms0.92Ma0.09 +0.92 +0.09 +0.02 +4803 +Ms0.98Ma0.32 +0.98 +0.32 +0.22 +4803 +Ms0.93Ma0.94 +0.93 +0.94 +2.0 +4803 +huge-0 +6.17 +0.22 +0.0025 +7923 +huge-1 +5.65 +0.42 +0.011 +7923 +huge-2 +5.81 +0.61 +0.022 +7923 +huge-3 +5.66 +0.82 +0.042 +7923 +huge-4 +5.62 +1.01 +0.065 +7923 +huge-5 +5.63 +1.19 +0.089 +7923 +huge-6 +5.70 +1.38 +0.12 +7923 +huge-7 +5.56 +1.55 +0.16 +7923 +huge-8 +5.50 +1.67 +0.18 +7923 +huge-9 +5.39 +1.71 +0.20 +7923 +e5r2 +0.13 +1.57 +292 +12003 +e5r3 +0.61 +0.52 +1.45 +12003 +e6r3 +5.45 +0.24 +0.0039 +12003 +Pencil Simulations +ma20 +0.91 +0.20 +1.56 +5123 +ma23 +0.22 +0.23 +2.17 +5123 +ma40 +0.20 +0.40 +8.00 +5123 +Pluto Simulations +ma56 +0.57 +0.59 +2.17 +5123 +ma68 +0.19 +0.68 +25.51 +5123 +ma75 +0.72 +0.75 +2.17 +5123 +ma86 +0.83 +0.86 +2.17 +5123 +Table 1. Description of MHD simulation cubes which some of +them have been used in the series of papers the authors have +worked on before (Yuen & Lazarian 2017b,a; Lazarian et al. 2018; +Makwana & Yan 2020; Zhang et al. 2020) the snapshots are taken. +4 APPLICATION (I): ADVANCEMENT OF THE +SPA TECHNIQUE +4.1 Review on the SPA technique +In Zhang et al. (2020) the authors discussed a novel im- +plementation called SPA in obtaining the modes from syn- +chrotron emission maps. Their argue that, since the tensor +structures for Alfven or compressible waves being different at +the synthesis of the Stokes parameter, the signature of the +dominance of the modes are left in the ”signature parame- +ter”. Let us first recap the formulation of Zhang et al. (2020) +and their main results (See Method section of Zhang et al. +2020). To start with, the authors consider the emissivity of +the synchrotron emissions under a locally defined reference +frame: +ϵxx = (I + Q)/2 += B2 +0,⊥ cos2 φs + 2B0,⊥ cos φsBiˆexi + (Biˆexi)2 +(16) +where our symbols follow the Zhang et al. (2020) notations +and we employ Einstein notation of summation. Very im- +portantly, the angle φs is the angle between the polarization +vector to the currently defined x-axis of the local magnetic +reference frame. In Zhang et al. (2020) they select a small area +and compute the change of ϵxx as the reference frame rotates. +They recognize that the variance of ϵxx contains factors that +could reflect the relative contributions of MHD modes. +The variance of the emissivity contains the linear term (the +first term) and the quadratic term depending on the power +of the tensor ˆT: +sxx = ⟨ϵ2 +x⟩ ∝ (2B0,⊥ cos φs)2 +� +dkF 2(k)eik·rˆexiˆexi ˆTij(ˆk) ++ 2 +�� +dkF 2(k)ˆexiˆexi ˆTij(ˆk) +�2 +(17) +Zhang et al. (2020) pointed out that the linear term, namely +the signature parameter: +sxx(φs) ∝ (2 cos φs)2 +� +dkF 2(k)eik·rˆexiˆexj ˆTij(ˆk) +(18) +can be expressed in the following format with some constants +axx, bxx defined according to MHD theory (See Zhang et al. +2020 for details): +sxx(φs) = (axx sin2 φs + bxx) cos2 φs, +φs ∈ [0, π] +(19) +where the classification parameter rxx is defined as +rxx = axx +bxx +(20) +In practice, we need to compute the parameter +sxx,tot = V ar(ϵxx) +4⟨ϵxx⟩ +(21) +This term contains both the ”linear” term (Eq.19) and the +”quadratic” term as defined in Zhang et al. (2020). +MNRAS 000, 1–18 (2022) + +Subsonic Ma = 0.2 +Subsonic Ma = 0.59 +Supersonic MA = 0.82 +Code: Pencil +Code: Pluto +Code: ZEUS-MP/3D +0.22 +0.22 +0.20 +Ldisp) +0.20 +C2 = sin2(c3) +0.18 +() +C² = sin2( +C = sin2(co) += +II +0.18 +0.18 +· BA/f(o : +0.16 +0.16 +0.16 +MA α3 +0.14 +0.14 +0.14 +0.12 +0.12 +0.12 +0.10 +0.10 +0.10 +=1 +1 +0.08 +0.08 +0.08 + 0.06 +0.06 +101 +2 × 101 +101 +3×101 +4×101 +2 × 101 +3× 101 +4×101 +101 +2 × 101 +3 ×101 +4×101 +Size of the local volume, +Size of the local volume, o +Size of the local volume, oAlfven leakage +7 +Figure 4. An illustration in showing how the observed features +of the synchrotron intensities are related to the different weight- +ing of spectrum, anisotropy and frame vectors (See Tab.A1) if a +strong guided field is given. From the top left: from a k−11/3 spec- +trum plus the Alfven frame vector; top right: a k−11/3 spectrum +with the Alfven anisotropic factor and also the Alfven frame vector +mimicking the Alfven mode; lower left: a k−11/3 spectrum with the +compressible frame vector, mimicking the Fast mode; lower right: +a k−11/3 spectrum with the slow mode (See Tab.A1) anisotropic +factor and also the compressible frame vector, mimicking the slow +mode. It is very apparent that both the anisotropy factor and also +the frame vector contributes to the observed anisotropy in 2D ob- +servables. From synthetic simulations assuming MA = 0.1 such +that the leakage effect is small. +4.2 Alternative ways to separate the Alfven and +compressible modes +As discussed in LP12, two-point turbulence statistics con- +tains three types of contributions: The spectrum which mea- +sures the cascade as a function of scales, the anisotropy which +records whether there is a preferred direction for the cascade +to happen, and the tensor structures which records the pro- +jection effect of the mode components. From the discussion +above, the SPA technique actually did not consider two-point +statistics. In particular, Eq.(21) is a one-point statistics, in +which the anisotropy of the observable does not play a role +in the value of the output. Since all scales are summed up +when computing the mean and variances, the spectrum also +do not play an important role for Eq.(21). As a result, a nat- +ural guess on how the SPA technique works is the projection +effect from the tensor structures. In this scenario using the +vector formulation (See Appendix) we can understand quan- +titatively better how the SPA technique works. Notice that, +in the case of one-point statistics, the 2D (i.e. the average op- +erator above) and 3D statistics (which we will consider later +below) should be the same. +Let us consider a 3D magnetic field line written as sum of +the mean and perturbed contribution in a selected volume +dΩ with MA,dΩ ≪ 1: +Hi(r) = ⟨Hi⟩ + +� +d3keik·r +� +X∈any frame +CX(ˆk, ˆλ)ˆζX(ˆk, ˆλ) (22) +From now on we are going to choose the frame to be the PCA +frame (See Appendix A2, Fig.A1), and assuming the line of +sight direction is at the z-axis and the magnetic field in the +plane of sky defines the x-axis. The x-component magnetic +field dispersion, which is just the mean value of the emissivity +subtracted by a constant (cf.Eq.(16)), is given by: +⟨δH2 +x⟩ = 2π +� +d3k +� +X∈C,A +C2 +X(ˆζX · ˆx)2 +(23) +Notice that the only difference between the compressible and +the Alfven component can be observed when we expand the +dot product for the above equation: +ˆζA · ˆx = (ˆλ · ˆz)(ˆk · ˆy) +|ˆk × ˆλ| +ˆζC · ˆx = −(ˆλ · ˆx)1 − (ˆk · ˆx)2 +|ˆk × ˆλ| +(24) +We can model Eq.(23) via the frame definition of φs in +Eq.(16), where the frame angle φs = 0 when the projection +of magnetic field is along the x-axis: +⟨δH2 +x⟩ = Axx cos2 γ + Bxx sin2 γ cos2 φs +(25) +where cos γ = ˆλ · ˆz is the line of sight angle, and Axx, Bxx +are: +Axx = 2π +� +d3kC2 +A,obs +� +(ˆk · ˆy) +|ˆk × ˆλ| +�2 +Bxx = 2π +� +d3kC2 +C,obs +� +1 − (ˆk · ˆx)2 +|ˆk × ˆλ| +�2 +(26) +The factors within the bracket of each equation above are the +geometric factors as discussed in LP12. Here we consider the +general case of the leakage, which applies to both Alfven and +compressible modes (See §3), i.e. the observed amplitudes +of Alfven and compressible modes CA,obs, CC,obs undergo an +orthogonal rotation of angle MA < 1 (See Eq.13): +CA,obs ≈ CA cos MA − CC sin MA +CC,obs ≈ CA sin MA + CC cos MA +(27) +The expressions inside the brackets of Axx, Bxx are the geo- +metric factors that considered in both LP12 and Zhang et al. +(2020). We can see from Eq.(25) that the contributions of +Alfven and compressible modes are separated when one con- +siders the frame rotation even for ⟨ϵ⟩. It is not necessary +to compute Eq.(21) in extracting the contributions of the +modes. Moreover, we can now quantify the contributions of +modes via Eq.(26) by using the modes for CA,C by simply +comparing the values of Axx and Bxx while analyzing the +observed synchrotron emission. In particular, if MA is small +and there is no compressible mode, then Bxx = 0. i.e. that +contribution of the Alfven mode to ⟨ϵ⟩ is frame independent +(i.e. rotating the x-y plane does not alter the result) since +(ˆλ · ˆz) cannot be changed due to frame rotation, while that +for compressible mode is a frame dependent quantity since +ˆλ · ˆx is a function of the reference frame. +4.3 The SPA technique for synchrotron emissions +with significant Faraday Rotation +In Zhang et al. (2020) they study the effects of Faraday ro- +tation to the SPA technique. Pictorially the Faraday depo- +larization effects shields information up to a certain distance +along the line of sight. This distance has been adequately +discussed in Lazarian & Pogosyan (2012) in the presence of +MNRAS 000, 1–18 (2022) + +No anisotropy factor is included +When anisotropy factor is included +Alfven, +Alfven , +Mimicking Alfven Mode +k-11/3 +k-11/3 × +anisotropy factor +Mimicking Slow Mod +Mimicking Fast Mode +Compressible , +Compressible, +k-11/3 × +k-11/3 +anisotropy factor8 +Yuen et.al +galactic MHD turbulence and is called the Faraday screen- +ing effect (Lazarian & Yuen 2018). Qualitatively, the SPA +technique can only determine the mode fraction before the +Faraday screen. However, we would like to perform the anal- +ysis based on the formalism of Lazarian & Yuen (2018). +In general, the synchrotron emission depends both on the +distribution of relativistic electrons +Ne(E)dE ∼ EαdE, +(28) +with intensity of the synchrotron emission being +Isync(X) ∝ +� +dzBγ +⊥(x) +(29) +where X = (x, y) is the 2D position of sky (POS) vector +and B⊥ = +� +B2x + B2y being the magnitude of the magnetic +field perpendicular to the LOS z. In general, γ = 0.5(α + +1) is a fractional power, which was a serious problem that +was successfully addressed in LP12. LP12 proves that the +statistics of I(α) is similar to that of I(α = 3). Therefore it +suffices to discuss the statistical properties of the case α = 3. +Per Lazarian & Pogosyan (2012), Synchrotron complex po- +larization function with Faraday rotation is given by: +Psynch(R) = +� +dzϵsynchρrelB2e2i(θ(R,z)+Cλ2Φ(R,z)) +(30) +where ϵsynch is the emissivity of synchrotron radiation, +Φ(R, z) = +� z +∞ +dz′(4π)−1/2ρthermal(R, z)Bz(R, z)rad m−2 +(31) +is the Faraday Rotation Measure 2. Notice that ρrel is the rel- +ativistic electron density, while ρthermal is the thermal elec- +tron density. The C-factor ≈ 0.81 (Lee et.al 2016). The pro- +jected magnetic field orientation is then given by: +θB = π +2 + 1 +2 tan−1 +2 (U +Q) +(32) +where tan−1 +2 +is the 2-argument arc-tangent function. +For frequencies lower than O(1GHz), the amplitude of the +Faraday Rotation measure will exceed 2π. The physical pic- +ture of the synchrotron polarization with Faraday rotation +measure can be understood as: photons that passes through +a section of ISM has to experience a certain amount of phase +shift. If this phase shift exceeds 2π, all information coming +from the source is completely lost. Therefore an important +concept called the Faraday screening emerges, which in- +dicates the maximal line of sight distance that the observed +synchrotron emissions can measure in the presence of line of +sight magnetic field. In the case of sub-Alfvenic turbulence, +the source term Pi = ρrel exp(2iθ(R, z)) is dominated by the +mean field rather than the fluctuating one. The two regimes: +(1) strong and (2) weak Faraday Rotation depend on whether +the ratio Leff/L, is smaller (strong) or larger (weak) than +unity: +Leff +L +∼ +1 +λ2L +1 +φ +(33) +where φ = max( +√ +2σφ, ¯Φ) with σφ is the dispersion of random +magnetic field. The difference between the two regimes are, +2 It is usually more convenient to use Hz = Bz/ +√ +4π for analysis. +Figure 5. A figure showing how the values of rxx varies as a +function of λ in the presence of Faraday Rotation ∝ λ2 � +dzρBz +for both Variance-driven Faraday Rotation (VFR) and mean-field +driven Faraday Rotation (MFR). +the Faraday rotation and the emission happens together in +the former regime (φ = +√ +2σφ), while the latter has the Fara- +day rotation happens after the emission of the polarization. +We shall name the two regimes ”Variance-driven Faraday Ro- +tation” (VFR) and ”Mean-field Faraday Rotation” (MFR), +respectively. Notice that both regimes have been considered +in Zhang et al. (2020). +Fig.5 shows a plot on how VFR and MFR could change +the value of rxx. For this current plot we intentionally plot +rxx with values that are not typically considered in previous +literature (See, e.g. Zhang et al. 2020, rxx ∈ [−1, 1]). This al- +lows us to better characterize whether the value of rxx came +from the effect of compressibility or from Faraday rotations. +We can observe from Fig.5 that there are two new regimes +of λ that could make rxx fluctuates well beyond the values +previously considered in Zhang et al. (2020). From Fig.5 we +classify the ranges of values of λ via the fluctuations of rxx +into three regimes: he ”weak” regime correspond to the case +where rxx is small (∈ [−1, 1] as in Zhang et al. 2020). The +intermediate regime correspond to the case where rxx start to +grow exponentially, and the strong regime correspond to the +case where the rxx basically loses traces on the compressibil- +ity. We can see that obviously the technique of SPA does not +work when we are in the strong regime. However an interest- +ing question is whether the SPA technique actually works in +the intermediate regime which will be the subject for future +studies. +5 APPLICATION (II): A SELF-CONSISTENT +LINE OF SIGHT ANGLE TRACING METHOD +VIA STRUCTURE FUNCTIONS OF I + Q AND +I − Q +The second application that we will deliver in this paper +would be the retrieval of the mean global line of sight an- +gle γ. In the case of synchrotron/dust polarization, we have +adequate information to estimate γ by considering the struc- +MNRAS 000, 1–18 (2022) + +40 - +20 - +0 +0-00-00-0-00-00-00-000-0000-00-00-0-0-00-00-0 +Intermediate +-20 - +Weak +FRregime +FR +regime +regime +-40- +. +VFR +MFR +10-3 +10~1 +101 +103 +105 +入 (cm)Alfven leakage +9 +ture functions of both I + Q ∝ +� +dzB2 +x and I − Q ∝ +� +dzB2 +y. +For the following subsections, we will assume that the global +mean field within the sampling area is ∥ to x-axis. One could +always rotate the frame in Stokes parameter space to have +the above condition satisfied. +5.1 Why γ is encoded in the statistics of I + Q and +I − Q? +The essence of on why γ is encoded in I + Q and I − Q is +based on the fact that tensor formulation (c.f. Eq.2) contains +different expressions for observables that ∥ B and ⊥ B. In +Fig. 6, we present a set of figures showing the anisotropy +of I + Q and I − Q for both A and F type contributions +(c.f. Fig.A1). We present two extreme cases for γ in Fig.6 3 +that is sufficient to illustrate the differences of behaviors for +the anisotropy of A and F type fluctuations. The left group +of figures in Fig.6 shows the case when γ = 89o, while the +group of figures on the right shows the case when γ = 9o. We +can observe from Fig.6 a few interesting phenomena which is +not covered in previous anisotropy studies: +(i) The anisotropies of A and F type tensor do not necessarily +align with the mean magnetic field direction. We discussed +this effect already from Fig.4. The reason behind is that both +the anisotropy and tensor contribution are anisotropic (c.f. +§4.2). However, the direction of anisotropy for the tensor con- +tribution (with the Alfven leakage in effect) does not neces- +sary be parallel to B-field and is a function of γ. Notice that +the change of anisotropy is highly tied with the γ value (See +Fig.7) +(ii) For the case of pure Alfven fluctuations, the anisotropy is +more or less parallel to magnetic field for I +Q, while ⊥ B for +I − Q. Yet, the compressible mode does not carry the same +trend as its Alfven counterpart: When γ ≈ 9o, the F-type +anisotropy for I + Q is actually ⊥ B, while that for I − Q is +∥ B. In contrast, when γ ≈ 89o the F-type anisotropy varies +very similarly to that of the Alfven counterpart. +(iii) The measurement of relative anisotropies between I +Q and +I − Q allows us to characterize the γ value. From Fig.6 we +can see that if we consider the anisotropies of I +Q and I −Q +at γ ≈ 89o, I +Q tends to be parallel to magnetic field, while +that for I − Q tends to be perpendicular to magnetic field. +We utilize the formulation in Appendix B that the minor- +to-major axis ratio l⊥/l∥ = +√ +1 − ϵ2, which the eccentricity +ϵ is related to the quadropole-to-monopole ratio |D2/D0| via +Eq.B3. The quadropole-to-monopole ratio is the key param- +eter in parametrizing the anisotropy in previous literature +(Lazarian & Pogosyan 2012, 2016; Kandel et al. 2016, 2017; +Lazarian et al. 2022b). +5.2 Formalism via §3 +We will start from the parameters I + Q and I − Q in which +we will assume the projected mean field is right now along +3 Notice that the projection of pure Alfven wave fluctuations when +γ is exactly 90o will vanish, see Lazarian et al. (2022b) for the +analysis. +Figure +6. A set of figures showing how the orientation of +anisotropy for I + Q and I − Q is related to the line of sight +angle γ for pure A (Alfven) and F (compressible, see Lazarian & +Pogosyan 2012) type tensor. The key difference between the case +of γ → π/2 (correspond to the case when B ⊥ LOS) and γ → 0 +is that, the anisotropies of I + Q and I − Q for pure A and F +tensors are similar for the former case, while for the latter case the +anisotropies of pure A and F tensors are exactly opposite. +the x direction 4. For the case of I+Q, we adopt the structure +function expression from Eq.(E20) of Lazarian et al. (2022b): +DI+Q(R) = ⟨(Bx(R + R′) − Bx(R′))2⟩R′ += +1 +2π2 +� +d2K +� +1 − eiK·R� +× +� +A(K, sin γ cos φK) +cos2 γ sin2 φK +1 − sin2 γ cos2 φK ++ +F(K, sin γ cos φK) +sin2 γ sin4 φK +1 − sin2 γ cos2 φK +� +(34) +where those factors are simply the expressions of ζAζA and +ζF ζF in the global frame of reference (i.e. after leakage). The +main takeaway here is, This D factor depends on the following +form +DI+Q(R) ∼ ¯A(R) cos2 γ + ¯F(R) sin2 γ +(35) +Similarly, the structure function for I −Q can be also mod- +elled similarly as: +DI−Q(R) ∼ ¯A(R) sin2 γ + ¯F(R) cos2 γ +(36) +Based on Fig.6 we can see that the construction: +¯y = Anisotropy(DI+Q) +Anisotropy(DI−Q) +(37) +contains the information on γ. Here we take the convention +that Anisotropy(D) > 1 when the anisotropy of structure +function is parallel to the global magnetic field direction, and +vice versa. In particular, from Fig.6 we expect that ¯yA > 1 +for all γ, while that for F-type contribution changes from +smaller than 1 to greater than 1. Detecting the value of ¯y for +4 For a general magnetic field configuration, one could always con- +sider the combination I + (Q cos(2φpol) − U sin(2φpol)), where we +perform an inverse orthogonal transform with twice of the polar- +ization angle 2φpol = tan−1 +2 +U/Q for this analysis. +MNRAS 000, 1–18 (2022) + +Pure A +Pure F +Pure A +Pure F +Y = 890 +06 =^ +Q +Mean B-field direction10 +Yuen et.al +Figure 7. A set of visualizations showing how the structure function of a certain variable D(R) can be visually decomposed as the linear +combination of the multipole moments Dn, and how the multipole moments should be physically correlated to the relative angle between +the line of sight and mean magnetic field γ. The multipole moments collects the relative weight on the shapes that are specifically defined +with the angular function exp(inθ). In particular, D0 records the weights of the isotropic components of the structure functions, while +D2 records the first order directionless anisotropy. Since empirically structure functions are mostly elliptical-like, |D2| must be non zero. +modern turbulence theory predicts that the observed anisotropy would be a function of γ. When ¯B ∥ LOS, then the structure function +should be isotropic. While ¯B ⊥LOS, the structure function should be anisotropic. Therefore under the framework of multipole moments, +the absolute amplitude of D2 should be a function of γ. +compressible modes (in global frame of reference) detected in +observation is the key to extract the value of γ. +The key reason why we consider the ratio of structure +functions instead of individual quantity is because, from our +expressions in the global frame of reference, the structure +function of some observables carries factors on spectrum, +anisotropy and tensors. For the case of structure functions +of I + Q and I − Q, their only difference is coming from the +tensor factor as spectrum and anisotropy factors are fixed +once the turbulence is set-up. +To proceed with our analysis, we consider the multipole +expansion up to quadrupole (See Appendix §B for the con- +dition for the expansion. In particularly, the expansion is +valid only for MA ∼ 0.5 − 1.0.). Formally we can express the +anisotropy function that we defined above with the monopole +and quadrupole coefficients D0, |D2|: +AnisotropyMA∈[0.5,1] ≈ sign(Anisotropy) × D0 − |D2| +D0 + |D2| +(38) +Recall from the previous discussion that the factors d0,2 can +be literally written as the spectrum, anisotropy and the ten- +sor contribution, and the first two contributions are cancelling +out under our treatment, we can formally write y, which is +the quadrupole approximation of ¯y to be (c.f. Eq.(E30) of +Lazarian et al. 2022b): +y = Anisotropy(I + Q) +Anisotropy(I − Q) += +�D0 − |D2| +D0 + |D2| +� +I+Q +�D0 − |D2| +D0 + |D2| +�−1 +I−Q +(39) +where we notice that under our current configurations, I + +MNRAS 000, 1–18 (2022) + +180° +exp(0i) +exp(2i0) +exp(4i0) +D(R) = +X ++ +X ++ +Do +D4 +D. +Isotropic term +Main term collecting anisotropy +ExampleStructureFunctions +B I L0S (y = 90°) +( =) SOT l +Mean B-Field Direction for LHS panel +↓→ID2↑Alfven leakage +11 +Figure 8. A figure showing the characterization of the relative +anisotropy index (y = anisotropy(I + Q)/anisotropy(I − Q)) as a +function of the line of sight angle γ. As we outlined in Fig.6, the rel- +ative anisotropies for A and F type fluctuations are different when +γ is different. For the case of γ → 90, we expect the anisotropies of +A and F type tensor fluctuate in the same way, which is illustrated +as the light blue box in the figure. However, when we are looking +at small γ limit, the anisotropies of A and F type tensor went com- +pletely opposite, which is highlighted by the red box in the figure. +We denote these two regimes the ”compressible” and the ”Alfven” +regime respectively. from numerical simulation ”e5r2” (See Tab.1.) +Q ∼ Lzb2 +x and I − Q ∼ Lzb2 +y where Lz is the length of the +integration. Keeping only the tensor term, we will have an +expression that is purely based on WI,L in Eq.15, and also +functions of γ (See Eqs.35 and 36). +Fig.8 shows how numerically the factor y depends on the +line of sight angle γ for Alfven mode (black) and the com- +pressible mode (green). We notice that the qualitative phe- +nomenon happened in Fig.6 is exactly described by y for com- +pressible modes: y < 1 for γ → 90o, while y > 1 for γ → 0o. +We recognize that there are fluctuations in terms of the vari- +ation of y relative to γ for the compressible case. Surprisingly, +the Alfven mode y also exhibits some interesting properties +that we can exploit in obtaining γ in observation. Notice that +y for Alfven mode stays < 1 from what we observe in Fig.6, +we see that the Alfven mode’s y has very similar trend when +γ ⪆ 55o , but when γ ⪅ 55o, the Alfven mode y-value went +exactly opposite to that of compressible mode. Moreover, we +observe that the change of values of y as a function of γ is +more or less monotonic if we consider γ ⪅ 55o and γ ⪆ 55o. +Notice that the modes that we are talking about here are all +in the global frame of reference. +To see whether the trend that we observed in Fig.8 is ro- +bust, we select some of the numerical cubes from Tab.1 and +to plot y as a function of γ for both A and F type contribu- +tion and plot it as Fig.9. The selected numerical cubes cover +a wide range of sonic and Alfvenic Mach numbers. We can +see from Fig.9 that the trends of the two curves are very sim- +ilar to that of Fig.8. Furthermore, the exact values of y are +also very similar across different turbulent conditions. Origi- +nally, the formalism of A and F type tensor applies only for +Ms,A < 1. However, we perform the calculation of y also for +supersonic sub-Alfvenic simulations, which is closer to the +environment of molecular clouds (See, e.g. Draine 2011) and +still observe the same trend. We therefore conclude that the ¯y +parameter tracers γ. In fact, we observe from Fig.9 that when +the plasma β ∝ M 2 +A/M 2 +s is smaller, it is easier to recover the +trend that we see in Fig.8. +At last, we provide the empirical formula (units in degrees) +for the case of low β (β < 1). For γ < 40o +y(F) ∼ 1.2 − γ/40 × 0.4 +y(A) ∼ 0.4 + γ/40 × 0.2 +(40) +for γ > 40 degrees +y(F) ∼ 0.8 − (γ − 40)/50 × 0.2 +y(A) ∼ 0.8 +(41) +The full study on how the y-parameter can be applied to +situation with different mixture of driving will be discussed +in Malik et al. (in prep). +6 DISCUSSION +6.1 The importance of Alfven leakage for mode +decomposition +The analysis of turbulence properties generally from observa- +tions requires the consideration of the local-to-global frame +problem, which is modelled as the ”magnetic field wander- +ing problem”. While the theory of MHD turbulence is well- +established, how the local scaling laws are projected glob- +ally is still mysterious, despite models have been proposed +from both Lazarian & Pogosyan (2012) and Lazarian et al. +(2020). Here, we propose the first physical model in explain- +ing how the wandering of magnetic field happens when pro- +jected along the line of sight, and how we could utilize the +magnetic field wandering in deducing a number of important +physical quantities such as the line of sight angle and also the +mode fractions. +The problem of the local-to-global frame transition in theo- +retical MHD turbulence studies have puzzled the community +for a while. While the anisotropic scaling k∥ ∼ k2/3 +⊥ +is well +motivated from the simple constant energy cascade and crit- +ical balance condition (GS95), we cannot retrieve the local +scaling from the global frame of reference. In fact, the global +correlation function usually gives a constant scaling rather +than a geometrically-driven, size-dependent scaling as pre- +dicted by GS95. Before the availability of MHD simulations +(e.g. Cho & Lazarian 2003; Beresnyak et al. 2005), it is not +yet possible to validate the GS95 relation even from numeri- +cal simulations. +The more puzzling effect comes when MA is very large. +Traditionally the numerical test on GS95 are done in small +MA systems and in small scales. However as we see from +the previous sections, in moderate and small k the Alfven +mode acquired from the Cho & Lazarian (2003) decomposi- +tion method contains non-negligible contributions along the +ˆζC vector, indicating the presence of anomalous compressive +wavevector even after Alfven mode decomposition. The only +plausible reason why this happens is because the mode de- +composition method from Cho & Lazarian (2003) is done on +a global frame of reference. As a result when we are looking +at small scales, in average the mean field is not very different +from its local field. Yet for larger scales the mean field is very +MNRAS 000, 1–18 (2022) + +1.4 +A +1.2 +(o+)Cdouosu +(-)doos +1.0 +Compressible (F) regime +0.8 +Alfven (A) regime +y +0.6 +0.4 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +LOS Angle y12 +Yuen et.al +Figure 9. A set of figures showing the universality of our finding (y as function of the line of sight angle γ in degrees) in Fig.8 in 6 +numerical simulations from Tab.1, which covers a large range of value of Ms,A. +different from the local field, so that there exists anomalous +compressible terms even the data are supposed to be ”Alfven +modes” according to Cho & Lazarian (2003). We named this +effect ”Alfven leakage” in our previous section since this ef- +fect happens even for Alfven waves as long as the Alfvenic +Mach number is not zero. +In this paper, we further show that the Alfven leakage effect +is a global function of MA. In fact, the presence of the leakage +effect suggests that the mode decomposition method by Cho +& Lazarian (2003) should subject to the a correction term for +moderate and small k. However since most of the calculation +from Cho & Lazarian (2003) are done in small scales, i.e. +large k, the results of their work are not affected. +6.2 The importance of tensor forms to the SPA +technique and general turbulence studies +The novel invention of the SPA technique (Zhang et al. +(2020)) utilizes the fact that the tensor projections have dif- +ferent contributions for Alfven and compressible modes to +identify them in observations. This work further strengthens +their argument through the use of Alfven leakage picture and +suggests a few important improvements to their method. For +instance, it is not necessary to compute the parameter sxx +as in Eq.21 to distinguish the modes. The tensor properties +are encoded in the Stokes parameters and thus ignoring the +tensor contribution would make dramatically different pre- +dictions in astrophysical applications. +One very important factor that is accounted by Zhang et al. +(2020) is the use of one point statistics under Stokes frame +transformation. The traditional turbulence statistical stud- +ies usually utilize multi-point statistics since they are either +directly related to the spectra (e.g. two-point) or is used to +validate scaling relations for higher order structure functions +(e.g. Kolmogorov 4/5 law). The reason of why single point +statistics was not useful before is because the spectrum and +anisotropy are the main characteristics of turbulence studies +for the past 60 years. However, how the tensor projection af- +fects the geometry of the structures for each of the turbulence +variable is not really explored. Tensor forms of turbulence +modes were not much explored beyond the physics of cosmic +rays(Schlickeiser 2002; Yan & Lazarian 2002, 2004). In fact, +the previous anisotropy analysis also did not consider what +is the statistics of a single component of a 3D turbulence, +i.e.tensor projection, after projection along the line of sight. +While the series of papers by Lazarian & Pogosyan started to +consider how the single component statistics works, not until +recently did both numerically (e.g. Lazarian et al. 2018) and +observationally (Zhang et al. 2020) found the effect of ten- +sors to be that important during single component projection. +The anisotropy of projected fast modes with the direction op- +posite to the Alfvenic anisotropy was shown in Lazarian & +Pogosyan (2012). In fact, one of the most common belief that +is circulating in the earlier studies of MHD turbulence the- +ory (e.g. the discussion section of Lazarian et al. (2017)) is +the presumption that the projection of the observables (e.g. +velocities, magnetic field) from fast modes will be isotropic +since the fast modes in 3D are. This is empirically proven +wrong by Lazarian et al. (2018) through the development of +velocity gradient and also utilized through the development +of SPA in Zhang et al. (2020). +In fact, in a number of astrophysical applications, the ob- +MNRAS 000, 1–18 (2022) + +Ms = 6.17, MA = 0.22 +Ms = 5.4, MA = 0.24 +Ms = 0.92, MA = 0.09 +Huge-o : ^ +e6r3: +Ms0.92Ma0.09 +1.2 +1.2 +1.0 +1.0 +0.8 +11 +0.6 +0.6 +304050 +70 +80 +90 +10 +20 +50 +60 +LOS Angle y +70 +80 +06 +LOS Angle y +LOS Angle Y +Ms = 0.93, Ma += 0.94 +Ms = 0.61,MA = 0.52 +Ms = 0.98, Ma = 0.32 +Ms0.93Ma0.94 +e5r3 +Ms0.98Ma0.32 +1.2 +(o-)doosr +1.2 - +1.0 +1.0 +0.8 +0.8 +10 +20 +40 +50 +60 +70 +80 +90 +10 +20 +40 +LOS Angle y +60 +70 +80 +10 +20 +30 +40 +So +60 +70 +80 +LOS Angle y +LOS Angle yAlfven leakage +13 +servables are constructed through not all three directions of +velocities or magnetic field, but just some of them. For in- +stance, the Davis-Chandrasekhar-Fermi (DCF) technique uti- +lizes both the line of sight velocity dispersion and the plane +of sky polarization angle dispersion to estimate the magnetic +field strength through the use of Alfven relation (See also Cho +& Yoo 2016). However, as found in Lazarian et al. (2022b), +the direction of the velocity and magnetic field fluctuations as +collected in DCF technique are exactly perpendicular to each +other. Moreover, in this work we also show both analytically +and numerically that the tensor term contains anisotropy and +can be dominant as long as the γ fulfills some conditions. As +a result, one should not ignore the contributions of the tensor +term in studying the properties of MHD turbulence. +6.3 The use of high pass filter? +In MHD turbulence studies, there are a few length scales +that determine whether the underlying turbulence is hydro- +dynamic or GS95-like. It is a general phenomenon that for 3D +saturated turbulence the small scale fluctuations are GS95 +like. However for both sub-Alfvenic and super-Alfvenic there +exist a transition scale that the turbulence becomes non-GS95 +like. For instance, for the sub-Alfvenic case there exist the +transition from weak to strong turbulence (Cho & Lazarian +2003; Makwana & Yan 2020) at the length scale LM 2 +A, while +for the super-Alfvenic case above the scale LM −3 +A +the turbu- +lence is hydrodynamic. This might suggest that the removal +of large scale fluctuations could allow observers to obtain the +desired GS95 statistics with the use of high pass filters. +However upon projection the high pass filter in 2D acts +a little bit differently compared to 3D. Fundamentally the +high pass filter (HPF) in 3D serves as the high frequency +extractor. As noticed in Lazarian et al. (2020) , HPF in 2D +acts as a lower bound of the HPF in 3D, i.e. if we explicitly +want K = +� +k2x + k2y > K0, this will automatically apply to +k = +� +k2x + k2y + k2z ≥ K > K0. However since the sampling +of turbulence statistics upon projection is not statistically +complete, meaning that the wavevectors with k > K0 but +K < K0 is not sampled, it is hard to determine whether we +will obtain back the same turbulence spectrum anisotropy +just by inspection here since we did have additional knowl- +edge on how the LOS direction is related to the inclination +angle. More importantly, if we are considering the case when +MA is not small, the randomness of the magnetic field fluc- +tuation will make the filtering in 2D in Stokes parameters +being completely different from that of the 3D. Fig.10 shows +an example on how different the Stokes Q look like. On the +left of Fig.10 we perform filtering after projection (i.e. 2D), +while on the right it is the projection after 3D filtering. We +can see that, while the statistical anisotropies for the two +maps are roughly the same, the differences of the features +are prominent. +7 CONCLUSION +In this paper, we introduce a vector-based framework in ex- +plaining the strength and the limitation of the recently intro- +duced techniques, namely SPA, CFA and VGT. In particular, +due to the use of the vector framework, we recognize that in +the presence of curved magnetic field Alfven waves will be +Figure 10. An illustration of the features of the Stokes Q after 2D +(left) and 3D (right) filtering. One can see that there is a significant +difference in terms of the structures of the features. +seen as the linear combination of Alfven and compressible +waves, which is named ”Alfven leakage”. In short, +(i) We recognize a straightforward transformation from the lo- +cal to global reference frame through the Alfven leakage +model. (Fig.3). Moreover, the projection parameters WI,L +that are introduced in LP12 are derived in an alternative +way in the picture of Alfven leakage. (Eq.(8), Fig.3) +(ii) The SPA technique, which allows the identifications of the +dominance of the Alfven and compressible waves in observed +synchrotron emissions, is the result of the one-point statis- +tics. The Alfven wave contribution is frame independent while +that for compressible waves are frame dependent. As a re- +sult, the quantitative contribution of Alfven and compressible +waves can be separated observationally (See Eq.(23)). +(iii) We suggest that the SPA technique is also applicable to +slightly Faraday rotated regime. (§4.3). +(iv) Based on the formulation of the Alfven leakage, we discover +a new γ tracing method that utilize the anisotropy fraction +of I + Q and I − Q in observations. We test the method in +numerical simulations and see universality of trends across a +wide range of turbulence parameters. (§5, Fig.8). +The expression of the vector frame formulation allows us to +visually understand and analyze the statistics of MHD tur- +bulence. Together with the theoretical establishment of the +Lazarian & Pogosyan series, how the turbulence statistics are +imprinted into observables will be better understood by ob- +servers. +APPENDIX A: THE MATHEMATICAL +DESCRIPTION ON VECTOR AND TENSOR +FORMULATIONS IN MHD STATISTICAL +TURBULENCE THEORY +For our analysis in this paper, we need to review some of +the required mathematical tools for the descriptions of the +MHD turbulence. The reason why we need them is because +some of the frame representations are advantageous in some +situations. Here we will first review the concept of the global +and local frame of reference, the leakage of modes due to the +Yuen & Lazarian (2020) of local magnetic field, and also the +mathematical establishments that are scattered in different +MNRAS 000, 1–18 (2022) + +Filtered after projection (2D filtering) +Filtered before projection (3D filtering) +Stokes Q14 +Yuen et.al +literature. The unified approach that we use in this paper +will lead to establishment of an analysis framework in under- +standing how the modes should behave in observations. +A1 Global and local frame of reference +The first important concept is the use of the local frame +of reference when computing the structure function of the +turbulence variable. The mathematical expression of the 3D +structure function of the turbulence variable v in the local +frame of reference is given by: +SF{v}(r) = +� � +(v(r′ + r) − v(r′)) · B(r′ + r) + B(r′) +|B(r′ + r) + B(r′)| +�2 � +(A1) +where in small r, the separation of the three eigenmodes +(Alfven, Fast, Slow) will give the correct spectrum and +anisotropy as predicted in GS95 and LV99. In particular, +the anisotropy will be scale dependent when observed locally +through the 3D structure functions. Table A1 summarizes +the spectral slopes and anisotropies that we expect from the +local structure functions. +However, we cannot deduce the expressions from Tab.A1 +due to the restriction of the local-to-global reference frame +transformation, which is the main topic of the current paper. +A more common method in computing the structure func- +tion is by simply computing the simplistic structure function +below, assuming Vz(R) = +� +dzˆz · v(r): +SF{Vz}(R) = ⟨(Vz(R′ + R) − Vz(R′))2⟩ +(A2) +which the spectrum and anisotorpy that is observed from this +variable could be different from what the local expressions. +In particular ,the anisotropy in the global frame of reference +becomes scale independent, meaning that there is no partic- +ular advantage in probing the anisotropy in smaller scale in +actual observations, aside from the standard LM −3 +A +scale. +. +A2 Tensor representation +In the global frame of reference, the spectral tensor for dif- +ferent modes can be represented by the sum of the three +linearly independent spectral tensors TP,C,A, which is given +by (Lazarian & Pogosyan (2012), cf Yan & Lazarian 2004): +TP,ij = ˆkiˆkj +TC,ij = (ˆk × (ˆk × ˆλ))i(ˆk × (ˆk × λ))j +|ˆk × ˆλ|2 += (λi − (ˆk · ˆλ)ki)(λj − (ˆk · ˆλ)kj) +|ˆk × ˆλ|2 +TA,ij = Iij − TP,ij − TC,ij += (ˆk × ˆλ)i(ˆk × λ)j +|ˆk × ˆλ|2 +(A3) +Notice that for Alfven mode vA,iTP,ij = 0 since ∇ · vA,i = 0. +Notice that TC + TA is isotropic. +A3 The ASF (CL03) frame with respect to the +PCA frame +For the actual numerical analysis, the realization of the in- +dividual MHD modes in the local frame of reference is not +achievable since obtaining the modes requires the perturba- +tion theory to start with. In this case, the expressions of the +modes are given in Fourier space by evaluating the perturba- +tion along a locally averaged mean field. In that case, for each +k ∈ R3, we can locally define the eigenvectors for the three +modes ˆζA,S,F given by Eq.1. Notice that the A(lfven)-S(low)- +F(ast) frame is a simple rotation of the ”magnetic frame” +along ˆζA given by the three eigenvectors (ˆλ, ˆk׈λ, ˆλ×(ˆk׈λ)) +by an angle φ: +tan φ = 2α cos2 θλ − (α + 1 + +√ +D) +2α cos2 θλ +tan θλ +(A4) +. +The ”magnetic field” is simply given by an additional ro- +tation of tan θλ from the P(otential)-C(ompressible)-A(lfven) +frame (ˆζP = ˆk, ˆζA = ˆk׈λ, ˆζC = ˆk×(ˆk׈λ)). The PCA frame +has its special advantage since the sampling of k is usually +complete in dΩk. That means we have the freedom to fix k +despite other unit vectors are changing. +From the tensor product we can always write the arbitrary +vector in the Fourier space as : +ζi(k) = CP ˆki + CC (ˆk × (ˆk × ˆλ))i +|ˆk × ˆλ| ++ CA (ˆk × ˆλ)i +|ˆk × ˆλ| +(A5) +which we will name the unit vector ζP,C,A now +From Cho & Lazarian (2003), in the global frame of refer- +ence the Alfven, slow and fast mode eigenvectors are: +ζA ∝ ˆk × ˆλ +ζS ∝ (−1 + α − +√ +D)(k · ˆλ)ˆλ + (1 + α − +√ +D)(ˆλ × (k × ˆλ)) +ζF ∝ (−1 + α + +√ +D)(k · ˆλ)ˆλ + (1 + α + +√ +D)(ˆλ × (k × ˆλ)) +(A6) +where α = βγ/2, D = (1 + α)2 − 4α cos2 θ, cos θ = ˆk · ˆλ. We +recognize that there is a frame rotation between the vector +ζP,C and ζS,F : +�ζS +ζF +� += +−1 +2 cos 2θ +√ +D +L(α, θ)R0(θ) +�ζP +ζC +� +(A7) +where R0(θ) is the standard two-dimensional rotation ma- +trix, the factor beforehand is just for normalization and: +L(α, θ) = +�(−1 + α − +√ +D) cos θ +(1 + α − +√ +D) sin θ +(−1 + α + +√ +D) cos θ +(1 + α + +√ +D) sin θ +� +(A8) +Then we can rewrite the tensors by +TS/F = ζS/F ⊗ ζS/F +(A9) +Notice that Tijζj = ζi if Tij = ζi ⊗ ζj. +MNRAS 000, 1–18 (2022) + +Alfven leakage +15 +Mode +Power spectra E(k), where E = +� +dkE(k) +Anisotropy factor +Frame vector +Alfven +k−5/3 +exp(−M−4/3 +A +k∥/k2/3 +⊥ ) +ζA +Slow (low β) +k−5/3 +varies (see Makwana & Yan 2020) +ζS +Slow (high β) +k−5/3 +exp(−M−4/3 +A +k∥/k2/3 +⊥ ) +ζS +Fast (low β) +k−3/2 +1 +ζF +Fast (high β) +k−3/2 +k2 +⊥ +ζF +Table A1. A summary of the theoretical expectations of the turbulence scaling laws. Summarized from CL03, Yan & Lazarian (2008) +and Makwana & Yan (2020). +Figure A1. The definition of the frames that we use in the current paper. From the left: The Potential-Compressible-Alfven Frame +(PCA, left) which is very convenient in analyzing the magnetic field perturbations. The Alfven-Slow-Fast Frame (ASF, right) was the +local reference frame for the three fundamental MHD eigenmodes derived in CL03 The magnetic frame (middle) is the frame that shares +similarity to the PCA frame defined through the local magnetic field direction ˆλ. To connect them, the PCA frame is a simple rotation +of θλ = cos−1(ˆk · ˆλ) from the magnetic frame, and the ASF frame is just a φ rotation from the magnetic field. +A4 Frenet-Serret frame +From Yuen & Lazarian (2020), the Frenet-Serret frame of the +the magnetic fields lines would be: +dˆt +ds = ++κˆn +dˆn +ds = −κˆt ++ τˆb +dˆb +ds = +−τ ˆn +(A10) +Here ˆt = ˆλ, representing the tangent vector of the magnetic +field line. (ˆt, ˆn,ˆb) forms a complete orthogonal set indepen- +dent of the choice of k. Notice that for mode decomposition, +the ”mean” field is selected before selecting (Fourier trans- +forming into) k, thus we can treat λ as k-independent and +uses its own position vector rλ). Notice that the unit vector +ˆn can be expressed as the linear combination of ˆζA and ˆλ× ˆζA +in the magnetic frame +A5 The relation between the tensor representation +(Lazarian & Pogosyan 2012) and vector +representation (this work) +In the local frame of reference, the Alfven mode magnetic +field is given by simply: +HA(r) = +� +d3kC(k)ˆζA(k) +(A11) +where C contains the isotropic and anisotropic factors from +its spectrum. However as we move from the local frame to +the global frame, the actual Alfven wave magnetic field real- +ization will contain both compressible and Alfven wave con- +tribution (here we simply pick an arbitrary k): +˜HA(k) = CWAˆζA + CWC ˆζC +(A12) +where WA,C are two factors yet to be found. LP12 branded +these two factors in the form of the direct tensor product +ζE = ζC + ζA and ζF = ζC, and TE,F = ˆζE,F ⊗ ˆζE,F . In +their case when Alfven mode is observed in the local frame +MNRAS 000, 1–18 (2022) + +P(otential)-C(ompressible)-A(Ifven) Frame +Magnetic Frame +A(Ifven)-s(low)-F(ast) Frame +F +=d +Sp = k +=d +0a +Φ +2D Rotation with Angle +2D Rotation with Angle Φ +Qa = cos-1k. +given by CL03 +kx +ZA +kx +k× +TA +[K +[K +× (×) +× (×) +[× (×) +[× (×) +k× (k×) += 2 +[k × (k × )I +2α cos2 - (α +1+vD) +tan Φ = +tan 0 +2α cos2 +D = (1 + α)² - 4α cos² ^16 +Yuen et.al +of reference, the Alfven mode correlation function in k-space +is given by: +˜Hi ˜Hj = C2(TE,ij − TF,ij) +(A13) +while in the global frame of reference +˜Hi ˜Hj = C2TE,ij − C2(WITE,ij + WLTF,ij) +(A14) +Some algebra will give +˜Hi ˜Hj = C2(1 − WI − WL)TC,ij + C2(1 − WI)TA,ij +(A15) +A6 Conversion between the frame of references of +velocity field and magnetic field +As derived by Cho & Lazarian (2002, 2003) the decomposed +Alfven-Slow-Fast frame was the frame for the displacement +vector ζ, which also applies to the velocity fluctuations. How- +ever the magnetic field fluctuations do not necessary follow +the ASF frame as defined in CL03. For Alfven wave, the fluc- +tuations of the magnetic field is in the same direction as that +of velocities, i.e. ˆk × ˆλ. For compressible modes, the prop- +agation of the magnetic field fluctuations ˜b(k) at a specific +wavevector k is given by the following relation: +˜b = ˆk × (˜v × ˆλ) +(A16) +where ˜v is the velocity fluctuation at k. Notice that the above +vector is parallel to the compressible vector ˆζC = ˆk×(ˆk׈λ). +APPENDIX B: THE FUNDAMENTALS OF +DESCRIBING THE ANISOTROPY IN +STRUCTURE FUNCTIONS +In this section we will discuss the essence of multipole ex- +pansions in analysing the statistics of turbulence under the +assumption of two-point closure5 based on the formalism of +Kandel et al. (2016). +It is visually compelling that the two-point structure func- +tions are concentric ellipses. Mathematically the structure +functions of anisotropic fundamental modes (e.g. Alfven, slow +modes) contains a dependence in the form of exp(−C| cos φ|) +for some constant C that carries a weak dependence on φ +(See, e.g. Lazarian & Pogosyan 2012; Kandel et al. 2016). +This exponent term is naturally elliptical like. The expression +of this term in the two-point statistics of any observables is +the main direction of theoretical study recently in literature +(Lazarian & Pogosyan 2012, 2016; Kandel et al. 2016, 2017; +Lazarian et al. 2022b,a). +There +are +a +few +choices +in +describing +ellip- +tical +features +on +the +sky +via +complete +basis: +Multipole expansions of even order: The spatial symme- +try of the function exp(−C| cos φ|) allows one to express the +5 The concept of two-point closure is simply to say that turbulence +variables can be ”adequately” described by the two-point structure +functions. This approximation is evidently incorrect in general tur- +bulence case as intermittency is a well-studied topic in the field. +However for equilibrium MHD turbulence that we are consider- +ing here, the two-point description contains ∼ 95% of the spectral +power. The prominent features that we are measuring (e.g. mode +fraction, γ etc) are therefore dominated by the two-point statistics. +See Yuen (thesis, 2022) +structure function of any observables X into the summation +the cosines with even orders: DX(φ) = � +m∈2Z+ Dm cos(mφ) +6. Visually we are expressing the structure function into +linear combination of cosines in polar coordinate. Notice that +for all m|4 = 2 the Dm term carries some anisotropy, however +for m ≥ 6 the multipole anisotropy has a upper limit. For +instance, the cos 6φ term has a maximum anisotropy of +1.15. Notice that the non-vanishing Dm≥6 will decrease the +anisotropy of the structure function. A typical treatment +of the multipole expansion is to truncate the series into +m = 0, 2, where the visual minor-to-major axis ratio for +the elliptical feature appeared in the structure function +χ = +� +1 − ϵ2 +ell (ϵell is the eccentricity of ellipse) is given by: +χ = D0 − D2 +D0 + D2 +(B1) +Notice that the multipole expansion fails when MA ≪ 1 or +MA > 1, as the Dm term is comparable ∀m. The empirical +limit where D4/D2 is comparable (∼ 0.5,Lazarian et al. +2022a) is roughly at MA ≈ 0.5. Therefore the multipole +expansion is suitable only for MA ∼ 0.5 − 1 (See Fig.B1) +Legendre Polynomial: The Legendre polynomial Pl(cos φ +is another popular choice in describing the statistics in 2D. +Similar to the multipole expansion, we express the structure +function DX(φ) += +� +l∈2Z+ alPl. al carries very similar +mathematical properties as Dm in multipole expansions +and therefore we would not discuss further. (See Fig.B1) +Elliptical basis: As the structure function look like ellipses, +it is natural to consider the function below to capture the +anisotropy of the structure function: +f(φ, ϵell) = +� +1 − ϵ2 +ell +(1 − +ϵ2 +ell +2 ++ +ϵ2 +ell +2 cos 2φ) +(B2) +The advantage of this basis is that (1) the eccentricity ϵell +is a direct measure of the minor-to-major axis ratio, which +allows one to quickly construct this function by simply mea- +suring the minor and major axis (2) due to the non-vanishing +higher-order multipole of Eq.B2, this functional form is still +applicable when MA ≪ 1. Notice that one can convert the +eccentricity ϵell to the D2/D0 via the formula: +���D2 +D0 +��� ≈ 1 +2 +2ϵ2 +ell +2 − ϵ2 +ell +(B3) +in which the approximation is valid when MA ∈ [0.5, 1] for +the case of linear (i.e. centroid, C ∝ +� +dzvz) or quadratically +projected observables (i.e. Stokes parameters). . The approx- +imation is valid for caustics (c.f. Yuen et al. 2021) for even +smaller values of MA. +Acknowledgments. K.H.Y. & A.L. acknowledge the sup- +port the NSF AST 1816234, NASA TCAN 144AAG1967 +and NASA ATP AAH7546. KHY thanks Dmitri Pogosyan +(U.Alberta) and Ka Wai Ho (UW-Madison) for their inspira- +tional comments. We thank Sunil Malik (DESY) and Parth +6 In previous literature (e.g. Kandel et al. 2016) they express +DX ∼ � +m∈2Z ¯Dmeimφ, where m can be both positive and nega- +tive. Typically structure functions are always real-valued. There- +fore for the sake of simplicity we adopt the cosine formalism. +MNRAS 000, 1–18 (2022) + +Alfven leakage +17 +Figure B1. Two figures showing how the change of MA affects the relative amplitude of Dl and al. We can see from the L.H.S. figure +that, when MA is large, the approximation til quadruple is very good since |D4| < min(D2, D0). However as MA becomes smaller, |D4| +is actually comparable to that of |D0,2|, and therefore the multipole approximation breaks down . Very similar result also happens for +Legendre expansion (right). +. +Pavaskar (DESY) for extensive discussions and cross-checks +on the validity of the y-parameter analysis. The main simula- +tions and the first version of the work is done during KHY’s +tenure in UW Madison. Research presented in this article was +supported by the Laboratory Directed Research and Devel- +opment program of Los Alamos National Laboratory under +project number(s) 20220700PRD1. +Code Availability The code can be found in https:// +github.com/kyuen2/MHD_modes +Data Availability The data underlying this article will be +shared on reasonable request to the corresponding author. +REFERENCES +Armstrong J. W., Rickett B. J., Spangler S. R., 1995, ApJ, 443, +209 +Begum A., Chengalur J. N., Bhardwaj S., 2006, MNRAS, 372, L33 +Beresnyak A., Lazarian A., 2019, Turbulence in Magnetohydrody- +namics +Beresnyak A., Lazarian A., Cho J., 2005, ApJ, 624, L93 +Biskamp D., 2003, Magnetohydrodynamic Turbulence +Brunetti G., Lazarian A., 2007, MNRAS, 378, 245 +Chandran B. D. G., 2000, Phys. Rev. Lett., 85, 4656 +Chepurnov A., Lazarian A., Stanimirovi´c S., Heiles C., Peek +J. E. G., 2010, ApJ, 714, 1398 +Cho J., Lazarian A., 2002, Phys. Rev. Lett., 88, 245001 +Cho J., Lazarian A., 2003, MNRAS, 345, 325 +Cho J., Lazarian A., 2005, Theoretical and Computational Fluid +Dynamics, 19, 127 +Cho J., Vishniac E. T., 2000, ApJ, 539, 273 +Cho J., Yoo H., 2016, ApJ, 821, 21 +Deshpande A. A., Dwarakanath K. S., Goss W. M., 2000, ApJ, +543, 227 +Dickey J. M., McClure-Griffiths N. M., Stanimirovi´c S., Gaensler +B. M., Green A. J., 2001, ApJ, 561, 264 +Draine B. T., 2009, in Henning T., Gr¨un E., Steinacker J., eds, +Astronomical Society of the Pacific Conference Series Vol. 414, +Cosmic Dust - Near and Far. p. 453 (arXiv:0903.1658) +Draine B. T., 2011, Physics of the Interstellar and Intergalactic +Medium +Elmegreen B. G., Scalo J., 2004, ARA&A, 42, 211 +Farmer A. J., Goldreich P., 2004, ApJ, 604, 671 +Fissel L. M., et al., 2016, ApJ, 824, 134 +Fu X., Li H., Gan Z., Du S., Steinberg J., 2022, ApJ, 936, 127 +Gan Z., Li H., Fu X., Du S., 2022, ApJ, 926, 222 +Goldreich P., Sridhar S., 1995, ApJ, 438, 763 +Green D. A., 1993, MNRAS, 262, 327 +Kandel D., Lazarian A., Pogosyan D., 2016, MNRAS, 461, 1227 +Kandel D., Lazarian A., Pogosyan D., 2017, MNRAS, 464, 3617 +Khalil A., Joncas G., Nekka F., Kestener P., Arneodo A., 2006, +ApJS, 165, 512 +Kowal G., Lazarian A., 2010, ApJ, 720, 742 +Kowal G., Lazarian A., Beresnyak A., 2007, ApJ, 658, 423 +Lazarian A., 2006, ApJ, 645, L25 +Lazarian A., 2016, ApJ, 833, 131 +Lazarian A., Pogosyan D., 2000, ApJ, 537, 720 +Lazarian A., Pogosyan D., 2004, ApJ, 616, 943 +Lazarian A., Pogosyan D., 2006, ApJ, 652, 1348 +Lazarian A., Pogosyan D., 2008, ApJ, 686, 350 +Lazarian A., Pogosyan D., 2012, ApJ, 747, 5 +Lazarian A., Pogosyan D., 2016, ApJ, 818, 178 +Lazarian A., Vishniac E. T., 1999, ApJ, 517, 700 +Lazarian A., Yuen K. H., 2018, ApJ, 865, 59 +Lazarian A., Yuen K. H., Lee H., Cho J., 2017, ApJ, 842, 30 +Lazarian A., Yuen K. H., Ho K. W., Chen J., Lazarian V., Lu Z., +Yang B., Hu Y., 2018, ApJ, 865, 46 +Lazarian A., Yuen K. H., Pogosyan D., 2020, arXiv e-prints, p. +arXiv:2002.07996 +Lazarian A., Yuen K. H., Pogosyan D., 2022a, arXiv e-prints +Lazarian A., Yuen K. H., Pogosyan D., 2022b, ApJ, accepted for +publication, p. arXiv: 2204.09731 +Lithwick Y., Goldreich P., 2001, in American Astronomical Society +Meeting Abstracts #198. p. 90.03 +Maiti S., Makwana K., Zhang H., Yan H., 2021, arXiv e-prints, p. +arXiv:2108.01936 +Makwana K. D., Yan H., 2020, Physical Review X, 10, 031021 +Maron J., Goldreich P., 2001, ApJ, 554, 1175 +McKee C. F., Ostriker E. C., 2007, ARA&A, 45, 565 +Narayan R., Medvedev M. V., 2001, ApJ, 562, L129 +MNRAS 000, 1–18 (2022) + +1.00 +2.0 +Ma=1.0,D2/Do=0.22 +Ma=0.5,Dz/Do=0.52 +Ma=1.0,az/ao=0.59 +Ma=0.5,az/ao=1.31 +M=0.9,Dz/Dg=0.25 +Ma=0.4,D2/Do=0.65 +Ma=0.9,az/ao=0.68 +Ma=0.4,az/ao=1.61 +0.75 +1.5 +Ma=0.8,Dz/Do=0.29 +Ma=0.3,Dz/Do=0.81 +Ma=0.8,az/ao=0.78 +Ma=0.3,az/ao=1.97 +Ma=0.7,Dz/Do=0.35 +Ma=0.2,D2z/Do=0.94 +Ma=0.7,az/ao=0.91 +Ma=0.2,az/ao=2.3 +0.50 +Ma=0.6,Dz/Do=0.42 +Ma=0.1,Dz/Do=0.99 +1.0 +Ma=0.6,az/ao=1.09 +Ma=0.1,az/ao=2.47 +0.25 +0.5 +D +0.00 +a +0.0 +-0.25 +0.5 +-0.50 +-1.0 - +-0.75 +-1.5 +-1.00 +2.0 +0 +2 +9 +8 +10 +12 +14 +16 +18 +20 +0 +8 +10 +12 +14 +16 +18 +20 +multipole +legendre l18 +Yuen et.al +Padoan P., Juvela M., Kritsuk A., Norman M. L., 2006, ApJ, 653, +L125 +Schekochihin +A. +A., +2022, +Journal +of +Plasma +Physics, +88, +155880501 +Schlickeiser R., 2002, Cosmic Ray Astrophysics +Yan H., Lazarian A., 2002, Phys. Rev. Lett., 89, 281102 +Yan H., Lazarian A., 2004, ApJ, 614, 757 +Yan H., Lazarian A., 2008, ApJ, 673, 942 +Yuen +K. +H., +Lazarian +A., +2017a, +arXiv +e-prints, +p. +arXiv:1703.03026 +Yuen K. H., Lazarian A., 2017b, ApJ, 837, L24 +Yuen K. H., Lazarian A., 2020, ApJ, 898, 66 +Yuen K. H., et al., 2018, ApJ, 865, 54 +Yuen K. H., Ho K. W., Lazarian A., 2021, ApJ, 910, 161 +Yuen K. H., Ho K. W., Law C. Y., Chen A., Lazarian A., 2022, +submitted, arXiv:2204.13760 +Zhang H., Chepurnov A., Yan H., Makwana K., Santos-Lima R., +Appleby S., 2020, Nature Astronomy, 4, 1001 +Zhao S. Q., Yan H., Liu T. Z., Liu M., Shi M., 2021, ApJ, 923, 253 +Zhao S. Q., Yan H., Liu T. Z., Liu M., Wang H., 2022, ApJ, in +press, p. arXiv:2204.05410 +This paper has been typeset from a TEX/LATEX file prepared by +the author. +MNRAS 000, 1–18 (2022) + diff --git a/bNFQT4oBgHgl3EQfhDYG/content/tmp_files/load_file.txt b/bNFQT4oBgHgl3EQfhDYG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..37971c71b5188133eb008f35995d38245a829fd7 --- /dev/null +++ b/bNFQT4oBgHgl3EQfhDYG/content/tmp_files/load_file.txt @@ -0,0 +1,1215 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf,len=1214 +page_content='MNRAS 000, 1–18 (2022) Preprint 1 February 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='0 Anomalous compressible mode generation by global frame projections of pure Alfven mode Ka Ho Yuen,1,2⋆ Huirong Yan,3,4† Alex Lazarian1‡ 1Department of Astronomy, University of Wisconsin-Madison, USA, 53715 2Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA 3Deutsches Elektronen-Synchrotron DESY 4Institut f¨ur Physik & Astronomie, Universit¨at Potsdam, Germany Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' in original form ZZZ ABSTRACT Alfven wave is the single most important physical phenomenon of magneto-hydrodynamic turbulence and has far- reaching impact to almost all studies related to astrophysical magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Yet the restoration of the Alfven wave fluctuations from a given magnetic field, aka the local Alfven wave problem, is never properly addressed in literature albeit its importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Previous works model the Alfven wave fluctuation as the perturbation along a straight-line, constant magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' However, Lazarian & Pogosyan (2012) suggested that the decomposition of Alfven wave along a straight line, aka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' the global frame decomposition, has a factor of discrepancy to the true local Alfven wave fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Here we provide a geometric interpretation on how the local Alfven wave is related to the global frame through the use of vector frame formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We prove both analytically and numerically that the local frame Alfven wave is an orthogonal transformation of that of the global frame and related by the local Alfvenic Mach number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In other words, when we observe Alfven wave in the global frame of reference, some of the Alfven wave will be mistaken as compressible waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The importance of frame choices have a far-reaching impact to the analytical studies of MHD turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Combining the frame formalism and the new techniques we can have accurate measurement to some of the fundamental turbulence properties like the inclination angle of mean magnetic field relative to the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Key words: turbulence – ISM: magnetic fields – ISM: structure — galaxies: ISM 1 INTRODUCTION Turbulence is ubiquitous in astrophysical environment and the interstellar gases are permeated by turbulent magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Magneto-hydrodynamic (MHD) turbulence plays a very important role in various astrophysical phenomena (see Armstrong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (1995);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Chepurnov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Biskamp (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Elmegreen & Scalo (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' McKee & Ostriker (2007)), including star formation (see McKee & Ostriker (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Fis- sel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2016)), propagation and acceleration of cosmic rays (see Chandran (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Yan & Lazarian (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Farmer & Gol- dreich (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Lazarian (2016)), as well as regulating heat and mass transport between different ISM phases (Green (1993);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Deshpande et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Dickey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Lazarian & Pogosyan (2004, 2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Khalil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Begum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Padoan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2006) see Draine (2009, 2011) for the list of the phases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' MHD turbulence is usually highly compressible, and has been thoughtfully studied by a number of authors in the community (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Kowal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2007)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' However, the com- ⋆ kyuen@lanl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='gov (Oppenheimer Fellow), ORCID: 0000-0003- 1683-9153 † huirong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='yan@desy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='de, ORCID: 0000-0003-2560-8066 ‡ alazarian@facstaff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='wisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='edu pressibility of the turbulence adds additional difficulty in the understanding of how the three fundamental MHD modes (namely Alfven, slow and fast modes) would behave in vari- ous astrophysical phenomena, each carrying different spectra and anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' For instance, it is believed that the Alfven mode plays a central role in making the cold neutral media aligned with the magnetic field (Lazarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2018) and controls the transport of heat and particles across magnetic fields (Narayan & Medvedev 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Lazarian 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Yan & Lazarian 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Maiti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In comparison, fast modes play an important role in the scattering and acceleration of cosmic rays (Yan & Lazarian 2002, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Cho & Lazarian 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Lazarian & Pogosyan 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Brunetti & Lazarian 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The modes composition strongly depends on the way of driv- ing Makwana & Yan (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' It is therefore essential to have a handy way in decomposing the three fundamental MHD modes in numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' A notable development is the statistical mode decomposi- tion developed by Cho & Lazarian (2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2003, latter here- after CL03), which allows one to obtain the realization of the three fundamental MHD modes in numerical simulations by considering a perturbation along a locally strong magnetic field direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The realization of MHD modes allowed the community to validate the theory of MHD turbulence (Gol- © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='13344v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='GA] 31 Jan 2023 2 Yuen et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='al dreich & Sridhar (1995) hereafter GS95, see also Lazarian & Vishniac (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Cho & Vishniac (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Maron & Gol- dreich (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Lithwick & Goldreich (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Cho & Lazarian (2002, 2003)) through numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In particular, the scaling relation of compressible modes were first verified through the realization of MHD modes using the mode de- composition algorithm developed by CL03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The realization of MHD modes also excites the development of different tech- niques in studying MHD turbulence in observations, includ- ing the Velocity Gradient Technique (VGT, Yuen & Lazarian (2017b,a)) which uses the anisotropy of different modes in re- trieving the magnetic field directions in spectroscopic data, and also the Synchrotron Polarization Analysis (SPA, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2020)) which utilizes the properties of the projected statistics in predicting the dominance of Alfven or compress- ible modes in observational synchrotron data, as well as de- tailed analysis of solar wind turbulence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' However, Goldreich & Sridhar (1995) model of MHD tur- bulence is of centre importance in the modern theory of MHD turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The latter is employs the concept of ”local frame of reference” that was added to the theory later (Lazarian & Vishniac 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Cho & Vishniac 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' This means that the eddies, which are usually elliptical in shape, are aligned to the local magnetic field rather than the mean magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' As pointed out by Kowal & Lazarian (2010), the de- composition of CL03 is a global frame decomposition, as op- posed to the local frame MHD theory stressed in the works that followed the original GS95 study (Lazarian & Vishniac 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' As described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='1, when one considers a different volume, the realization of the three fundamental modes will be different due to the change of the mean magnetic field directions under the CL03 decomposition algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The dif- ficulty of obtaining the statistics of three modes in a local- ized manner has been attempted, including abandoning the realization of modes but focusing on the structure functions Beresnyak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2005), decomposing the MHD quantities into linear combination of fundamental localized patches be- fore performing the CL03 decomposition (Kowal & Lazarian 2010), or the introduction of the frame changing parame- ters in the framework of correlation functions (Lazarian & Pogosyan 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Yet, how the three fundamental modes are realized in the local system of reference is still an unsolved problem for numerical community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In this paper, we explore how the Alfven and compressible modes in the local system of reference are expressed globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In §2 we review the CL03 method and its possible improve- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In particular, in §3 we discuss about the generation of ”compressible waves signature” due to the wrong choice of lo- cal frame of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' From §4 to §5, we describe a few appli- cations that utilize the concept of Alfven leakage, namely the applications of the Synchrotron Polarization Analysis Tech- nique to regimes with strong Faraday rotation (§4) and the determination of the line of sight angle γ (§5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In §6 we discuss about the possible impacts of our method and the caveats of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In §7 we conclude our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2 MODE DECOMPOSITION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='1 Review of the MHD mode decomposition methods In this section we review the underlying assumptions of the mode decomposition method as introduced by CL03 and the development since then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In CL03 they consider a volume dΩ in which the perturbation of magnetic field is small compared to the mean field δB(dΩ) < ⟨B⟩, so does the density fluctua- tions δρ/⟨ρ⟩ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='1 shows how the volume dΩ is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Readers should be careful that once the volume is selected the mean magnetic field direction ˆλ is also defined respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In this scenario, the small perturbation in the presence of a strong mean magnetic field will provide a linearized set of MHD equations in which three non-trivial eigenvectors would be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In this localized box, the Alfven, slow and fast mode eigenvectors are1: ζA(ˆk, ˆλ) ∝ ˆk × ˆλ ζS(ˆk, ˆλ) ∝ (−1 + α − √ D)(k · ˆλ)ˆλ + (1 + α − √ D)(ˆλ × (k × ˆλ)) ζF (ˆk, ˆλ) ∝ (−1 + α + √ D)(k · ˆλ)ˆλ + (1 + α + √ D)(ˆλ × (k × ˆλ)) (1) where α = βγ/2, D = (1 + α)2 − 4α cos2 θλ, cos θλ = ˆk · ˆλ, plasma β ≡ Pgas/Pmag measures the compressibility and γ = ∂P/∂ρ is the polytropic index of the adiabatic equa- tion of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The presence of ˆk suggests that the direction of the three mode vectors are changing as k changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In this scenario, the perturbed quantities, say for the velocity fluc- tuations v1 = v − ⟨v⟩ can be written as: v1(r) = � d3keik·r � X∈A,S,F F0,X(k)F1,X(k, ˆλ)CXζX(ˆk, ˆλ) (2) for some power spectrum Ev(k) = F 2 0 = k−n/2 (Yuen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2022), some anisotropy weighting function F1 and the mode constants CX denoting the relative weight of the three modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Notice that the decomposition (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (1)) is a global decom- position method since the magnetic field fluctuations within the volume dΩ is not considered when computing the eigen- vectors of the three modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' One of the most important conse- quences of performing global decomposition is the loss of the GS95 scaling for small k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In fact, Cho & Lazarian (2002) (see also Lazarian & Pogosyan (2012)) pointed out that in the global system of reference the anisotropic scaling is scale in- dependent, meaning that the elongation of turbulence eddies is fixed and does not change as the eddies cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Another important consequence of the concept of dΩ is that, when one changes the sampling volume, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' from the volume in blue to that of yellow or orange in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='1, the weighting of the 1 However, recent publication from Gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2022) suggests that a significant portion of the projected spectral powers are not in the form of propagating waves, but fluctuations with miniature frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The nature of the non-wave fluctuations as dubbed in Gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2022) requires further clarifications from the theory of MHD turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' See Beresnyak & Lazarian (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Schekochihin (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' MNRAS 000, 1–18 (2022) Alfven leakage 3 three modes will also change due to the change of the mean direction of magnetic field from volume to volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Notice that the selection of the volume dΩ has to fulfill the conditions as assumed in CL03: Both the sonic and Alfven Mach number within the volume must be smaller than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Notice that if the volume is smaller than the volume defined by the correlation length of the turbulence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' the dispersion of the turbulence observables will be scaled as a function of distance according to their respective turbulence statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' meaning that if ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' v follows GS95,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' δρ2(r) ∝ r2/3 δv2(r) ∝ r2/3 (3) To address the issue of the locality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' the community has ex- plored a number of ways to include the local fluctuations of magnetic field during the calculation of statistics of MHD modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' For instance, one of the most notable ways of ob- taining the statistics of MHD modes is to compute the local structure functions Beresnyak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The mathemati- cal expression of the 3D structure function of the turbulence variable v in the local frame of reference is given by: SF{v}(r) = � � (v(r′ + r) − v(r′)) · ˆλ(r, r′) �2 � r′ (4) where ˆλ(r, r′) = B(r′ + r) + B(r′) |B(r′ + r) + B(r′)| (5) The anisotropy computed throughout this manner is scale dependent and exhibit the GS95 scaling r∥ ∝ r2/3 ⊥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The use of the local structure function correctly recovers the GS95 statistics, yet it is not possible to obtain the realization of the modes in this manner, meaning that further study of the features of the modes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' how does the mode look like when projected, are prohibited when using the structure functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Another important way of improving the CL03 is to rewrite the turbulence variables into the linear sum of small local- ized patches through the wavelet transform (Kowal & Lazar- ian 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Physically, they are looking for specific functional forms obeying the orthogonality requirement and represent the volumes as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='By considering the set of or- thogonal wavelets φ, one can write the velocity field v(r) as: ˜v(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' a) = a−3/2 � d3xφ(r − w a )v(r) (6) where φ is the set of wavelet functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' For (Kowal & Lazarian 2010) they select the 12-tap Daubechies wavelet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' To perform mode decomposition, they first convert the velocity field into the linear combinations of wavelets, and then proceed with the procedures of CL03 for the wavelet transformed variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The contributions for all wavelets for a specific mode are added before the inverse Fourier transform takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' No- tice that the wavelet functions is simply a mathematical con- struction that may contain spatially dispatched regions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' D4 and D12 of the Daubechies wavelet), for which taking the statistical calculations within the wavelet might not be physically justifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Nevertheless, the improved mode de- composition method proposed by Kowal & Lazarian (2010) still retrieve seemingly the correct GS95 statistics from the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='2 The locality of the mode decomposition problem due to magnetic field wandering effect In realistic MHD simulations the magnetic field lines are fluc- tuating within any selected volume, which is named wander- ing effect as in CL03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' However the mode decomposition algo- rithm available in the community had not considered any of these wandering effect, which makes the estimation of modes be rather unrealistic for larger MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' To model the additional effect when decomposing the modes in the global frame of reference, Lazarian & Pogosyan (2012) model the magnetic field correlation function as the linear combination of the isotropic tensor ˆTE and the axis-symmetric tensor ˆTF (See Yan & Lazarian 2004 also our Appendix for a detailed math- ematical construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In general the direct tensor of the magnetic field in the Fourier space at a given wavevector k in the local frame of reference can be written as: ˜Hi ˜Hj ∝ E(k) ˆTE,ij + F(k) ˆTF,ij (7) The transformation from the local to global frame as mod- elled by LP12 is: ˆTE,local → ˆTE,global ˆTF,local → WI ˆTE,global + WL ˆTF,global (8) where WI,L are two modelling constants that are functions of MA and also k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 3 ALFVEN LEAKAGE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='1 Description of the problem Since ˆTE = ˆTC + ˆTA and ˆTF = ˆTC (See the Appendix A2) one can actually write the magnetic field in the Fourier space in the global frame of reference using vector notations due to the orthogonality of the base vectors: ˜H(k) = CCζC + CAζA (9) where ζC = (ˆk × (ˆk × ˆλ))i |ˆk × ˆλ| ζA = (ˆk × ˆλ)i |ˆk × ˆλ| (10) (See Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The modelling of LP12 simply means that, assuming if in the local frame of reference the magnetic field only has the Alfven component ˜H(k) = � E(k)C0ζA, then the transformation from local to global reference frame sim- ply means a vector projection: ˜H(k) = � E(k)C0ζA global −−−−→ � E(k)(CCζC + CAζA) (11) where CC = C0 sin(θf), CA = C0 cos(θf) for some θf that we will explore in the coming subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' For Alfven waves, E(k) = −F(k), and the relation between CC,A and WI,L can be easily derived: C2 A = 1 − WI C2 C = 1 − WI − WL (12) Notice that when MA ≪ 1, there is no magnetic field wander- ing effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In this limit, CA(MA → 0) = 1, CC(MA → 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The vector notation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (9)) allows us to think of this MNRAS 000, 1–18 (2022) 4 Yuen et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='al Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The concept of mode decomposition in CL03: (1) By selecting a volume dΩ, a local mean magnetic field direction ˆλ would then be defined for later decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2) All the wavevector k that is contained in this volume dΩ are used for the decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (3) For each k there is a local reference frame (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='A1) that decomposes the magnetic field into the three eigenmodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The change of the selected volume dΩ will lead to different mean field vector ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' As a result, the local decomposition result would be functions of both λ and the wavevector k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' An illustrative figure showing how the Alfven Leakage phenomenon happens during the mode decomposition process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Here the red line represents the B-field line, point X is the origin of the volume dΩ, that is represented by the dash orange circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The vector ˆλ represents the mean field averaged over dΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' For a given point within dΩ, assuming the k-vector points inside the plane, the local Alfven wave unit vector (green) makes an angle to that of the Alfven wave unit vector defined by ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' This effect will be stronger if the magnetic field fluctuations within dΩ is larger, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' problem geometrically and relates to a very important phys- ical effect that mentioned in both Yuen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2018) and more recently Yuen & Lazarian (2020): The Alfven leak- age effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Alfven leakage describes the effect that the locally Alfven component of the turbulent variables are projected as linear combinations of Alfven and non-Alfven components in the presence of a curved magnetic field averaged over se- lected volume (local mean magnetic field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In Yuen & Lazarian (2020) we consider the effect that the gravitational forces creates extra compression to the Alfven waves and thus some of the Alfven waves from the self-gravitating systems are transferred into compressible components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In fact, in the MNRAS 000, 1–18 (2022) Local Volume dΩ B-field line Wavevector k Wavevector k Local Local Mean Field Local Mean αr Mean Field d3r B(r) d2 JrEd2 Field Wavevector kGreen: Real, local {A = k X H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' = CA,oA,local Blue: From the decomposition algorithm what A = k × looks like k > Cc,global = CA,local sin SQdn Point X Local volume d2 CA,global = CA,local cos SQdn Ssα = Dispersion of B within the local yolume B-field lineAlfven leakage 5 presence of non-trivial magnetic field structures, there exist non-zero (Yuen & Lazarian 2020) field lines in any volume due to magnetic field wandering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Similar effect has been con- sidered in Maron & Goldreich (2001) but being corrected by Cho & Vishniac (2000) as the ”rotation effect” by recogniz- ing that the global frame anisotropy axis ratio is a function of MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' However the actual relation between the local and global frame of reference has yet to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='2 gives a pictorial illustration on how the Alfven leakage happens during mode decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' For a given magnetic field Hi(r), its Fourier transform ˜Hi(k) can be written as the linear sum of Alfven and compressible components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' To sim- plify our argument, we consider a pure Alfven wave B-field in the local frame of reference, which ˜Hi is simply proportional to the local Alfven vector ζA,local, which is represented by the green vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' However if the volume dΩ is selected (the orange dash circle), the mean field is defined in dΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In this case, if we consider a pure Alfven mode magnetic field in the local frame of reference as we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='2, the projection of the Alfven mode in the global frame of reference will be an linear combination of Alfven and compressible modes as written in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' By selecting a k-vector that points inside the plane, the local Alfven wave unit vector (green) , which is defined by the cross product between the k-vector and the local magnetic field direction, makes an angle to that of the Alfven wave unit vector defined by ˆk × ˆλ which we will name that angle δθdΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The projection effect causes artificial com- pressible modes to be detected within dΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' It is very apparent that if the magnetic field line is aligned with the mean field vector ˆλ in dΩ, then there is no artificial compressible modes due to the projection effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We call this effect Alfven Leak- age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' This effect is artificial and does not involve the change of the cascade laws or anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='2 Modelling of the Alfven leakage problem We can further model the leakage phenomenon through the use of δθdΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='2 we see that δθdΩ is a measure of the average magnetic field angle dispersion in this selected volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We can postulate that the dispersion of angles are related to the Alfven Mach number measured within dΩ: δθdΩ ∼ MA,dΩ (13) Notice that the dispersion of magnetic field angle can scale up to MA ≈ 2 as shown in the appendix of Lazarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Notice that according to the definition of the Alfven Mach number and the self-similarity of turbulence cascade, any localized calculation of statistical observables in a tur- bulence system can be approximated by the scale relation through the observable’s structure function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' For the case of Alfven Mach number, the corresponding observable is δB/B, then MA,dΩ ≈ � 1 2B2 ⟨δB(r + r′) − δB(r′)⟩r′ �1/2 ∼ MA,global � r Linj �1/3 (14) where the last equality comes from the fact that the magnetic field fluctuation is small and Kolmogorov: ⟨δB2⟩ ∼ r2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We can then relate the CC,A with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (13) and (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='3 Numerical tests We can perform a very simple numerical test to illustrate the behavior of the Alfven leakage, which can be done by the following steps: (i) We select the local frame vectors ζA,C, where they can be approximated by selecting a very small volume in the numer- ical simulations and use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We shall call this volume 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (ii) We then perform Alfven wave decomposition using ζA from simulations so that we can put CA,local = 1 in all our case (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (iii) We then select a larger volume with size σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' According to §2, if there is indeed the effect of Alfven leakage happening, then CC = sin MA ∼ sin(Cσ1/3), where C = MA,globalL−1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In this new volume, there will be another pair of ζA,C being defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We shall call this volume 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (iv) We plot the quantity C2 C = 1 − (⟨ ˆBAlf,1 · ˆBAlf,2)2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Notice we only compare the directions of B within volume 1, since this calculation only makes sense there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (v) There could be three outcome from this test (a) If there is no such leakage effect, CC should be a constant zero as we already removed all non-compressible compo- nents in the previous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (b) If there is indeed CC but our model in the previous section is incorrect, then there should not be a dependence of CC ∼ sin(Cσ1/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (c) If the Alfven leakage effect indeed exists, we expect CC ∼ sin(Cσ1/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='3 shows how the CC behave as a function of the size of the volume σ, where we plot the regime when σ is not in the dissipation range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Notice that here we intentionally pick simu- lation cubes from various numerical codes and with different conditions to show that the leakage effect is universal and rather independent to what choices of MHD code one works with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' For each figure, we draw the predicted proportional- ity CC ∼ sin2(Cσ1/3) as the red dash curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' As we can see from these three subplots, our prediction CC ∼ sin2(Cσ1/3), which came analytically from previous sections and not from a fitting algorithm, follows the trend reasonably well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' This indicates that (1) the Alfven leakage effect actually ex- ists even in incompressible mode turbulence (2) the leakage is smaller when one goes to smaller scales (3) our postulate that δθdΩ ∼ MA,local is a good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' These three consequences indicate that the mode decomposition as pro- posed by CL03 requires an additional update in addressing the contributions of large scale magnetic field wandering to the relative composition of modes within the volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' As a direct consequence of this section, the WI,L constants originated from Lazarian & Pogosyan (2012) as a function of MA,local (≪ 1) are given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (8) are : WI = 1 − C2 A ∝ M 2 A WL = C2 A − C2 C ∝ 1 − 2M 2 A (15) MNRAS 000, 1–18 (2022) 6 Yuen et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='al Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Three figures showing the ratio of compressible modes CC as a function of the size of the volume selected σ for three simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' From the left: ma20, ma59, huge-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (See Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Model MS MA β = 2M2 A/M2 S Resolution ZEUS-MP Simulations Ms0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='92Ma0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='02 4803 Ms0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='98Ma0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='22 4803 Ms0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='93Ma0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='94 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='0 4803 huge-0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='0025 7923 huge-1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='011 7923 huge-2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='022 7923 huge-3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='042 7923 huge-4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='065 7923 huge-5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='63 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='089 7923 huge-6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='12 7923 huge-7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='56 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='16 7923 huge-8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='18 7923 huge-9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='39 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='20 7923 e5r2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='57 292 12003 e5r3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='52 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='45 12003 e6r3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='0039 12003 Pencil Simulations ma20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='56 5123 ma23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='23 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='17 5123 ma40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='40 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='00 5123 Pluto Simulations ma56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='59 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='17 5123 ma68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='68 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='51 5123 ma75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='17 5123 ma86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='86 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='17 5123 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Description of MHD simulation cubes which some of them have been used in the series of papers the authors have worked on before (Yuen & Lazarian 2017b,a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Lazarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Makwana & Yan 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2020) the snapshots are taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 4 APPLICATION (I): ADVANCEMENT OF THE SPA TECHNIQUE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='1 Review on the SPA technique In Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2020) the authors discussed a novel im- plementation called SPA in obtaining the modes from syn- chrotron emission maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Their argue that, since the tensor structures for Alfven or compressible waves being different at the synthesis of the Stokes parameter, the signature of the dominance of the modes are left in the ”signature parame- ter”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Let us first recap the formulation of Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2020) and their main results (See Method section of Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' To start with, the authors consider the emissivity of the synchrotron emissions under a locally defined reference frame: ϵxx = (I + Q)/2 = B2 0,⊥ cos2 φs + 2B0,⊥ cos φsBiˆexi + (Biˆexi)2 (16) where our symbols follow the Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2020) notations and we employ Einstein notation of summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Very im- portantly, the angle φs is the angle between the polarization vector to the currently defined x-axis of the local magnetic reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2020) they select a small area and compute the change of ϵxx as the reference frame rotates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' They recognize that the variance of ϵxx contains factors that could reflect the relative contributions of MHD modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The variance of the emissivity contains the linear term (the first term) and the quadratic term depending on the power of the tensor ˆT: sxx = ⟨ϵ2 x⟩ ∝ (2B0,⊥ cos φs)2 � dkF 2(k)eik·rˆexiˆexi ˆTij(ˆk) + 2 �� dkF 2(k)ˆexiˆexi ˆTij(ˆk) �2 (17) Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2020) pointed out that the linear term, namely the signature parameter: sxx(φs) ∝ (2 cos φs)2 � dkF 2(k)eik·rˆexiˆexj ˆTij(ˆk) (18) can be expressed in the following format with some constants axx, bxx defined according to MHD theory (See Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2020 for details): sxx(φs) = (axx sin2 φs + bxx) cos2 φs, φs ∈ [0, π] (19) where the classification parameter rxx is defined as rxx = axx bxx (20) In practice, we need to compute the parameter sxx,tot = V ar(ϵxx) 4⟨ϵxx⟩ (21) This term contains both the ”linear” term (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='19) and the ”quadratic” term as defined in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' MNRAS 000, 1–18 (2022) Subsonic Ma = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='2 Subsonic Ma = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='59 Supersonic MA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='82 Code: Pencil Code: Pluto Code: ZEUS-MP/3D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='20 Ldisp) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='20 C2 = sin2(c3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='18 () C² = sin2( C = sin2(co) = II 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='18 BA/f(o : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='16 MA α3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='10 =1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='06 101 2 × 101 101 3×101 4×101 2 × 101 3× 101 4×101 101 2 × 101 3 ×101 4×101 Size of the local volume, Size of the local volume, o Size of the local volume, oAlfven leakage 7 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' An illustration in showing how the observed features of the synchrotron intensities are related to the different weight- ing of spectrum, anisotropy and frame vectors (See Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='A1) if a strong guided field is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' From the top left: from a k−11/3 spec- trum plus the Alfven frame vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' top right: a k−11/3 spectrum with the Alfven anisotropic factor and also the Alfven frame vector mimicking the Alfven mode;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' lower left: a k−11/3 spectrum with the compressible frame vector, mimicking the Fast mode;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' lower right: a k−11/3 spectrum with the slow mode (See Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='A1) anisotropic factor and also the compressible frame vector, mimicking the slow mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' It is very apparent that both the anisotropy factor and also the frame vector contributes to the observed anisotropy in 2D ob- servables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' From synthetic simulations assuming MA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='1 such that the leakage effect is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='2 Alternative ways to separate the Alfven and compressible modes As discussed in LP12, two-point turbulence statistics con- tains three types of contributions: The spectrum which mea- sures the cascade as a function of scales, the anisotropy which records whether there is a preferred direction for the cascade to happen, and the tensor structures which records the pro- jection effect of the mode components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' From the discussion above, the SPA technique actually did not consider two-point statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In particular, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (21) is a one-point statistics, in which the anisotropy of the observable does not play a role in the value of the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Since all scales are summed up when computing the mean and variances, the spectrum also do not play an important role for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='(21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' As a result, a nat- ural guess on how the SPA technique works is the projection effect from the tensor structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In this scenario using the vector formulation (See Appendix) we can understand quan- titatively better how the SPA technique works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Notice that, in the case of one-point statistics, the 2D (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' the average op- erator above) and 3D statistics (which we will consider later below) should be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Let us consider a 3D magnetic field line written as sum of the mean and perturbed contribution in a selected volume dΩ with MA,dΩ ≪ 1: Hi(r) = ⟨Hi⟩ + � d3keik·r � X∈any frame CX(ˆk, ˆλ)ˆζX(ˆk, ˆλ) (22) From now on we are going to choose the frame to be the PCA frame (See Appendix A2, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='A1), and assuming the line of sight direction is at the z-axis and the magnetic field in the plane of sky defines the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The x-component magnetic field dispersion, which is just the mean value of the emissivity subtracted by a constant (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (16)), is given by: ⟨δH2 x⟩ = 2π � d3k � X∈C,A C2 X(ˆζX · ˆx)2 (23) Notice that the only difference between the compressible and the Alfven component can be observed when we expand the dot product for the above equation: ˆζA · ˆx = (ˆλ · ˆz)(ˆk · ˆy) |ˆk × ˆλ| ˆζC · ˆx = −(ˆλ · ˆx)1 − (ˆk · ˆx)2 |ˆk × ˆλ| (24) We can model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (23) via the frame definition of φs in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (16), where the frame angle φs = 0 when the projection of magnetic field is along the x-axis: ⟨δH2 x⟩ = Axx cos2 γ + Bxx sin2 γ cos2 φs (25) where cos γ = ˆλ · ˆz is the line of sight angle, and Axx, Bxx are: Axx = 2π � d3kC2 A,obs � (ˆk · ˆy) |ˆk × ˆλ| �2 Bxx = 2π � d3kC2 C,obs � 1 − (ˆk · ˆx)2 |ˆk × ˆλ| �2 (26) The factors within the bracket of each equation above are the geometric factors as discussed in LP12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Here we consider the general case of the leakage, which applies to both Alfven and compressible modes (See §3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' the observed amplitudes of Alfven and compressible modes CA,obs, CC,obs undergo an orthogonal rotation of angle MA < 1 (See Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='13): CA,obs ≈ CA cos MA − CC sin MA CC,obs ≈ CA sin MA + CC cos MA (27) The expressions inside the brackets of Axx, Bxx are the geo- metric factors that considered in both LP12 and Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We can see from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (25) that the contributions of Alfven and compressible modes are separated when one con- siders the frame rotation even for ⟨ϵ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' It is not necessary to compute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (21) in extracting the contributions of the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Moreover, we can now quantify the contributions of modes via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (26) by using the modes for CA,C by simply comparing the values of Axx and Bxx while analyzing the observed synchrotron emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In particular, if MA is small and there is no compressible mode, then Bxx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' that contribution of the Alfven mode to ⟨ϵ⟩ is frame independent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' rotating the x-y plane does not alter the result) since (ˆλ · ˆz) cannot be changed due to frame rotation, while that for compressible mode is a frame dependent quantity since ˆλ · ˆx is a function of the reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='3 The SPA technique for synchrotron emissions with significant Faraday Rotation In Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2020) they study the effects of Faraday ro- tation to the SPA technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Pictorially the Faraday depo- larization effects shields information up to a certain distance along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' This distance has been adequately discussed in Lazarian & Pogosyan (2012) in the presence of MNRAS 000, 1–18 (2022) No anisotropy factor is included When anisotropy factor is included Alfven, Alfven , Mimicking Alfven Mode k-11/3 k-11/3 × anisotropy factor Mimicking Slow Mod Mimicking Fast Mode Compressible , Compressible, k-11/3 × k-11/3 anisotropy factor8 Yuen et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='al galactic MHD turbulence and is called the Faraday screen- ing effect (Lazarian & Yuen 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Qualitatively, the SPA technique can only determine the mode fraction before the Faraday screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' However, we would like to perform the anal- ysis based on the formalism of Lazarian & Yuen (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In general, the synchrotron emission depends both on the distribution of relativistic electrons Ne(E)dE ∼ EαdE, (28) with intensity of the synchrotron emission being Isync(X) ∝ � dzBγ ⊥(x) (29) where X = (x, y) is the 2D position of sky (POS) vector and B⊥ = � B2x + B2y being the magnitude of the magnetic field perpendicular to the LOS z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In general, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='5(α + 1) is a fractional power, which was a serious problem that was successfully addressed in LP12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' LP12 proves that the statistics of I(α) is similar to that of I(α = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Therefore it suffices to discuss the statistical properties of the case α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Per Lazarian & Pogosyan (2012), Synchrotron complex po- larization function with Faraday rotation is given by: Psynch(R) = � dzϵsynchρrelB2e2i(θ(R,z)+Cλ2Φ(R,z)) (30) where ϵsynch is the emissivity of synchrotron radiation, Φ(R, z) = � z ∞ dz′(4π)−1/2ρthermal(R, z)Bz(R, z)rad m−2 (31) is the Faraday Rotation Measure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Notice that ρrel is the rel- ativistic electron density, while ρthermal is the thermal elec- tron density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The C-factor ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='81 (Lee et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='al 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The pro- jected magnetic field orientation is then given by: θB = π 2 + 1 2 tan−1 2 (U Q) (32) where tan−1 2 is the 2-argument arc-tangent function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' For frequencies lower than O(1GHz), the amplitude of the Faraday Rotation measure will exceed 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The physical pic- ture of the synchrotron polarization with Faraday rotation measure can be understood as: photons that passes through a section of ISM has to experience a certain amount of phase shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' If this phase shift exceeds 2π, all information coming from the source is completely lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Therefore an important concept called the Faraday screening emerges, which in- dicates the maximal line of sight distance that the observed synchrotron emissions can measure in the presence of line of sight magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In the case of sub-Alfvenic turbulence, the source term Pi = ρrel exp(2iθ(R, z)) is dominated by the mean field rather than the fluctuating one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The two regimes: (1) strong and (2) weak Faraday Rotation depend on whether the ratio Leff/L, is smaller (strong) or larger (weak) than unity: Leff L ∼ 1 λ2L 1 φ (33) where φ = max( √ 2σφ, ¯Φ) with σφ is the dispersion of random magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The difference between the two regimes are, 2 It is usually more convenient to use Hz = Bz/ √ 4π for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' A figure showing how the values of rxx varies as a function of λ in the presence of Faraday Rotation ∝ λ2 � dzρBz for both Variance-driven Faraday Rotation (VFR) and mean-field driven Faraday Rotation (MFR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' the Faraday rotation and the emission happens together in the former regime (φ = √ 2σφ), while the latter has the Fara- day rotation happens after the emission of the polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We shall name the two regimes ”Variance-driven Faraday Ro- tation” (VFR) and ”Mean-field Faraday Rotation” (MFR), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Notice that both regimes have been considered in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='5 shows a plot on how VFR and MFR could change the value of rxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' For this current plot we intentionally plot rxx with values that are not typically considered in previous literature (See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2020, rxx ∈ [−1, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' This al- lows us to better characterize whether the value of rxx came from the effect of compressibility or from Faraday rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We can observe from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='5 that there are two new regimes of λ that could make rxx fluctuates well beyond the values previously considered in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='5 we classify the ranges of values of λ via the fluctuations of rxx into three regimes: he ”weak” regime correspond to the case where rxx is small (∈ [−1, 1] as in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The intermediate regime correspond to the case where rxx start to grow exponentially, and the strong regime correspond to the case where the rxx basically loses traces on the compressibil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We can see that obviously the technique of SPA does not work when we are in the strong regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' However an interest- ing question is whether the SPA technique actually works in the intermediate regime which will be the subject for future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 5 APPLICATION (II): A SELF-CONSISTENT LINE OF SIGHT ANGLE TRACING METHOD VIA STRUCTURE FUNCTIONS OF I + Q AND I − Q The second application that we will deliver in this paper would be the retrieval of the mean global line of sight an- gle γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In the case of synchrotron/dust polarization, we have adequate information to estimate γ by considering the struc- MNRAS 000, 1–18 (2022) 40 - 20 - 0 0-00-00-0-00-00-00-000-0000-00-00-0-0-00-00-0 Intermediate 20 - Weak FRregime FR regime regime 40- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' VFR MFR 10-3 10~1 101 103 105 入 (cm)Alfven leakage 9 ture functions of both I + Q ∝ � dzB2 x and I − Q ∝ � dzB2 y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' For the following subsections, we will assume that the global mean field within the sampling area is ∥ to x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' One could always rotate the frame in Stokes parameter space to have the above condition satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='1 Why γ is encoded in the statistics of I + Q and I − Q?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The essence of on why γ is encoded in I + Q and I − Q is based on the fact that tensor formulation (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='2) contains different expressions for observables that ∥ B and ⊥ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 6, we present a set of figures showing the anisotropy of I + Q and I − Q for both A and F type contributions (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We present two extreme cases for γ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='6 3 that is sufficient to illustrate the differences of behaviors for the anisotropy of A and F type fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The left group of figures in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='6 shows the case when γ = 89o, while the group of figures on the right shows the case when γ = 9o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We can observe from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='6 a few interesting phenomena which is not covered in previous anisotropy studies: (i) The anisotropies of A and F type tensor do not necessarily align with the mean magnetic field direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We discussed this effect already from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The reason behind is that both the anisotropy and tensor contribution are anisotropic (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' However, the direction of anisotropy for the tensor con- tribution (with the Alfven leakage in effect) does not neces- sary be parallel to B-field and is a function of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Notice that the change of anisotropy is highly tied with the γ value (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='7) (ii) For the case of pure Alfven fluctuations, the anisotropy is more or less parallel to magnetic field for I +Q, while ⊥ B for I − Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Yet, the compressible mode does not carry the same trend as its Alfven counterpart: When γ ≈ 9o, the F-type anisotropy for I + Q is actually ⊥ B, while that for I − Q is ∥ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In contrast, when γ ≈ 89o the F-type anisotropy varies very similarly to that of the Alfven counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (iii) The measurement of relative anisotropies between I +Q and I − Q allows us to characterize the γ value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='6 we can see that if we consider the anisotropies of I +Q and I −Q at γ ≈ 89o, I +Q tends to be parallel to magnetic field, while that for I − Q tends to be perpendicular to magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We utilize the formulation in Appendix B that the minor- to-major axis ratio l⊥/l∥ = √ 1 − ϵ2, which the eccentricity ϵ is related to the quadropole-to-monopole ratio |D2/D0| via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The quadropole-to-monopole ratio is the key param- eter in parametrizing the anisotropy in previous literature (Lazarian & Pogosyan 2012, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Kandel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2016, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Lazarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='2 Formalism via §3 We will start from the parameters I + Q and I − Q in which we will assume the projected mean field is right now along 3 Notice that the projection of pure Alfven wave fluctuations when γ is exactly 90o will vanish, see Lazarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2022b) for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' A set of figures showing how the orientation of anisotropy for I + Q and I − Q is related to the line of sight angle γ for pure A (Alfven) and F (compressible, see Lazarian & Pogosyan 2012) type tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The key difference between the case of γ → π/2 (correspond to the case when B ⊥ LOS) and γ → 0 is that, the anisotropies of I + Q and I − Q for pure A and F tensors are similar for the former case, while for the latter case the anisotropies of pure A and F tensors are exactly opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' the x direction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' For the case of I+Q, we adopt the structure function expression from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (E20) of Lazarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2022b): DI+Q(R) = ⟨(Bx(R + R′) − Bx(R′))2⟩R′ = 1 2π2 � d2K � 1 − eiK·R� × � A(K, sin γ cos φK) cos2 γ sin2 φK 1 − sin2 γ cos2 φK + F(K, sin γ cos φK) sin2 γ sin4 φK 1 − sin2 γ cos2 φK � (34) where those factors are simply the expressions of ζAζA and ζF ζF in the global frame of reference (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' after leakage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The main takeaway here is, This D factor depends on the following form DI+Q(R) ∼ ¯A(R) cos2 γ + ¯F(R) sin2 γ (35) Similarly, the structure function for I −Q can be also mod- elled similarly as: DI−Q(R) ∼ ¯A(R) sin2 γ + ¯F(R) cos2 γ (36) Based on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='6 we can see that the construction: ¯y = Anisotropy(DI+Q) Anisotropy(DI−Q) (37) contains the information on γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Here we take the convention that Anisotropy(D) > 1 when the anisotropy of structure function is parallel to the global magnetic field direction, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In particular, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='6 we expect that ¯yA > 1 for all γ, while that for F-type contribution changes from smaller than 1 to greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Detecting the value of ¯y for 4 For a general magnetic field configuration, one could always con- sider the combination I + (Q cos(2φpol) − U sin(2φpol)), where we perform an inverse orthogonal transform with twice of the polar- ization angle 2φpol = tan−1 2 U/Q for this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' MNRAS 000, 1–18 (2022) Pure A Pure F Pure A Pure F Y = 890 06 =^ Q Mean B-field direction10 Yuen et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='al Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' A set of visualizations showing how the structure function of a certain variable D(R) can be visually decomposed as the linear combination of the multipole moments Dn, and how the multipole moments should be physically correlated to the relative angle between the line of sight and mean magnetic field γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The multipole moments collects the relative weight on the shapes that are specifically defined with the angular function exp(inθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In particular, D0 records the weights of the isotropic components of the structure functions, while D2 records the first order directionless anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Since empirically structure functions are mostly elliptical-like, |D2| must be non zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' modern turbulence theory predicts that the observed anisotropy would be a function of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' When ¯B ∥ LOS, then the structure function should be isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' While ¯B ⊥LOS, the structure function should be anisotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Therefore under the framework of multipole moments, the absolute amplitude of D2 should be a function of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' compressible modes (in global frame of reference) detected in observation is the key to extract the value of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The key reason why we consider the ratio of structure functions instead of individual quantity is because, from our expressions in the global frame of reference, the structure function of some observables carries factors on spectrum, anisotropy and tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' For the case of structure functions of I + Q and I − Q, their only difference is coming from the tensor factor as spectrum and anisotropy factors are fixed once the turbulence is set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' To proceed with our analysis, we consider the multipole expansion up to quadrupole (See Appendix §B for the con- dition for the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In particularly, the expansion is valid only for MA ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='5 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Formally we can express the anisotropy function that we defined above with the monopole and quadrupole coefficients D0, |D2|: AnisotropyMA∈[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='5,1] ≈ sign(Anisotropy) × D0 − |D2| D0 + |D2| (38) Recall from the previous discussion that the factors d0,2 can be literally written as the spectrum, anisotropy and the ten- sor contribution, and the first two contributions are cancelling out under our treatment, we can formally write y, which is the quadrupole approximation of ¯y to be (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (E30) of Lazarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2022b): y = Anisotropy(I + Q) Anisotropy(I − Q) = �D0 − |D2| D0 + |D2| � I+Q �D0 − |D2| D0 + |D2| �−1 I−Q (39) where we notice that under our current configurations, I + MNRAS 000, 1–18 (2022) 180° exp(0i) exp(2i0) exp(4i0) D(R) = X + X + Do D4 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Isotropic term Main term collecting anisotropy ExampleStructureFunctions B I L0S (y = 90°) ( =) SOT l Mean B-Field Direction for LHS panel ↓→ID2↑Alfven leakage 11 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' A figure showing the characterization of the relative anisotropy index (y = anisotropy(I + Q)/anisotropy(I − Q)) as a function of the line of sight angle γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' As we outlined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='6, the rel- ative anisotropies for A and F type fluctuations are different when γ is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' For the case of γ → 90, we expect the anisotropies of A and F type tensor fluctuate in the same way, which is illustrated as the light blue box in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' However, when we are looking at small γ limit, the anisotropies of A and F type tensor went com- pletely opposite, which is highlighted by the red box in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We denote these two regimes the ”compressible” and the ”Alfven” regime respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' from numerical simulation ”e5r2” (See Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=') Q ∼ Lzb2 x and I − Q ∼ Lzb2 y where Lz is the length of the integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Keeping only the tensor term, we will have an expression that is purely based on WI,L in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='15, and also functions of γ (See Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='35 and 36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='8 shows how numerically the factor y depends on the line of sight angle γ for Alfven mode (black) and the com- pressible mode (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We notice that the qualitative phe- nomenon happened in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='6 is exactly described by y for com- pressible modes: y < 1 for γ → 90o, while y > 1 for γ → 0o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We recognize that there are fluctuations in terms of the vari- ation of y relative to γ for the compressible case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Surprisingly, the Alfven mode y also exhibits some interesting properties that we can exploit in obtaining γ in observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Notice that y for Alfven mode stays < 1 from what we observe in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='6, we see that the Alfven mode’s y has very similar trend when γ ⪆ 55o , but when γ ⪅ 55o, the Alfven mode y-value went exactly opposite to that of compressible mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Moreover, we observe that the change of values of y as a function of γ is more or less monotonic if we consider γ ⪅ 55o and γ ⪆ 55o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Notice that the modes that we are talking about here are all in the global frame of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' To see whether the trend that we observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='8 is ro- bust, we select some of the numerical cubes from Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='1 and to plot y as a function of γ for both A and F type contribu- tion and plot it as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The selected numerical cubes cover a wide range of sonic and Alfvenic Mach numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='9 that the trends of the two curves are very sim- ilar to that of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Furthermore, the exact values of y are also very similar across different turbulent conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Origi- nally, the formalism of A and F type tensor applies only for Ms,A < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' However, we perform the calculation of y also for supersonic sub-Alfvenic simulations, which is closer to the environment of molecular clouds (See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Draine 2011) and still observe the same trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We therefore conclude that the ¯y parameter tracers γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In fact, we observe from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='9 that when the plasma β ∝ M 2 A/M 2 s is smaller, it is easier to recover the trend that we see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' At last, we provide the empirical formula (units in degrees) for the case of low β (β < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' For γ < 40o y(F) ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='2 − γ/40 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='4 y(A) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='4 + γ/40 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='2 (40) for γ > 40 degrees y(F) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='8 − (γ − 40)/50 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='2 y(A) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='8 (41) The full study on how the y-parameter can be applied to situation with different mixture of driving will be discussed in Malik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 6 DISCUSSION 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='1 The importance of Alfven leakage for mode decomposition The analysis of turbulence properties generally from observa- tions requires the consideration of the local-to-global frame problem, which is modelled as the ”magnetic field wander- ing problem”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' While the theory of MHD turbulence is well- established, how the local scaling laws are projected glob- ally is still mysterious, despite models have been proposed from both Lazarian & Pogosyan (2012) and Lazarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Here, we propose the first physical model in explain- ing how the wandering of magnetic field happens when pro- jected along the line of sight, and how we could utilize the magnetic field wandering in deducing a number of important physical quantities such as the line of sight angle and also the mode fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The problem of the local-to-global frame transition in theo- retical MHD turbulence studies have puzzled the community for a while.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' While the anisotropic scaling k∥ ∼ k2/3 ⊥ is well motivated from the simple constant energy cascade and crit- ical balance condition (GS95), we cannot retrieve the local scaling from the global frame of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In fact, the global correlation function usually gives a constant scaling rather than a geometrically-driven, size-dependent scaling as pre- dicted by GS95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Before the availability of MHD simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Cho & Lazarian 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Beresnyak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2005), it is not yet possible to validate the GS95 relation even from numeri- cal simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The more puzzling effect comes when MA is very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Traditionally the numerical test on GS95 are done in small MA systems and in small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' However as we see from the previous sections, in moderate and small k the Alfven mode acquired from the Cho & Lazarian (2003) decomposi- tion method contains non-negligible contributions along the ˆζC vector, indicating the presence of anomalous compressive wavevector even after Alfven mode decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The only plausible reason why this happens is because the mode de- composition method from Cho & Lazarian (2003) is done on a global frame of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' As a result when we are looking at small scales, in average the mean field is not very different from its local field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Yet for larger scales the mean field is very MNRAS 000, 1–18 (2022) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='4 A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='2 (o+)Cdouosu (-)doos 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='0 Compressible (F) regime 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='8 Alfven (A) regime y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='4 0 10 20 30 40 50 60 70 80 90 LOS Angle y12 Yuen et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='al Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' A set of figures showing the universality of our finding (y as function of the line of sight angle γ in degrees) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='8 in 6 numerical simulations from Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='1, which covers a large range of value of Ms,A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' different from the local field, so that there exists anomalous compressible terms even the data are supposed to be ”Alfven modes” according to Cho & Lazarian (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We named this effect ”Alfven leakage” in our previous section since this ef- fect happens even for Alfven waves as long as the Alfvenic Mach number is not zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In this paper, we further show that the Alfven leakage effect is a global function of MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In fact, the presence of the leakage effect suggests that the mode decomposition method by Cho & Lazarian (2003) should subject to the a correction term for moderate and small k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' However since most of the calculation from Cho & Lazarian (2003) are done in small scales, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' large k, the results of their work are not affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='2 The importance of tensor forms to the SPA technique and general turbulence studies The novel invention of the SPA technique (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2020)) utilizes the fact that the tensor projections have dif- ferent contributions for Alfven and compressible modes to identify them in observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' This work further strengthens their argument through the use of Alfven leakage picture and suggests a few important improvements to their method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' For instance, it is not necessary to compute the parameter sxx as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='21 to distinguish the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The tensor properties are encoded in the Stokes parameters and thus ignoring the tensor contribution would make dramatically different pre- dictions in astrophysical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' One very important factor that is accounted by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2020) is the use of one point statistics under Stokes frame transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The traditional turbulence statistical stud- ies usually utilize multi-point statistics since they are either directly related to the spectra (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' two-point) or is used to validate scaling relations for higher order structure functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Kolmogorov 4/5 law).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The reason of why single point statistics was not useful before is because the spectrum and anisotropy are the main characteristics of turbulence studies for the past 60 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' However, how the tensor projection af- fects the geometry of the structures for each of the turbulence variable is not really explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Tensor forms of turbulence modes were not much explored beyond the physics of cosmic rays(Schlickeiser 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Yan & Lazarian 2002, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In fact, the previous anisotropy analysis also did not consider what is the statistics of a single component of a 3D turbulence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='tensor projection, after projection along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' While the series of papers by Lazarian & Pogosyan started to consider how the single component statistics works, not until recently did both numerically (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Lazarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2018) and observationally (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2020) found the effect of ten- sors to be that important during single component projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The anisotropy of projected fast modes with the direction op- posite to the Alfvenic anisotropy was shown in Lazarian & Pogosyan (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In fact, one of the most common belief that is circulating in the earlier studies of MHD turbulence the- ory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' the discussion section of Lazarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2017)) is the presumption that the projection of the observables (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' velocities, magnetic field) from fast modes will be isotropic since the fast modes in 3D are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' This is empirically proven wrong by Lazarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2018) through the development of velocity gradient and also utilized through the development of SPA in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In fact, in a number of astrophysical applications, the ob- MNRAS 000, 1–18 (2022) Ms = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='17, MA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='22 Ms = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='4, MA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='24 Ms = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='92, MA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='09 Huge-o : ^ e6r3: Ms0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='92Ma0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='8 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='6 304050 70 80 90 10 20 50 60 LOS Angle y 70 80 06 LOS Angle y LOS Angle Y Ms = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='93, Ma = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='94 Ms = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='61,MA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='52 Ms = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='98, Ma = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='32 Ms0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='93Ma0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='94 e5r3 Ms0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='98Ma0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='2 (o-)doosr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='2 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='8 10 20 40 50 60 70 80 90 10 20 40 LOS Angle y 60 70 80 10 20 30 40 So 60 70 80 LOS Angle y LOS Angle yAlfven leakage 13 servables are constructed through not all three directions of velocities or magnetic field, but just some of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' For in- stance, the Davis-Chandrasekhar-Fermi (DCF) technique uti- lizes both the line of sight velocity dispersion and the plane of sky polarization angle dispersion to estimate the magnetic field strength through the use of Alfven relation (See also Cho & Yoo 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' However, as found in Lazarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2022b), the direction of the velocity and magnetic field fluctuations as collected in DCF technique are exactly perpendicular to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Moreover, in this work we also show both analytically and numerically that the tensor term contains anisotropy and can be dominant as long as the γ fulfills some conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' As a result, one should not ignore the contributions of the tensor term in studying the properties of MHD turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='3 The use of high pass filter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In MHD turbulence studies, there are a few length scales that determine whether the underlying turbulence is hydro- dynamic or GS95-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' It is a general phenomenon that for 3D saturated turbulence the small scale fluctuations are GS95 like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' However for both sub-Alfvenic and super-Alfvenic there exist a transition scale that the turbulence becomes non-GS95 like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' For instance, for the sub-Alfvenic case there exist the transition from weak to strong turbulence (Cho & Lazarian 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Makwana & Yan 2020) at the length scale LM 2 A, while for the super-Alfvenic case above the scale LM −3 A the turbu- lence is hydrodynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' This might suggest that the removal of large scale fluctuations could allow observers to obtain the desired GS95 statistics with the use of high pass filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' However upon projection the high pass filter in 2D acts a little bit differently compared to 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Fundamentally the high pass filter (HPF) in 3D serves as the high frequency extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' As noticed in Lazarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2020) , HPF in 2D acts as a lower bound of the HPF in 3D, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' if we explicitly want K = � k2x + k2y > K0, this will automatically apply to k = � k2x + k2y + k2z ≥ K > K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' However since the sampling of turbulence statistics upon projection is not statistically complete, meaning that the wavevectors with k > K0 but K < K0 is not sampled, it is hard to determine whether we will obtain back the same turbulence spectrum anisotropy just by inspection here since we did have additional knowl- edge on how the LOS direction is related to the inclination angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' More importantly, if we are considering the case when MA is not small, the randomness of the magnetic field fluc- tuation will make the filtering in 2D in Stokes parameters being completely different from that of the 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='10 shows an example on how different the Stokes Q look like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' On the left of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='10 we perform filtering after projection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2D), while on the right it is the projection after 3D filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We can see that, while the statistical anisotropies for the two maps are roughly the same, the differences of the features are prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 7 CONCLUSION In this paper, we introduce a vector-based framework in ex- plaining the strength and the limitation of the recently intro- duced techniques, namely SPA, CFA and VGT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In particular, due to the use of the vector framework, we recognize that in the presence of curved magnetic field Alfven waves will be Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' An illustration of the features of the Stokes Q after 2D (left) and 3D (right) filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' One can see that there is a significant difference in terms of the structures of the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' seen as the linear combination of Alfven and compressible waves, which is named ”Alfven leakage”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In short, (i) We recognize a straightforward transformation from the lo- cal to global reference frame through the Alfven leakage model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Moreover, the projection parameters WI,L that are introduced in LP12 are derived in an alternative way in the picture of Alfven leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (8), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='3) (ii) The SPA technique, which allows the identifications of the dominance of the Alfven and compressible waves in observed synchrotron emissions, is the result of the one-point statis- tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The Alfven wave contribution is frame independent while that for compressible waves are frame dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' As a re- sult, the quantitative contribution of Alfven and compressible waves can be separated observationally (See Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (23)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (iii) We suggest that the SPA technique is also applicable to slightly Faraday rotated regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (iv) Based on the formulation of the Alfven leakage, we discover a new γ tracing method that utilize the anisotropy fraction of I + Q and I − Q in observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We test the method in numerical simulations and see universality of trends across a wide range of turbulence parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (§5, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The expression of the vector frame formulation allows us to visually understand and analyze the statistics of MHD tur- bulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Together with the theoretical establishment of the Lazarian & Pogosyan series, how the turbulence statistics are imprinted into observables will be better understood by ob- servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' APPENDIX A: THE MATHEMATICAL DESCRIPTION ON VECTOR AND TENSOR FORMULATIONS IN MHD STATISTICAL TURBULENCE THEORY For our analysis in this paper, we need to review some of the required mathematical tools for the descriptions of the MHD turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The reason why we need them is because some of the frame representations are advantageous in some situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Here we will first review the concept of the global and local frame of reference, the leakage of modes due to the Yuen & Lazarian (2020) of local magnetic field, and also the mathematical establishments that are scattered in different MNRAS 000, 1–18 (2022) Filtered after projection (2D filtering) Filtered before projection (3D filtering) Stokes Q14 Yuen et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='al literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The unified approach that we use in this paper will lead to establishment of an analysis framework in under- standing how the modes should behave in observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' A1 Global and local frame of reference The first important concept is the use of the local frame of reference when computing the structure function of the turbulence variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The mathematical expression of the 3D structure function of the turbulence variable v in the local frame of reference is given by: SF{v}(r) = � � (v(r′ + r) − v(r′)) · B(r′ + r) + B(r′) |B(r′ + r) + B(r′)| �2 � (A1) where in small r, the separation of the three eigenmodes (Alfven, Fast, Slow) will give the correct spectrum and anisotropy as predicted in GS95 and LV99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In particular, the anisotropy will be scale dependent when observed locally through the 3D structure functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Table A1 summarizes the spectral slopes and anisotropies that we expect from the local structure functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' However, we cannot deduce the expressions from Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='A1 due to the restriction of the local-to-global reference frame transformation, which is the main topic of the current paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' A more common method in computing the structure func- tion is by simply computing the simplistic structure function below, assuming Vz(R) = � dzˆz · v(r): SF{Vz}(R) = ⟨(Vz(R′ + R) − Vz(R′))2⟩ (A2) which the spectrum and anisotorpy that is observed from this variable could be different from what the local expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In particular ,the anisotropy in the global frame of reference becomes scale independent, meaning that there is no partic- ular advantage in probing the anisotropy in smaller scale in actual observations, aside from the standard LM −3 A scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' A2 Tensor representation In the global frame of reference,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' the spectral tensor for dif- ferent modes can be represented by the sum of the three linearly independent spectral tensors TP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' which is given by (Lazarian & Pogosyan (2012),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' cf Yan & Lazarian 2004): TP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='ij = ˆkiˆkj TC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='ij = (ˆk × (ˆk × ˆλ))i(ˆk × (ˆk × λ))j |ˆk × ˆλ|2 = (λi − (ˆk · ˆλ)ki)(λj − (ˆk · ˆλ)kj) |ˆk × ˆλ|2 TA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='ij = Iij − TP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='ij − TC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='ij = (ˆk × ˆλ)i(ˆk × λ)j |ˆk × ˆλ|2 (A3) Notice that for Alfven mode vA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='iTP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='ij = 0 since ∇ · vA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Notice that TC + TA is isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' A3 The ASF (CL03) frame with respect to the PCA frame For the actual numerical analysis, the realization of the in- dividual MHD modes in the local frame of reference is not achievable since obtaining the modes requires the perturba- tion theory to start with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In this case, the expressions of the modes are given in Fourier space by evaluating the perturba- tion along a locally averaged mean field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In that case, for each k ∈ R3, we can locally define the eigenvectors for the three modes ˆζA,S,F given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Notice that the A(lfven)-S(low)- F(ast) frame is a simple rotation of the ”magnetic frame” along ˆζA given by the three eigenvectors (ˆλ, ˆk׈λ, ˆλ×(ˆk׈λ)) by an angle φ: tan φ = 2α cos2 θλ − (α + 1 + √ D) 2α cos2 θλ tan θλ (A4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The ”magnetic field” is simply given by an additional ro- tation of tan θλ from the P(otential)-C(ompressible)-A(lfven) frame (ˆζP = ˆk, ˆζA = ˆk׈λ, ˆζC = ˆk×(ˆk׈λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The PCA frame has its special advantage since the sampling of k is usually complete in dΩk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' That means we have the freedom to fix k despite other unit vectors are changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' From the tensor product we can always write the arbitrary vector in the Fourier space as : ζi(k) = CP ˆki + CC (ˆk × (ˆk × ˆλ))i |ˆk × ˆλ| + CA (ˆk × ˆλ)i |ˆk × ˆλ| (A5) which we will name the unit vector ζP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='A now From Cho & Lazarian (2003),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' in the global frame of refer- ence the Alfven,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' slow and fast mode eigenvectors are: ζA ∝ ˆk × ˆλ ζS ∝ (−1 + α − √ D)(k · ˆλ)ˆλ + (1 + α − √ D)(ˆλ × (k × ˆλ)) ζF ∝ (−1 + α + √ D)(k · ˆλ)ˆλ + (1 + α + √ D)(ˆλ × (k × ˆλ)) (A6) where α = βγ/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' D = (1 + α)2 − 4α cos2 θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' cos θ = ˆk · ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We recognize that there is a frame rotation between the vector ζP,C and ζS,F : �ζS ζF � = −1 2 cos 2θ √ D L(α, θ)R0(θ) �ζP ζC � (A7) where R0(θ) is the standard two-dimensional rotation ma- trix, the factor beforehand is just for normalization and: L(α, θ) = �(−1 + α − √ D) cos θ (1 + α − √ D) sin θ (−1 + α + √ D) cos θ (1 + α + √ D) sin θ � (A8) Then we can rewrite the tensors by TS/F = ζS/F ⊗ ζS/F (A9) Notice that Tijζj = ζi if Tij = ζi ⊗ ζj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' MNRAS 000, 1–18 (2022) Alfven leakage 15 Mode Power spectra E(k), where E = � dkE(k) Anisotropy factor Frame vector Alfven k−5/3 exp(−M−4/3 A k∥/k2/3 ⊥ ) ζA Slow (low β) k−5/3 varies (see Makwana & Yan 2020) ζS Slow (high β) k−5/3 exp(−M−4/3 A k∥/k2/3 ⊥ ) ζS Fast (low β) k−3/2 1 ζF Fast (high β) k−3/2 k2 ⊥ ζF Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' A summary of the theoretical expectations of the turbulence scaling laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Summarized from CL03, Yan & Lazarian (2008) and Makwana & Yan (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The definition of the frames that we use in the current paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' From the left: The Potential-Compressible-Alfven Frame (PCA, left) which is very convenient in analyzing the magnetic field perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The Alfven-Slow-Fast Frame (ASF, right) was the local reference frame for the three fundamental MHD eigenmodes derived in CL03 The magnetic frame (middle) is the frame that shares similarity to the PCA frame defined through the local magnetic field direction ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' To connect them, the PCA frame is a simple rotation of θλ = cos−1(ˆk · ˆλ) from the magnetic frame, and the ASF frame is just a φ rotation from the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' A4 Frenet-Serret frame From Yuen & Lazarian (2020), the Frenet-Serret frame of the the magnetic fields lines would be: dˆt ds = +κˆn dˆn ds = −κˆt + τˆb dˆb ds = −τ ˆn (A10) Here ˆt = ˆλ, representing the tangent vector of the magnetic field line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (ˆt, ˆn,ˆb) forms a complete orthogonal set indepen- dent of the choice of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Notice that for mode decomposition, the ”mean” field is selected before selecting (Fourier trans- forming into) k, thus we can treat λ as k-independent and uses its own position vector rλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Notice that the unit vector ˆn can be expressed as the linear combination of ˆζA and ˆλ× ˆζA in the magnetic frame A5 The relation between the tensor representation (Lazarian & Pogosyan 2012) and vector representation (this work) In the local frame of reference, the Alfven mode magnetic field is given by simply: HA(r) = � d3kC(k)ˆζA(k) (A11) where C contains the isotropic and anisotropic factors from its spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' However as we move from the local frame to the global frame, the actual Alfven wave magnetic field real- ization will contain both compressible and Alfven wave con- tribution (here we simply pick an arbitrary k): ˜HA(k) = CWAˆζA + CWC ˆζC (A12) where WA,C are two factors yet to be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' LP12 branded these two factors in the form of the direct tensor product ζE = ζC + ζA and ζF = ζC, and TE,F = ˆζE,F ⊗ ˆζE,F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' In their case when Alfven mode is observed in the local frame MNRAS 000, 1–18 (2022) P(otential)-C(ompressible)-A(Ifven) Frame Magnetic Frame A(Ifven)-s(low)-F(ast) Frame F =d Sp = k =d 0a Φ 2D Rotation with Angle 2D Rotation with Angle Φ Qa = cos-1k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' given by CL03 kx ZA kx k× TA [K [K × (×) × (×) [× (×) [× (×) k× (k×) = 2 [k × (k × )I 2α cos2 - (α +1+vD) tan Φ = tan 0 2α cos2 D = (1 + α)² - 4α cos² ^16 Yuen et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='al of reference,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' the Alfven mode correlation function in k-space is given by: ˜Hi ˜Hj = C2(TE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='ij − TF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='ij) (A13) while in the global frame of reference ˜Hi ˜Hj = C2TE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='ij − C2(WITE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='ij + WLTF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='ij) (A14) Some algebra will give ˜Hi ˜Hj = C2(1 − WI − WL)TC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='ij + C2(1 − WI)TA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='ij (A15) A6 Conversion between the frame of references of velocity field and magnetic field As derived by Cho & Lazarian (2002,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2003) the decomposed Alfven-Slow-Fast frame was the frame for the displacement vector ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' which also applies to the velocity fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' How- ever the magnetic field fluctuations do not necessary follow the ASF frame as defined in CL03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' For Alfven wave, the fluc- tuations of the magnetic field is in the same direction as that of velocities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' ˆk × ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' For compressible modes, the prop- agation of the magnetic field fluctuations ˜b(k) at a specific wavevector k is given by the following relation: ˜b = ˆk × (˜v × ˆλ) (A16) where ˜v is the velocity fluctuation at k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Notice that the above vector is parallel to the compressible vector ˆζC = ˆk×(ˆk׈λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' APPENDIX B: THE FUNDAMENTALS OF DESCRIBING THE ANISOTROPY IN STRUCTURE FUNCTIONS In this section we will discuss the essence of multipole ex- pansions in analysing the statistics of turbulence under the assumption of two-point closure5 based on the formalism of Kandel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' It is visually compelling that the two-point structure func- tions are concentric ellipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Mathematically the structure functions of anisotropic fundamental modes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Alfven, slow modes) contains a dependence in the form of exp(−C| cos φ|) for some constant C that carries a weak dependence on φ (See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Lazarian & Pogosyan 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Kandel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' This exponent term is naturally elliptical like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The expression of this term in the two-point statistics of any observables is the main direction of theoretical study recently in literature (Lazarian & Pogosyan 2012, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Kandel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2016, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Lazarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2022b,a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' There are a few choices in describing ellip- tical features on the sky via complete basis: Multipole expansions of even order: The spatial symme- try of the function exp(−C| cos φ|) allows one to express the 5 The concept of two-point closure is simply to say that turbulence variables can be ”adequately” described by the two-point structure functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' This approximation is evidently incorrect in general tur- bulence case as intermittency is a well-studied topic in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' However for equilibrium MHD turbulence that we are consider- ing here, the two-point description contains ∼ 95% of the spectral power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The prominent features that we are measuring (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' mode fraction, γ etc) are therefore dominated by the two-point statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' See Yuen (thesis, 2022) structure function of any observables X into the summation the cosines with even orders: DX(φ) = � m∈2Z+ Dm cos(mφ) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Visually we are expressing the structure function into linear combination of cosines in polar coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Notice that for all m|4 = 2 the Dm term carries some anisotropy, however for m ≥ 6 the multipole anisotropy has a upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' For instance, the cos 6φ term has a maximum anisotropy of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Notice that the non-vanishing Dm≥6 will decrease the anisotropy of the structure function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' A typical treatment of the multipole expansion is to truncate the series into m = 0, 2, where the visual minor-to-major axis ratio for the elliptical feature appeared in the structure function χ = � 1 − ϵ2 ell (ϵell is the eccentricity of ellipse) is given by: χ = D0 − D2 D0 + D2 (B1) Notice that the multipole expansion fails when MA ≪ 1 or MA > 1, as the Dm term is comparable ∀m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The empirical limit where D4/D2 is comparable (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='5,Lazarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2022a) is roughly at MA ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Therefore the multipole expansion is suitable only for MA ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='5 − 1 (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='B1) Legendre Polynomial: The Legendre polynomial Pl(cos φ is another popular choice in describing the statistics in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Similar to the multipole expansion, we express the structure function DX(φ) = � l∈2Z+ alPl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' al carries very similar mathematical properties as Dm in multipole expansions and therefore we would not discuss further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='B1) Elliptical basis: As the structure function look like ellipses,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' it is natural to consider the function below to capture the anisotropy of the structure function: f(φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' ϵell) = � 1 − ϵ2 ell (1 − ϵ2 ell 2 + ϵ2 ell 2 cos 2φ) (B2) The advantage of this basis is that (1) the eccentricity ϵell is a direct measure of the minor-to-major axis ratio,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' which allows one to quickly construct this function by simply mea- suring the minor and major axis (2) due to the non-vanishing higher-order multipole of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='B2, this functional form is still applicable when MA ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Notice that one can convert the eccentricity ϵell to the D2/D0 via the formula: ���D2 D0 ��� ≈ 1 2 2ϵ2 ell 2 − ϵ2 ell (B3) in which the approximation is valid when MA ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='5, 1] for the case of linear (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' centroid, C ∝ � dzvz) or quadratically projected observables (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Stokes parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The approx- imation is valid for caustics (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Yuen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2021) for even smaller values of MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' acknowledge the sup- port the NSF AST 1816234, NASA TCAN 144AAG1967 and NASA ATP AAH7546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' KHY thanks Dmitri Pogosyan (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='Alberta) and Ka Wai Ho (UW-Madison) for their inspira- tional comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We thank Sunil Malik (DESY) and Parth 6 In previous literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Kandel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 2016) they express DX ∼ � m∈2Z ¯Dmeimφ, where m can be both positive and nega- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Typically structure functions are always real-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' There- fore for the sake of simplicity we adopt the cosine formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' MNRAS 000, 1–18 (2022) Alfven leakage 17 Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Two figures showing how the change of MA affects the relative amplitude of Dl and al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' We can see from the L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' figure that, when MA is large, the approximation til quadruple is very good since |D4| < min(D2, D0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' However as MA becomes smaller, |D4| is actually comparable to that of |D0,2|, and therefore the multipole approximation breaks down .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Very similar result also happens for Legendre expansion (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Pavaskar (DESY) for extensive discussions and cross-checks on the validity of the y-parameter analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' The main simula- tions and the first version of the work is done during KHY’s tenure in UW Madison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Research presented in this article was supported by the Laboratory Directed Research and Devel- opment program of Los Alamos National Laboratory under project number(s) 20220700PRD1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Code Availability The code can be found in https:// github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='com/kyuen2/MHD_modes Data Availability The data underlying this article will be shared on reasonable request to the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' REFERENCES Armstrong J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Rickett B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Spangler S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 1995, ApJ, 443, 209 Begum A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Chengalur J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Bhardwaj S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2006, MNRAS, 372, L33 Beresnyak A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2019, Turbulence in Magnetohydrody- namics Beresnyak A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Cho J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2005, ApJ, 624, L93 Biskamp D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2003, Magnetohydrodynamic Turbulence Brunetti G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2007, MNRAS, 378, 245 Chandran B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2000, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 85, 4656 Chepurnov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Stanimirovi´c S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Heiles C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Peek J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2010, ApJ, 714, 1398 Cho J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2002, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 88, 245001 Cho J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2003, MNRAS, 345, 325 Cho J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2005, Theoretical and Computational Fluid Dynamics, 19, 127 Cho J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Vishniac E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2000, ApJ, 539, 273 Cho J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Yoo H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2016, ApJ, 821, 21 Deshpande A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Dwarakanath K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Goss W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2000, ApJ, 543, 227 Dickey J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', McClure-Griffiths N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Stanimirovi´c S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Gaensler B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Green A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2001, ApJ, 561, 264 Draine B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2009, in Henning T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Gr¨un E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Steinacker J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', eds, Astronomical Society of the Pacific Conference Series Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 414, Cosmic Dust - Near and Far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' 453 (arXiv:0903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='1658) Draine B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2011, Physics of the Interstellar and Intergalactic Medium Elmegreen B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Scalo J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2004, ARA&A, 42, 211 Farmer A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Goldreich P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2004, ApJ, 604, 671 Fissel L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2016, ApJ, 824, 134 Fu X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Li H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Gan Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Du S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Steinberg J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2022, ApJ, 936, 127 Gan Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Li H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Fu X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Du S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2022, ApJ, 926, 222 Goldreich P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Sridhar S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 1995, ApJ, 438, 763 Green D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 1993, MNRAS, 262, 327 Kandel D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Pogosyan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2016, MNRAS, 461, 1227 Kandel D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Pogosyan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2017, MNRAS, 464, 3617 Khalil A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Joncas G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Nekka F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Kestener P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Arneodo A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2006, ApJS, 165, 512 Kowal G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2010, ApJ, 720, 742 Kowal G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Beresnyak A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2007, ApJ, 658, 423 Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2006, ApJ, 645, L25 Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2016, ApJ, 833, 131 Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Pogosyan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2000, ApJ, 537, 720 Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Pogosyan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2004, ApJ, 616, 943 Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Pogosyan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2006, ApJ, 652, 1348 Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Pogosyan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2008, ApJ, 686, 350 Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Pogosyan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2012, ApJ, 747, 5 Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Pogosyan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2016, ApJ, 818, 178 Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Vishniac E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 1999, ApJ, 517, 700 Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Yuen K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2018, ApJ, 865, 59 Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Yuen K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Lee H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Cho J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2017, ApJ, 842, 30 Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Yuen K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Ho K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Chen J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Lazarian V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Lu Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Yang B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Hu Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2018, ApJ, 865, 46 Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Yuen K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Pogosyan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2020, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='07996 Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Yuen K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Pogosyan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2022a, arXiv e-prints Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Yuen K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Pogosyan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2022b, ApJ, accepted for publication, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' arXiv: 2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='09731 Lithwick Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Goldreich P.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Yan H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2021, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='01936 Makwana K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Yan H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2020, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='al Padoan P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Juvela M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Kritsuk A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Norman M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2006, ApJ, 653, L125 Schekochihin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2022, Journal of Plasma Physics, 88, 155880501 Schlickeiser R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2002, Cosmic Ray Astrophysics Yan H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2002, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 89, 281102 Yan H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2004, ApJ, 614, 757 Yan H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2008, ApJ, 673, 942 Yuen K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2017a, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' arXiv:1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='03026 Yuen K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2017b, ApJ, 837, L24 Yuen K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2020, ApJ, 898, 66 Yuen K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2018, ApJ, 865, 54 Yuen K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Ho K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2021, ApJ, 910, 161 Yuen K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Ho K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Law C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Chen A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Lazarian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2022, submitted, arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='13760 Zhang H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Chepurnov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Yan H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Makwana K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Santos-Lima R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Appleby S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2020, Nature Astronomy, 4, 1001 Zhao S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Yan H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Liu T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Liu M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Shi M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2021, ApJ, 923, 253 Zhao S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Yan H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Liu T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Liu M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', Wang H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=', 2022, ApJ, in press, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content='05410 This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} +page_content=' MNRAS 000, 1–18 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFQT4oBgHgl3EQfhDYG/content/2301.13344v1.pdf'} diff --git a/c9AyT4oBgHgl3EQfjPg-/content/2301.00410v1.pdf b/c9AyT4oBgHgl3EQfjPg-/content/2301.00410v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a6a1ccaf0b595f3432861d3b4feec8b65cc9c329 --- /dev/null +++ b/c9AyT4oBgHgl3EQfjPg-/content/2301.00410v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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C. Jebaraj1, 2, A. Kouloumvakos3, N. Dresing4, A. Warmuth5, N. Wijsen6, 7, C. Palmroos4, J. Gieseler4, R. +Vainio4, V. Krupar7, 8, J. Magdalenic1, 2, T. Wiegelmann9, F. Schuller5, A. F. Battaglia10, 11, and A. Fedeli4 +1 Center for mathematical Plasma Astrophysics-CmPA, Department of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 +Leuven, Belgium. +2 Solar–Terrestrial Centre of Excellence—SIDC, Royal Observatory of Belgium, 1180 Brussels, Belgium +3 The Johns Hopkins University Applied Physics Laboratory, 11101 Johns Hopkins Road, Laurel, MD 20723, USA. +4 Department of Physics and Astronomy, University of Turku, Finland. +5 Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany. +6 Department of Astronomy, University of Maryland, College Park, MD 20742, USA. +7 Heliospheric Physics Laboratory, Heliophysics Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA. +8 Goddard Planetary Heliophysics Institute, University of Maryland, Baltimore County, Baltimore, MD 21250, USA. +9 Max-Planck-Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany. +10 Institute for Data Science (I4DS), University of Applied Sciences and Arts Northwestern Switzerland, Bahnhofstrasse 6, 5210 +Windisch, Switzerland. +11 Institute for Particle Physics and Astrophysics (IPA), Swiss Federal Institute of Technology in Zurich (ETHZ), Wolfgang-Pauli- +Strasse 27, 8039 Zurich, Switzerland. +ABSTRACT +Context. We study the solar energetic particle (SEP) event observed on 9 October 2021, by multiple spacecraft including Solar Orbiter +(SolO). The event was associated with an M1.6 flare, a coronal mass ejection (CME) and a shock wave. During the event, high-energy +protons and electrons were recorded by multiple instruments located within a narrow longitudinal cone. +Aims. An interesting aspect of the event was the multi-stage particle energization during the flare impulsive phase and also what +appears to be a separate phase of electron acceleration detected at SolO after the flare maximum. We aim to investigate and identify +the multiple sources of energetic electron acceleration. +Methods. We utilize SEP electron observations from the Energetic Particle Detector (EPD) and hard X-ray (HXR) observations from +the Spectrometer/Telescope for Imaging X-rays (STIX) on-board SolO, in combination with radio observations at a broad frequency +range. We focus on establishing an association between the energetic electrons and the different HXR and radio emissions associated +with the multiple acceleration episodes. +Results. We have found that the flare was able to accelerate electrons for at least 20 minutes during the nonthermal phase observed in +the form of five discrete HXR pulses. We also show evidence that the shock wave has contributed to the electron acceleration during +and after the impulsive flare phase. The detailed analysis of EPD electron data shows that there was a time difference in the release of +low- and high-energy electrons, with the high-energy release delayed. Also, the observed electron anisotropy characteristics suggest +different connectivity during the two phases of acceleration. +Key words. type III radio emission; Energetic electrons; Solar flares +1. Introduction +Acceleration of solar energetic particles (SEPs) during eruptive +events may be associated with several different physical phe- +nomena and mechanisms, such as solar jets and flares, CMEs, +or shock waves (e.g. see Klein & Dalla 2017; Anastasiadis et al. +2019; Vlahos et al. 2019; Reames 2021), and they can accelerate +particles to energies ranging from a few tens of keV to several +GeV. The origins of SEPs measured in-situ have been a long- +standing debate. This is because it is difficult to distinguish be- +tween several possible processes and interpretations using in-situ +observations only near 1 AU. Previous studies suggest that both +flare- and shock-related physical processes can contribute to the +acceleration of SEPs (e.g., Kouloumvakos et al. 2015; Papaioan- +nou et al. 2016); however, it remains an open issue to quantify +the contribution of each process to each species and a broad en- +ergy range. Energetic electrons are thought to be primarily accel- +erated in the low corona, and the physical mechanisms respon- +sible for the acceleration of electrons may then be constrained +to reconnection, for example at solar jets (e.g. see Krucker et al. +2011; Glesener et al. 2012; Musset et al. 2020; Zhang et al. 2022) +or at current sheets forming at the wake of CMEs (Kahler & +Hundhausen 1992; Klein et al. 1999). It is not yet well under- +stood if shock waves can have any role in the acceleration at +high energies (i.e. >1 MeV) through a drift acceleration at shock +fronts (Ball & Melrose 2001). +While it is commonly believed that both protons and elec- +trons can be accelerated to very high energies at magnetic recon- +nection sites in the low corona, it is still not clear which condi- +tions can facilitate the escape of the accelerated particles from +the acceleration site to the interplanetary space. The direct injec- +tion and escape of the energised electrons can be described in +three scenarios, namely, the electrons released 1) by propagating +shock fronts (e.g., Kouloumvakos et al. 2022a), 2) CME interac- +Article number, page 1 of 20 +arXiv:2301.03650v1 [astro-ph.SR] 9 Jan 2023 + +A&A proofs: manuscript no. aanda +tion with the ambient magnetic field lines, 3) by open magnetic +field lines rooted directly to the active region that gives direct +access to interplanetary space (Masson et al. 2019). Depending +on the proximity of the acceleration site to open magnetic field +lines, the electrons can escape directly into the interplanetary +space via open magnetic field lines and usually manifest as type +III radio bursts (see, Reid & Ratcliffe 2014, for a review). +Type III radio emission is produced when beams of ener- +getic electrons resonantly generate Langmuir/slow-electrostatic +waves which are linearly (Krasnoselskikh et al. 2019; Tkachenko +et al. 2021; Jebaraj et al. 2022) or non-linearly (Ginzburg & +Zhelezniakov 1958; Cairns 1987; Melrose 2017) transformed +into electromagnetic waves at the plasma frequency and/or its +harmonics. Type III radio bursts are known to be associated with +flares, jets, and other solar eruptive phenomena. There is also +a very good connection of energetic electron events with type +III bursts at energies below 300 keV (e.g., Krucker et al. 2007, +2008; Klassen et al. 2011a,b, 2012, 2018). +On the other hand, efficient shock acceleration of electrons, +while possible, may be constrained to the shock drift acceler- +ation mechanism (SDA; Ball & Melrose 2001, and references +therein) in regions where a quasi-perpendicular shock geome- +try is present. However, the efficiency of energy gain from the +mechanism itself is rather limited since most electrons are trans- +mitted through the shock transition regardless of electron energy +when the shock geometry tends towards perpendicular to the up- +stream magnetic field (θBn ≈ 90◦). Evidence of shock waves ac- +celerating beams of energetic electrons is common during solar +eruptions and can be seen manifesting as drifting type II radio +emissions. Numerous studies of type II radio bursts have shown +that the emission most likely arises from the upstream regions of +the shock wave, implying an active acceleration process at the +shock wave (Krasnoselskikh et al. 1985; Thejappa 1987; Jebaraj +et al. 2021; Kouloumvakos et al. 2021) at multiple acceleration +sites in some events. Aurass et al. (1998) have suggested that +coronal shock waves are able to accelerate electrons to energies +considerably higher than the background thermal population. +A recent statistical study (Dresing et al. 2022) has suggested +that the acceleration of mildly-relativistic and relativistic elec- +trons correlate rather well with critical shock parameters close +to the Sun, namely, the shock strength (fast-magnetosonic Mach +number). Similarly, good correlations have been found for high- +energy protons as well (Kouloumvakos et al. 2019). This good +correlation for electrons also suggests that efficient acceleration +at shock regions with oblique and quasi-parallel shock geometry +may also be present. However, the mechanism of electron accel- +eration in oblique shocks follows a diffusive shock acceleration +mechanism (DSA; Bell 1978, and references therein), which is +highly dependent on the shock wave’s ability to accelerate ions +(for review; Treumann & Jaroschek 2008) and the subsequent +generation of upstream wave turbulence. The electrons may then +be trapped by the upstream waves and be accelerated through +a Fermi acceleration mechanism similar to that of the protons +(e.g., Tsytovich 1973; Vaisberg et al. 1983; Galeev 1984; Galeev +et al. 1995; McClements et al. 1997; Gieseler et al. 2000). +In this study, we investigate an SEP event observed on 9 Oc- +tober 2021, by Solar Orbiter (SolO; Müller et al. 2020). The +event was associated with a CME/shock wave and an M-class +flare. During the event, high-energy protons and electrons were +observed by multiple observers, such as, near-Earth spacecraft +and Parker Solar Probe (PSP; Fox et al. 2016). Our motivation is +to analyse what appears to be a second phase of electron accel- +eration as suggested by SolO SEP electron observations, in com- +bination with the complex radio observations that show multiple +stages of particle energization during and after the flare impul- +sive phase. We find that this is an interesting aspect of the event, +hence, our analysis is focused on establishing an association be- +tween the energetic electrons and the different radio emissions +during the impulsive phase of the flare and understanding the +origin of the apparent second phase of electron energization af- +ter the flare maximum. Specifics on the energetic proton obser- +vations for this event can be found in Lario et al. (2022). +This multi-faceted study is organised as follows. We start +with a brief introduction to the space-based and ground-based +instrumentation and an overview of the event in Section 2 and +3, respectively. In Section 4, we show observations of the so- +lar event with a special emphasis on the X-rays (Section 4.1), +the radio waves (Section 4.2), and the solar energetic electrons +(Section 4.3). In this section we also analyse the different obser- +vations. The results of our analysis and our conclusions about +the two apparent phases of electron energization are provided in +Section 5 and 6, respectively. +2. Instrumentation +For this study, we utilize data from instruments on-board SolO, +PSP, Solar TErrestrial RElations Observatory Ahead (STEREO- +A Kaiser 2005; Kaiser et al. 2008), SOlar and Heliospheric Ob- +servatory (SOHO; Domingo et al. 1995), Solar Dynamics Obser- +vatory (SDO; Pesnell et al. 2012), and the Wind (Harten & Clark +1995) spacecraft, as well as measurements from ground-based +instruments. Below we give a summary of the data used in this +study. +• Energetic Particle observations: From SolO, we utilize en- +ergetic particle measurements from different sensors of the +Energetic Particle Detector (EPD; Rodríguez-Pacheco et al. +2020) instrument suite, namely Electron Proton Telescope +(EPT), High Energy Telescope (HET), and the Suprathermal +Electrons and Protons (STEP), in the energy range from a +few keV to a few MeV for electrons. From STEREO-A, we +utilize SEP measurements from the High Energy Telescope +(HET; von Rosenvinge et al. 2008) and the Solar Electron +and Proton Telescope (SEPT; Müller-Mellin et al. 2008). In +addition, electron measurements from the 3DP (Lin et al. +1995) instrument on-board Wind have been used. +• Hard and soft X-rays observations: We utilize hard X-ray +(HXR) spectra and images from the Spectrometer/Telescope +for Imaging X-rays (STIX; Krucker et al. 2020) on-board +SolO and soft X-ray (SXR) observations from the Geosta- +tionary Operational Environmental Satellite (GOES; Garcia +1994). +• Radio observations: We utilize radio observations from both, +ground-based and space-borne instruments. For the inter- +planetary part of the dynamic radio spectrum we utilize ob- +servations from the Radio and Plasma Waves instrument +(RPW; Maksimovic et al. 2020) on-board SolO and from +the Radio Frequency Spectrometer (RFS; Pulupa et al. 2017) +part of the FIELDS electric antennas (Bale et al. 2016) on- +board PSP. Observations from legacy instrumentation such +as the SWAVES (Bougeret et al. 2008) instrument on-board +STEREO-A and the WAVES experiment on-board the Wind +spacecraft (Bougeret et al. 1995) are also employed. We also +utilize ground-based radio observations from the Yamagawa +radio spectrograph (9 GHz - 70 MHz, +Iwai et al. 2017) +and the e-Callisto network of radio telescopes (in particular, +ASSA (Astronomical Society of South Australia; 80 MHz +- 16 MHz), Benz et al. 2009). These observations cover a +Article number, page 2 of 20 + +Immanuel. C. Jebaraj et al.: Solar energetic electrons on 9 October 2021 +Fig. 1. The right panel shows a view of the heliographic equatorial plane from the north, in the Stonyhurst coordinate system, and the spacecraft +constellation on 9 October 2021 at 06:30 UT. The different coloured squares indicate the location of all observing spacecraft, namely, L1 (Earth, +green), STEREO-A (red), SolO (blue), and PSP (purple). The nominal Parker spiral (curved line) and line of sight (dotted lines) from the Sun to +each spacecraft is shown for each observer. The small back arrow and the dashed black line indicate the flare location and the reference Parker +spiral. The large black arrows help locate the in-situ omni-directional electron recordings at SolO and STEREO-A, which are presented in the two +rectangular panels on the left (note that STEREO-A/HET data is not omni-directional, but pointing almost perpendicular to the nominal Parker +spiral direction in the ecliptic plane; cf. Sec. 4.4.1). +broad range of the radio wavelengths, from the millimetric +to the decametric domain. +• Extreme ultra violet observations (EUV): We utilize observa- +tions from the Atmospheric Imaging Assembly (AIA; Lemen +et al. 2012) on-board SDO and the Extreme Ultra Violet Im- +agers (EUVI), part of the Sun Earth Connection Coronal and +Heliospheric Investigation (SECCHI; Howard et al. 2008) in- +strument suite on-board STEREO-A. +• Coronagraphs white light observations (WL): We utilize +coronagraphic observations in WL, from the Large An- +gle and Spectroscopic Coronagraphs (C2 & C3) (LASCO; +Brueckner et al. 1995) on-board SOHO and the two coron- +agraphs (COR1 & COR2), part of the SECCHI instrument +suite on-board STEREO-A. +3. Event overview +The event on 9 October 2021, was one of the first major erup- +tions of Solar Cycle 25 and was associated with an M1.6 flare +in GOES classification, a filament eruption, and a halo CME +originating from the NOAA AR 12882. The active region had +a βγ configuration of its photospheric magnetic field 1 and was +located at the central meridian (N20W01). Soft X-ray (SXR) +observations from GOES showed that the solar flare started +at 06:19 UT, peaked at 06:38 UT, and decayed to flux levels +between the maximum and the pre-flare background level at +06:53 UT. The flare was also observed in hard X-rays (HXR) +by the STIX instrument on-board SolO, which at that time was +at a heliocentric distance of 0.68 AU and was separated from the +1 based +on +the +classification +provided +in: +https://www. +solarmonitor.org/index.php?date=20211009®ion=12882 +Sun-Earth line by 15.2◦ to the east (see Figure 1). Hence, the +flare was close to the disk centre for both SolO and near-Earth +assets. Radio observations were also very rich during the event, +showing several complex spectral features. We provide details +about the X-ray and radio observations in Sections 4.1 and 4.2, +respectively. +Figure 1 shows an overview of the energetic electron obser- +vations from SolO (EPD-EPT) and STEREO-A (SEPT) that ob- +served the SEP event on 9 October 2021. All the spacecraft, and +in particular SolO and STEREO-A, clearly observed an intensity +increase of electrons at energies >1 MeV. The heliospheric view +on the right panel of Figure 1 shows the heliographic equato- +rial plane from solar north and illustrates the spacecraft loca- +tions close to the start of the solar event, on 9 October 2021 at +06:30 UT. The new solar mission spacecraft (SolO and PSP), +STEREO-A, and near-Earth spacecraft were closely positioned +in a quadrant and covered a narrow range of heliolongitudes of +about 50◦. During the SEP event, SolO (located at 0.68 AU), and +STEREO-A (located at 1 AU) were both trailing Earth for about +15◦ and ∼40◦, respectively. In the heliospheric view of Figure 1, +we also show the Parker spiral field lines connecting each space- +craft with the Sun. For the illustration of the spacecraft location, +we used the Solar-MACH tool2 (Gieseler et al. 2022). For each +spacecraft, we used a solar wind speed of about 350 km/s to cal- +culate the Parker spiral. We see that theinterplanetary magnetic +field (IMF) lines connecting STEREO-A and SolO with the Sun +are spatially close. We estimated the longitudinal difference of +the magnetic footpoints at the solar surface to be around 4.9◦. +From the electron recordings (left panel of Figure 1, we see that +at SolO there is a rapid rise of the electron flux that lasts for +2 https://solar-mach.github.io +Article number, page 3 of 20 + +180° +OMNI-DIRECTIONAL ELECTRONS +105- +r s MeV) +225° +135° +103. +SOLO EPD +Flux / (cm² sr +EPT 43.2-45.9 keV +EPT 62.7-67.3 keV +101. +EPT 93.4-101.1 keV +1.0 +EPT-142.3-154.1 keV. +EPT 218.1-237.1 keV +10-1 +HET 0.45-1.04 MeV +0.5 +HET 1.05-2.4 MeV +270° +.06 +105. +2021-10-09 +(cm² sr s MeV) +103. +06:30/UT +STEREO-A +SEPT 45.0-55.0 keV +STEREO A +101 +SEPT 105.0-125.0 keV +Earth +) /xn +SEPT 225.0-255.0 keV +BepiColombo +Parker Solar Probe + SEPT 375.0-425.0 keVL +tu +315° +Solar Orbiter +HET 0.7-1.4 MeVM +10-1. +field line connecting to +HET 1.4-2.8 MeV +ref. long. (vsw=350 km/s) +06:00 +07:00 +08:00 +00:60 +10:00 +11:00 + reference long. +Time [UT] of 2021-10-09 +.0A&A proofs: manuscript no. aanda +Fig. 2. (a) GOES soft X-ray fluxes shows the thermal emission of the +M1.6 flare. (b) STIX hard X-ray count rates in five broad energy bands. +The STIX times have been shifted to be consistent with the GOES ob- +servations from 1 AU. Results of the spectral fitting of STIX spectra +show: (c) the background-subtracted STIX count rate in the 25–50 keV +range, (d) the spectral index of the injected electrons, δ, (e) the low- +energy cutoff, ELC, and (f) the total injected electron flux, as well as +the fluxes above 50 and 100 keV. The five major nonthermal peaks are +indicated by red dashed lines. +about 15 minutes until the maximum, and then there is a second +more gradual increase after the maximum. This second increase, +however, was not observed by the closely connected STEREO-A +spacecraft. +4. Observations and data analysis +4.1. X-ray observations +Figure 2 (panels (a) and (b)) shows the GOES soft X-ray fluxes +in comparison with the STIX hard X-ray count rates in five wide +energy bands (note that 160 s have been added to the STIX times +to account for the light travel time difference between SolO and +GOES). While the count rates below 15 keV show the smooth +time evolution characteristics for the thermal bremsstrahlung +produced by hot plasma, the multiple spikes visible at higher +energies suggest several episodes of electron acceleration. To in- +vestigate this quantitatively, we have forward-fitted a time se- +ries of observed STIX count spectra with the combination of +an isothermal model and a thick-target nonthermal component +(Brown 1971), using the OSPEX spectral analysis tool3. +The fit results for the nonthermal component are shown in +Fig. 2 (panels (c), (d), (e), and (f)). Similarly to panels (a) and +(b) of the same figure, the times have been shifted to be con- +sistent with a viewpoint at 1 AU. The background-subtracted +count rates in the range of 25–50 keV (panel (c)), show at least +five major peaks, indicated by dashed vertical lines. The spec- +tral index δ of the injected electron flux, in panel (d), shows a +clear anti-correlation with the nonthermal count rates. This is +the well-known soft-hard-soft behaviour (e.g., Grigis & Benz +2004), and it is indeed present in each peak. Overall, the spec- +trum gets harder over time (soft-hard-harder evolution). This be- +haviour was observed in some solar flares (Grigis & Benz 2008), +particularly in those associated with interplanetary proton events +(Kiplinger 1995). Next, panel (e) shows the low-energy cutoff +ELC, which lies mostly in the range of 12–16 keV. This quantity +is rather ardous to constrain (note the large uncertainties), and +we can only determine the highest ELC that is consistent with +the data because the true cutoff is usually masked by the thermal +emission (see e. g. Warmuth & Mann 2020). Adopting this, panel +(f) shows the total injected electron flux, which reflects a lower +estimate for this quantity. Additionally, we show the fluxes above +50 keV and 100 keV in red and blue, respectively. Note that the +HXR count rate peaks are indeed associated with higher electron +fluxes, which are most clearly seen at higher energies. +To investigate electron acceleration in more detail, we per- +formed image reconstruction of the hard X-ray sources using the +STIX pixelated science data (cf. Krucker et al. 2020; Battaglia +et al. 2021; Massa et al. 2022). Figure 3 shows the evolution +of the thermal (6–10 keV, red contours) and nonthermal X- +ray sources (22–70 keV, blue contours) reconstructed with the +MEM_GE imaging algorithm (Massa et al. 2020). Countour lev- +els start at 20% of the maximum intensity. The image times cor- +respond to the five major nonthermal peaks. While the integra- +tion time was one minute for the thermal images, it varied be- +tween one and four minutes for the nonthermal peaks in order to +optimize counting statistics. The sources are overplotted on AIA +1600 Å images that have been rotated to conform with the STIX +viewpoint (see Battaglia et al. 2021). The inverted colour table +shows the flaring ribbons and kernels in dark. The UV images +show that the flare had an unusually complex morphology with +multiple ribbons and kernels, lying partly at right angles to each +other. +The X-ray sources were co-aligned with the rotated AIA im- +ages using the pointing information provided by the STIX As- +pect System (SAS; Warmuth et al. 2020), which has a nominal +positioning accuracy of ±4". We see that during the first two +nonthermal peaks, two HXR footpoints (FPs) are located at the +southern end of the eastern flaring ribbon and the western flaring +kernel. At the second peak, an additional weaker FP appears near +the central part of the eastern ribbon. At the third nonthermal +peak, the southeastern FP has faded away, and we have again the +classical two-footpoint configuration, which is also maintained +3 http://hesperia.gsfc.nasa.gov/ssw/packages/spex/doc/ +Article number, page 4 of 20 + +10-4 +a) +1-8 A +10~5 +0.5-4 A +/ m"] +M +10*6 +flux +GOES +10° +10 +10 +105 +4-10 keV +10-15 keV +104 +15-25keV +count rate +25-50keV +103 +50-84 keV +102 +XIIS +101 +L +10° +06:30 +06:40 +06:50 +07:00 +07:10 +07:20 +200 +25-50 keV +c) +150 +rate +100 E +count +50 +0 +8 +d) +7 +5盒 +3 +e) +[keV] +10 +ELc +5日 +0 +total +1035 +>50 keV +1034 +>100 keV +flux +1033 +electron +1032 +1031 +06:30 +06:36 +06:42 +06:48 +06:54 +9-Oct-2021 [UT]Immanuel. C. Jebaraj et al.: Solar energetic electrons on 9 October 2021 +Fig. 3. Flare evolution as seen in a series of SDO/AIA 1600 Å images. Due to the inverted colour table, the flare ribbons and kernels are dark. +The AIA frames have been rotated to conform to the view from SolO. The provided times represent the recording times of the AIA frames (UT at +Earth). STIX images showing the thermal and nonthermal X-ray sources are overlaid as red and blue contours, respectively. +during the subsequent two peaks. While the western FP remains +stationary, the eastern FP moves along the flare ribbon to the +northeast. Note that the eastern FP has nearly faded away in the +fifth peak, so that in contrast to the other panels, the nonthermal +contour lines shown here start already at the 5% level. +In an effort to visualise the complex STIX observations with +respect to the magnetic field topology of the active region, we +have performed a non-linear force free field extrapolation. The +details of the analysis are provided in Appendix. A. The HXR +sources and their movement correspond well with the western +part of the AR and the dominant footpoint rooted in the positive +polarity region. The shifting of the HXR footpoints is seen in the +south-eastern parts of the AR. +In summary, we conclude that the flare was able to accelerate +electrons from at least 06:28 to 06:54 UT, mostly in the form of +five discrete pulses. A change of magnetic topology appears to +have taken place between the first and third nonthermal burst, +shown by a significant shift of the nonthermal emission in the +eastern part of the flare. +4.2. Radio observations +In Figure 4, we show a composite dynamic radio spectrum +constructed from the observations of several ground-based and +space-borne instruments. Such a composite spectrum provides +uninterrupted high time and frequency resolution observations, +starting from microwave wavelengths that are generated at alti- +tudes very close to the chromosphere and up to the kilometric +wavelengths, correspond to the interplanetary space. The com- +bined observations indicate a wide variety of coronal processes +related to the acceleration and propagation of nonthermal elec- +trons. +The solar radio event is rich with different types of radio +emissions i.e., type II, III, radio bursts, IV continuum. In mi- +crowave wavelengths (9 GHz – 1 GHz) we observe a diffuse ra- +dio emission which is most-likely gyro-resonant in nature (e.g., +Bastian et al. 1998) and is emitted by near-relativistic electrons +(≈100 keV) trapped in the flaring loops (e.g., Nindos 2020). The +low-decimetric to decametric wavelengths (i.e., from ∼1 GHz to +∼20 MHz) are mostly dominated by plasma emission (e.g., type +II, III, IV radio emissions) produced by fast electron beams that +are associated with flares, propagating shock waves, and elec- +trons trapped within flare loops or in flux rope CMEs (McLean +& Labrum 1985). From the composite dynamic radio spectrum +of these wavelengths (Figure 4), we distinguish different type +III radio bursts, some of them associated with the HXR pulses, +multiple parts of type II burst that exhibit various fine structures, +and a type IV continuum. At hecto-kilometric wavelengths, the +interplanetary radio emissions associated with the event were ob- +served by all spacecraft namely, PSP, SolO, STEREO-A, Wind. +We observe two main type III radio burst groups and two patchy +parts of type II radio burst. +The event of 9 October 2021, was first observed in radio +wavelengths with the emission starting at around 06:30 UT. +The diffuse radio emission observed in microwave wavelengths +lasted for the entire duration of the flare and corresponds well +with the HXR pulses as shown in Fig. 2, from 06:30 to 06:54 UT. +At the beginning of the event, we observe the first type III ra- +dio burst (TIII-G1), which was associated with the first peak in +HXR during the flare impulsive phase. The first properly dis- +tinguishable signature (although faint) of G1 was observed at +06:31:40 UT, starting at ∼40 MHz. Apart from the TIII-G1 and +the microwave continuum, near the very beginning of the event, +we observe structured and narrowband spike-like emission at +around 350 MHz, at 06:30 UT. Similar type of emission was +also observed at higher frequencies, around 1000 MHz, at the +same time. The morphology and the apparent drift of the fea- +tures suggest that they could be precursors (Fárník et al. 2003; +Pohjolainen 2008) to the type II radio burst observed a few min- +utes later. +The first part of the type II (TII(1a)) radio burst was observed +at 300 MHz around 06:33 UT. This 2nd harmonic emission lane +(2fpe) shows rather patchy morphology. The fundamental lane +( fpe) of the same type II was observed later (06:34:20 UT) start- +ing at about 90 MHz, and it was considerably patchier than the +second harmonic and consisted of distinguishable fine structures. +The nature of the narrow-band patchy features indicates rather +localized source regions with rapidly changing plasma condi- +tions. Such morphological characteristics are common in the +case of metric-decametric type II bursts (e.g., Cairns et al. 2003; +Kouloumvakos et al. 2021; Jebaraj et al. 2021). The harmonic +component was observed at ∼250 MHz meaning that a shock +wave was formed at ∼1.4 R⊙ (applying a 2-fold Newkirk coronal +electron density model; Newkirk 1961). The coronal electron +density is much higher in the corona close to the flaring region +and at regions that are probably dominated by closed magnetic +field lines. The radio emission ended at around 06:42 UT, and at +23 MHz and 50 MHz for the fundamental and harmonic lanes, +respectively. +Recently, Ramesh et al. (2022) imaged the 80 MHz harmonic +component of TII(1a) at ≈ 06:38 UT, which corresponds to the +end of the inferred injection time of the low-energy electrons. +The source of TII(1a) according to this study was located close +to the southwest periphery of the flaring active region. This po- +sition roughly coincides with the direction of the strong EUV +wave expansion, toward the southwest direction from the source +region, where the open magnetic field lines were connected to +SolO and STEREO-A (Sec. 5). +Simultaneously with the TII(1a), we traced also a second, +more diffuse and broadband radio emission possibly also type II +Article number, page 5 of 20 + +AIA 1600 A - 9-Oct-2021 06:31:02.130 +AIA 1600 A - 9-Oct-2021 06:33:02.120 +AIA 1600 A - 9-Oct-2021 06:37:02.130 +AIA 1600 A - 9-Oct-2021 06:40:38.130 +AIA 1600 A - 9-Oct-2021 06:45:50.130 +500 +500 +500 +500 +500 +450 +450 +450 +450 +450 +400 +Y +350 +(0 +350 +350 +350 +350 +6-10 keV +-22-70 keV +300 +300 +300 L +50 +100 +150 +200 +250 +50 +100 +150 +200 +250 +50 +100 +150 +200 +250 +50 +100 +150 +200 +250 +50 +100 +150 +200 +250 +X (arcsec) +X (arcsec) +X (arcsec) +X (arcsec) +X (arcsec)A&A proofs: manuscript no. aanda +Fig. 4. The radio event associated with the flare/CME on 9 October 2021. Panel (a) shows the full radio spectrum from the Earth vantage point +including both ground-based and space-borne observations in the range of 10 GHz to 20 kHz. The different types of radio bursts are marked on the +spectra with their respective abbreviations. Panel (b) presents a zoomed in view of the decimetric-metric-decametric observations that showcase +different structured radio emission, such as herringbones and a stationary flare continuum. The details of the structured radio emission, i.e. type II +herringbones(TII(HB)) and very narrow-band wavy part of the type II (TII(W)), are presented in panels (c1) and (c2), respectively. +burst (TII(2a)). This emission lane had a comparable drift rate +to the one of TII(1a), and it was observed starting at high fre- +quency ∼600 MHz, at around 06:36 UT. Approximately at the +same time, also a continuum emission was observed, starting at +∼1 GHz (at around 06:31 UT). The continuum emission seems +to be superposed with the TII(2a) burst. Such a synchronous +appearance makes the separation of the two different types of +radio emission very difficult. A continuation of the broad-band +continuum was also observed after the cessation of TII(2a), but +with variable intensity. The intense part of the broadband contin- +uum with fine structures (marked in Fig. 4a,b as TIV) started at +06:47 UT. The stationary TIV extended from ∼600 MHz to 200 +MHz and consisted of intense broadband pulsation, extending +till about 08:00 UT. +The second and most intense group of TIII-G2 radio burst +was observed after the flare impulsive phase at around 06:37 UT, +starting at ∼70 MHz and corresponding well to one of the HXR +pulses. We find no clear evidence to confirm that the type III +bursts emanate from the type II burst, so called type II-associated +bursts (Cane et al. 1981; Dulk et al. 2000). However, concur- +rently with the appearance of the TIII-G2 radio burst a clear +brightening of a type II was observed. The distinguishable her- +ringbone features (TII(HB)) at the decametric wavelengths (40 +MHz fundamental emission) can be observed. TII(HB) was ob- +served together with TIII-G2 starting from 06:41 UT and marks +a period of very efficient electron acceleration at the shock wave. +It is possible that some of the electron beams generating TIII-G2 +were accelerated by the shock wave. We show the details of the +corresponding dynamic spectrum in Figure 4.(c1). The fast drift- +ing herringbone bursts originating from both sides of the type II +backbone are generated by the fast electron beams accelerated +in the upstream region of the shock wave. The geometry of the +shock wave in these regions is close to θBn ≈ 90◦, which is evi- +dent by the lack of a bright backbone (Stewart & Magun 1980). +The drift rate of the herringbone features and the irregular mor- +phology of the backbone may reflect the physical characteristics +of either the shock wave or the variations in the local plasma +frequency close to the shock wave. The spectral irregularities of +the backbone structure are somewhat to be expected when the +shock wave interacts with an electron-rich environment (e.g., a +streamer) and therefore a reverse drift into higher plasma fre- +quency. +A few minutes (∼5 min.) after the end of the TII(HB), we ob- +served a similar (wavy) type II-like burst (TII(W)). The TII(W) +burst starts at 06:50 UT, and as it is unclear if it is the decamet- +ric continuation of TII(2a) ,we distinguish it just as TII(W). A +zoomed-in view of TII(W) is presented in Figure 4.(c2), which +shows that the second harmonic of the emission was brighter +than the fundamental, which is opposite to the fundamental +brightening in the case of TII(HB). While TII(W) also appears +to have some herringbone-like features, they are not as clear as +those observed in TII(HB). Another important characteristic of +TII(W) is also the presence of a bright backbone together with +the herringbones, indicating a shock wave that is probably not as +perpendicular as TII(HB). +The two main parts of the type II radio burst, namely TII(1a) +and TII(2a), observed in the metric wavelengths were also ob- +served in the hectometer wavelengths. The hectometric counter- +Article number, page 6 of 20 + +Composite radio dynamic spectra +a) +Ic1) +b) +C2) +0.1 +MHz +TII(W) +TII(HB) +1 +TII(G1) +TII(G2) +TII(2b) +(1b) +TIV +Tll(2a) +b +1000 +6:35 +6:40 +6:45 +6:50 +6:55 +7:00 +100 +Il(1al +c1) +c2) +Pulsations +AII +1000 +4 +HB +H +STIX25-50keV +8 +6:42 +6:44 +6:46 +6:51 +6:53 +6:55 +6:30 +7:00 +7:30 +8:00 +Time (UT)Immanuel. C. Jebaraj et al.: Solar energetic electrons on 9 October 2021 +part of the two parts of the type II burst were also noticeably +patchier and bursty compared to their decametric counterparts. +The continuation of the two parts of the metric type II radio burst +was recorded by the space-based instruments and it is marked as +TII(1b) and TII(2b) in Fig. 4. +The TII(2b) was observed in the hectometer range starting +at 7:08 UT, which was about 12 minutes after the appearance +of the TII(W). The relative bandwidth of the hectometric type II +burst was comparable to that of the decametric, suggesting that +the source of the type II bursts may have been located in a rela- +tively small region of the shock wave (Schmidt & Cairns 2016; +Kouloumvakos et al. 2021; Jebaraj et al. 2021). Both bursts, +namely, TII(1b) and TII(2b) come to a cessation at 07:07 UT +(6 MHz) and 07:40 UT (4.5 MHz), respectively. +Around the same time that the type II radio emission ap- +peared in the hectometric wavelengths, we continued to observe +also the TIV with sporadic broad-band radio pulsations in the +metric wavelengths centred at 150 MHz. This emission was most +likely produced by electrons trapped in post-flare loops. +Hecto-kilometric observations of the two type III groups, +namely, TIII-G1 and TIII-G2 were recorded by all available +space-borne observers. At kilometric wavelengths, both TIII-G1 +and G2 were also observed together with Langmuir waves at +SolO and STEREO-A. Langmuir waves are fundamental plasma +waves that are generated by the electron beam, which can be +subsequently converted into type III radio emission observed at +large angles (Jebaraj et al. 2022). In-situ Langmuir waves are +an indication of the electron beams propagating at close prox- +imity to the observing spacecraft. Despite the fact that all the +spacecraft observed the different type III bursts, only STEREO- +A and SolO observe Langmuir waves. This confirms the passage +of the type III generating electron beams through the position +of the spacecraft and indicates that the magnetic connectivity of +the two spacecraft with the electron beam was most likely simi- +lar. On the other hand, PSP and Wind did not observe Langmuir +waves, and this indicates that the electron beams did not prop- +agate in their vicinity. An in-depth analysis of the directivity of +both TIII groups is presented in Appendix B. +4.3. Remote sensing observations in EUV and WL +The solar event on 9 October 2021, was associated with a CME +and a large-scale propagating coronal wave that was observed in +both EUV and WL. While there has been an extended debate on +the physical nature of such large-scale coronal disturbances (for +a review, see Warmuth 2015), they are now generally interpreted +as signature of fast-mode waves or shocks (e.g. Long et al. 2017). +The expansion of the EUV wave was first observed by SDO/AIA +at 06:30 UT. Figure 5 presents SDO/AIA running difference im- +ages at 193 Å, from different times of the EUV wave evolu- +tion. Due to the presence of a small coronal hole and a streamer +south of the flaring active region, the evolution of the wave was +rather constrained and deformed in the low corona. The pres- +ence of pre-existing magnetic structures such as coronal holes +and streamers can affect the propagation and characteristics of +large amplitude waves (Vršnak & Luli´c 2000a,b). This mainly +occurs because, in these regions, there is a rapid change in the +characteristic speed of the medium (e.g., the fast-magnetosonic +speed). The evolution of the low-coronal EUV wave shown in +Fig. 5 was observed to be rather strong in the southwest quad- +rant away from the eruptive source region. Such an intense evo- +lution of the EUV wave in this direction was also accompanied +by a propagating pressure wave at higher altitudes, which was +observed by the coronagraphs. +We utilise the information from radio observations in tandem +with EUV images to understand the evolution of the pressure +wave and the formation of a shock wave in the low corona. The +presence of multiple type II radio bursts (described in Sec. 4.2) +suggests that a shock wave was present very early during the +event at multiple locations. TII(1a) was observed at a lower fre- +quency, while TII(2a) was observed at a much higher frequency +3 minutes later (≈ 06:36 UT, Sec. 4.2). Such a disparity in start- +ing frequencies suggests that the source of TII(1a) was possibly +closer to the leading edge of the pressure wave, while the source +region of TII(2a) could have been located closer to the flanks. +Near the start of both TII(1a) and TII(2a), the propagation and +interaction of the wave with closed field magnetic structures at +the periphery of the active region most likely led to favourable +conditions for electron acceleration and the subsequent genera- +tion of type II radio emission in regions close to the apex and the +flanks of the wave (e.g., Kong et al. 2016, 2017; Kouloumvakos +et al. 2021). +After 06:41 UT, the wave evolution was rather strong in the +southwest direction (Fig. 5a). Close to the start of the TIII-G2 +and also TII(HB), we observed an interaction of the EUV wave’s +southwest flank with open field lines that are probably part of +a streamer. In Fig. 5a, we show the open magnetic field lines +that connect to STEREO-A, SolO, and Earth. These field lines +were derived using the Potential Field Source Surface (PFSS) +model (Schrijver & De Rosa 2003) and Air Force Data Assim- +ilative Photospheric Flux Transport (ADAPT) magnetograms +(Arge et al. 2010). The interaction of the pressure wave with the +magnetic structures in this region was most likely an important +catalyst for efficient electron acceleration that we observed in +radio wavelengths (TIII-G2 and TII(HB) in Fig. 4c1) and in-situ +energetic electrons observed by SolO and STEREO-A. Further- +more, during the time of these interactions, both spacecraft were +connected to the EUV wave. +When the TII(HB) first appears in the radio spectrum at +06:41 UT, we observe that the EUV wave propagated past the +streamer towards the boundaries of a coronal hole southwest of +the active region. This area is marked roughly by the red ar- +rows in Fig. 5b at 06:43 UT. When the wave interacted with +this coronal hole, part of the wave was reflected by the coro- +nal hole boundary at ∼06:50 UT (Fig. 5c). A faint part of the +wave transmitted across the coronal hole boundary and also re- +fracted into regions where it was more favourable for the shock +to exist, i.e., higher in the middle corona where the local speed +of the medium decreases (Uchida et al. 1973; Warmuth & Mann +2005). Due to the low density of the coronal hole region and +the high magnetic field strength (increased characteristic speed +of the medium), the large amplitude wave may propagate faster +and also dampen at an increased rate. During and well after this +part of the interaction at the coronal hole boundary, we observed +the TII(W) presented in Fig. 4.c2. +In +WL, +the +pressure +wave +was +first +observed +by +SOHO/LASCO C2 coronagraph at 07:12 UT as a halo-shaped +coronal wave propagating higher in the middle solar corona +along the position angle (PA) 263◦ (i.e., in the south-west quad- +rant). This part of the WL shock wave corroborates well with +the fastest component of the EUV wave as we discussed earlier. +The observations also validate the aforementioned refraction of +the pressure wave higher in the corona, where there may exist +more favourable conditions for the formation of a shock wave +(e.g., lower Alfven speed). As the event originated close to the +central meridian, the pressure wave was observed as a halo event +Article number, page 7 of 20 + +A&A proofs: manuscript no. aanda +Fig. 5. The evolution of the EUV wave as observed by SDO/AIA 193 Å at three time instances. Panel (a) shows the EUV wave close to the SXR +peak time. The open magnetic field lines connecting to both SolO, STEREO-A, and Wind are drawn in blue, red, and green, respectively. The red +line across the Sun face represents the heliospheric current sheet. Panels (b) and (c) show the EUV wave expansion at 06:43 UT and 06:51 UT +roughly corresponding to TII(HB) and TII(W). The red arrows indicate the region of interest where the wave also undergoes reflection at the +coronal hole boundary. The blue, red, and green markings in (b) and (c) represent the spacecraft connections shown in (a). +encircling the solar disk from the Earth’s point of view. Due to +projection effects, tracking the shock wave becomes increasingly +difficult at later times in both LASCO C2 and C3 coronagraphs. +At STEREO-A, the event was observed ∼40◦ closer to the +west limb. The pressure wave was first observed in the COR1 +coronagraph at 06:46:18 UT and in the COR2 coronagraph at +07:08:45 UT. At the time when the pressure wave enters the +COR1 field of view, all the observed radio emission was in the +decametric wavelengths (mid-coronal plasma) and the TII(1a), +TII(HB), and TIII-G2 were near their cessation. This shows +that the most efficient electron acceleration took place when the +shock wave was still located low in the middle corona. The pres- +sure wave evolved fastest along PA 266◦ which suggests that this +was almost the same leading-edge identified in the LASCO C2 +observations. The wave was highly deformed by the presence +of two streamers at PA 230◦ and at PA 280◦ further suggest- +ing the interaction between the pressure wave and the various +density structures on the Sun. These interactions may also have +contributed to the patchy and bursty yet continuous observations +of the TII(1b) and TII(2b) at hectometric wavelengths until their +cessation at 07:07 UT and 07:40 UT, respectively. The kinemat- +ics of the shock wave and their association with TII(1a), TII(ab), +TII(2a), and TII(2b) are presented in Appendix C. +4.4. Energetic electrons +4.4.1. Pitch-angle distributions +Figure 6a shows the electron event observed by SolO/EPT in +the 40–50 keV energy channel. The top panel, which shows the +intensities as measured by the four viewing directions of EPT, +reveals a strongly anisotropic event starting at ∼6:40 UT fol- +lowed by an isotropic phase starting around 7:15 UT. The second +panel shows the pitch-angle coverage of EPT’s four viewing di- +rections, which is ideal and exceptionally stable over the course +of the event. The good pitch-angle coverage allows not only to +determine the strong first order parallel anisotropy (shown in the +bottom panel) in the early phase of the event, but also to verify +that the later isotropic phase is real and not caused by poor-pitch +angle coverage. +Figure 6b shows the electron event at 1.1–2.4 MeV measured +by SolO/HET, which provides the same four viewing directions, +allowing us to determine the pitch-angle distribution also at MeV +energies. The initial phase of the MeV-electron event shows a +significantly smaller anisotropy. However, the time series still +reveals a two-component picture with the later, almost isotropic +phase starting like the one observed by EPT around 7:15 UT. +A smaller anisotropy for higher electron energies is expected be- +cause of the energy-dependence on pitch-angle scattering, which +is stronger for higher energy electrons than for the lower en- +ergy ones (Dröge 2000; Agueda et al. 2014; Strauss et al. 2020). +However, in contrast to the low energy electrons (Fig. 6a), the +intensity level of the two components is rather similar at MeV +energies, with the later component being even slightly more in- +tense. At 40–50 keV (Fig. 6a), the first, anisotropic component +is much more intense than the second, isotropic component, in- +dicating a softer spectrum of the first component. This is con- +firmed by analysing the peak spectra of both components (see +Section 4.4.2). +Figure 6c shows the 45–55 keV electron event observed in +the four viewing directions of STEREO-A/SEPT. Because the +spacecraft was put upside down after its superior solar conjunc- +tion in 2014, the SEPT sun and anti-sun sectors do not point +anymore along a nominal Parker spiral field line but perpendicu- +lar to it. This causes often non-ideal pitch-angle coverage, which +is, however, not too bad during the electron event of 9 October +2021, as shown by the 2nd and 3rd panels of Fig. 6c. The onset +of the event is observed by the sun sector of SEPT. However, +during the rise phase of the event, the magnetic field direction +changes at ∼7 UT, so that the anti-sunward propagating beam +is then observed best in the anti-sun telescope. We note that the +pointing of the SEPT instrument has changed after the superior +conjunction in 2015, since when the spacecraft was put upside +down. Therefore, the north and south telescopes are swapped +and the sun and anti-sun telescopes point perpendicular to the +nominal Parker spiral (cf. Fig. 8 in Badman et al. 2022) At the +time of the onset the pitch-angle coverage is ideal showing a +smaller anisotropy than at SolO. The rise phase of the event also +shows a smaller anisotropy than observed by SolO, with no in- +dication of two distinct components. However, this could be po- +Article number, page 8 of 20 + +AIA 193 A 2021-10-09 06:37:04 +AIA 193 A 2021-10-09 06:43:04 +AIA 193 A 2021-10-09 06:51:04 +1000" +1000" +b) +1000" +a +C +Helioprojective Latitude (Solar-Y) +500" +500" +500" +.0 +-500" +-500" +-500" +-1000" +-1000" +-1000" +-1000" +-500" +0" +500" +1000" +-1000" +-500" +0" +500" +1000" +-1000" +-500" +0" +500" +1000" +Helioprojective Longitude (Solar-X) +Helioprojective Longitude (Solar-X) +Helioprojective Longitude (Solar-X)Immanuel. C. Jebaraj et al.: Solar energetic electrons on 9 October 2021 +Fig. 6. In-situ electron recordings by multiple observers. Energetic electron event observed by SolO/EPT (a) and HET (b). The top panels of (a) +and (b) show the intensities of 40–50 keV and 1.1–2.4 MeV electrons as observed by the four different sensors of the EPT and HET instruments. +The second panels show the corresponding pitch-angle coverage, and the third panels show the pitch-angle dependent intensities with intensity +level marked by colour coding. The bottom panels show the first order anisotropies. Energetic electron event observed by STEREO-A/SEPT in the +45–55 keV energy channel (c) and Wind/3DP (d) in the 50–82 keV channel. The panels have the same content as for (a) and (b), but (d) shows a +longer time period. +tentially masked by the non-ideal pitch-angle coverage during +the rising phase of the event. Unfortunately, STEREO-A/HET +provides only one viewing direction, which is aligned with the +pointing of SEPT-sun so that we cannot infer the anisotropy at +MeV energies for STEREO-A. +Sectored electron observations by Wind/3DP are shown in +Fig. 6d with a longer time period than shown in the other pan- +els. The event is much more gradual and less intense. How- +ever, the initial phase shows some anisotropy, followed by a sec- +ond step with less anisotropy. At a first glance, this might ap- +pear like the same two-component picture as observed by SolO, +but the time periods are significantly different. While the first, +anisotropic component lasts about 30 min at SolO, it has a du- +ration of about five hours at Wind so that a one-to-one corre- +spondence is unlikely. The second increase coincides with the +crossing of a magnetic sector boundary and is therefore likely +related with a change of magnetic connectivity rather than with +a new injection at the Sun. +4.4.2. Energy spectra observed by SolO +SolO is the only spacecraft that clearly observes a two compo- +nent electron event as described above. Figure 7 shows peak +electron spectra observed by the three EPD instruments STEP +(orange), EPT (red), and HET (brown). For EPT and HET, we +use the sunward-looking telescopes that align with the viewing +direction of STEP and cover the usual incoming direction of +SEPs. The figure shows a spectrum for each of the two compo- +nents, with circles (squares) denoting the first (second) compo- +Article number, page 9 of 20 + +a) +b) +Solar Orbiter (R=0.68 au) +Solar Orbiter (R=0.68 au) +0.0406 -0.0497 MeV +EPT Electrons +MeV) +1.0530-2.4010MeV +HET Electrons +105 +sun +sun +100 +asun +asun +103 +north +north +s)/ +south +south +180 +180 +Pitch angle / +HET +135 +Pitch angle / +135 +sun +sun +90 +asun +06 +asun +45 +north +45 +north +south +south +0 +0 +180 +180 +MeV) +Pitch angle / ° +MeV) +135 +105 +135 +100 +06 +104 +06 +45 +103 +45 +10-1 +0 +0 +2 +anisotropy +0 +0 +-2 +-2 +10-09 +10-09 +10-09 +10-09 +10-09 +10-09 +10-09 +10-09 +10-09 +10-09 +10-09 +10-09 +05:00 +06:00 +07:00 +08:00 +00:60 +10:00 +05:00 +06:00 +07:00 +08:00 +00:60 +10:00 +Date / Time in year 2021 +d) +Date / Time in year 2021 +c) +STEREO A SEPT +Wind/3DP +MeV) +45-55keV +Electrons +66 keV e +104 +sun +asun +102 +102 +north +101 +W +/(s +south +180 +180 +135 +sun +135 +90 +asun +90 +north +45 +south +45 +180 +MeV) +18 +/(s cm2 sr MeV) +135 +135 +103 +104 +06 +90 +102 +cm2 +45 +45 +101 +0 +0 +anisotropy +2 +2 +1st order +1st order +0 +0 +-2 +-2 +10-09 +10-09 +10-09 +10-09 +10-09 +10-09 +10-09 +10-09 +10-09 +10-09 +10-09 +10-10 +05:00 +06:00 +07:00 +08:00 +09:00 +10:00 +04:00 +08:00 +12:00 +16:00 +20:00 +00:00 +Date / Time in year 2021 +Date / Time in year 2021A&A proofs: manuscript no. aanda +Fig. 7. Electron peak intensity energy spectra determined for the two SEP components of the electron event observed by SolO. +nent. The light-grey points represent the pre-event background +spectrum that has been subtracted from each of the two spectra. +Points with dark-grey colour represent energy bins that did not +show a significant increase above the background level (HET) +or which were contaminated by ions (STEP). Both spectra have +been fit with a broken power law using the methods described in +Dresing et al. (2020); Strauss et al. (2020). The resulting spectral +indices below and above the break as well as the break energy are +provided in the figure legend. We only use part of the STEP en- +ergy channels in the fit to avoid fitting the upturn in the very low +energy range observed by STEP, which could be caused by mix- +ing with another low-energy event. We also exclude the first en- +ergy channel of HET from the fit because of instrumental effects +that lead to too low intensity measurement, which has been fixed +in a patch uploaded to the spacecraft only after this event. Com- +pared to the first component, which has a spectral index above +the break energy of (δ2 = −3.64±0.18), the spectrum of the sec- +ond component is clearly harder (δ2 = −2.67±0.19). At energies +≳ 200 keV, the spectra even intersect, meaning that the second +component is more intense than the first one at these energies. +This suggests that the second component is formed by a distinct +particle injection, rather than being a part of the first compo- +nent. Furthermore, the significantly harder spectrum of the sec- +ond component suggests a more efficient acceleration process. +4.4.3. Release times +The electron event is well observed at SolO, especially in the +near-relativistic range covered by EPT. We determine the onset +time at the spacecraft of each energy channel using a statisti- +cal quality control scheme that is designed to decide when the +quality of the monitored variable changed from being in con- +trol to out of control. There are many different quality control +schemes, and cumulative sum (CUSUM) schemes are used in +many industries due to their capability to give an early warn- +ing of changes in the monitored process (Page 1954). A tradi- +tional CUSUM method assumes that the variable in question +is normally distributed, and in the case of the monitored vari- +able having a Poisson distribution, the quality control scheme +used should be the Poisson-CUSUM (Lucas 1985; Huttunen- +Heikinmaa et al. 2005), which we apply here for the determi- +nation of the onset times. +For the parameters of the method we use the mean and stan- +dard deviation of intensity measurements in the chosen pre-event +background. We average the original measurements from 1 sec- +ond resolution to 30 second resolution, in order to minimise the +effects of counting noise. Furthermore, to avert false positives in +the determination of onset times, we demand that at least 60 con- +secutive out-of-control signals must be counted by the method +before identifying the onset of the event as the first of these sig- +nals. This limit corresponds to 30 minutes of intensity measure- +ments exceeding the threshold of Poisson-CUSUM method. +Using these onset times, we perform a velocity dispersion +analysis (VDA, e.g., Vainio et al. 2013) to infer the common so- +lar injection time of the observed electrons. Figure 8 shows the +onset times as a function of the inverse unit-less speed (1/β = +c/v) of the electrons. Blue symbols correspond to the onset times +observed in the different energy channels of EPT, which is capa- +ble of observing electrons in the energy range 31.2–471.4 keV, +and red symbols mark HET observations, which measures elec- +trons in the MeV range. For some channels the onset could not +be resolved, which is why there are fewer data points in the plot +than there are total channels in EPT and HET combined. +The VDA method assumes that particles of all energies were +injected at the Sun into the interplanetary space at the same time +and that their propagation path length is identical. Consequently, +the onset times as a function of inverse beta should align them- +selves to a line with a rising slope. This slope then allows to de- +termine the length of the path travelled by the electrons, and the +intersection with the vertical time axis tells the time of the com- +mon particle injection. The grey lines in the background of Fig. 8 +represent slopes that one would get with the nominal Parker spi- +ral path length of 0.736 AU, assuming a solar wind speed of +vsw = 360 km/s and taking into account SolO’s heliocentric dis- +Article number, page 10 of 20 + +61=-0.39±0.22 +62 = -3.64±0.18 +α= 12.48 +Eb = 41.0 +106 +±2.0 keV +61=-1.14±0.09 +62 = -2.67±0.19 +α= 6.17 +oreg +Eb = 53.5 +104 +±6.0 keV + sr MeV) +STEP 1st comp. +EPT sun 1st comp. +HET sun 1st comp. ++ +not significant +cm² +102 +申 +STEP 2nd comp. +串 +EPT sun 2nd comp. +串 +HET sun 2nd comp. +申 +not significant +100 +10-2 +10-2 +10-1 +100 +101 +Energy (MeV)Immanuel. C. Jebaraj et al.: Solar energetic electrons on 9 October 2021 +tance. As can be seen in Fig. 8, the onset times do not follow this +trend, nor do they form a straight line. Instead, the higher energy +channels of EPT show more and more delayed onsets. +Above an energy of about 142 keV, it becomes unreasonable +to include further data points into the VDA, since the general +trend from that energy channel upward are more and more de- +layed onsets. We therefore perform a fit only to the first 18 EPT +lowest energy channels (shown by the green line). However, this +results in a path length of L=0.56 AU, which is nonphysical (i.e., +too short) since the spacecraft’s radial distance is 0.68 AU. We +therefore apply another fit only to the 10 lowest energy chan- +nels (31.2–62.7 keV), which is shown by the orange line. This fit +yields a reasonable path length of L = 0.845 ± 0.380 AU and an +inferred injection time at the Sun for electrons of these energies +at 6:25:46 UT ± 476 s. +Like for the high energy channels of EPT, the onset times of +MeV electrons as observed by SolO/HET (red points in Fig. 8) +are also systematically delayed and do not fit the velocity dis- +persion trend as indicated by either fit. This suggests a delayed +solar injection with respect to the lower energy electrons. We +apply a time shift analysis (TSA, e.g., Vainio et al. 2013) to the +onset time of the 1.05–2.41 MeV energy channel using a nomi- +nal Parker spiral length of L = 0.736 AU and a solar wind speed +of 360 km/s in order to infer the solar injection time of these +electrons. This channel was selected because it has the clear- +est and most resolvable onset. We obtain an injection time of +06:35:56 UT at the Sun, which is about 10 minutes later than that +for the ∼30–60 keV energy electrons using TSA. For same low +energy channel, using the path length determined by the VDA +(L = 0.845 AU), we only obtain a one minute earlier injection +time. This strongly suggests the delayed solar injection of the +high-energy electrons compared to the low-energy electrons. An +application of the TSA is also presented in Fig. 9 for a number +of energy channels assuming a path length larger than normal +(L = 0.85 AU). +We also performed a VDA analysis for STEREO-A. From +the onset times determined from the STEREO-A/SEPT electron +channels, a trend similar to SolO/EPT arises: The onset times +in the highest channels, namely those above 195 keV, are con- +sistently more delayed. On the other hand, a fit to the lower en- +ergy channels in the range 45–195 keV yields a nonphysical path +length of L = 0.53 AU. If we instead choose the onset time as the +moment when the measured electron intensity reaches 1% of the +event peak intensity, then a fit to the same energy range yields a +longer, but still nonphysical path length of L = 0.72 AU. If we +instead use TSA on onset time of the 45–55 keV channel to infer +the solar injection time, we get t j = 06:28:50, assuming the nom- +inal Parker spiral arm for a solar wind speed of 390 km/s. Simi- +larly, for 0.7–1.4 MeV electrons measured by STEREO/HET we +obtain a solar release time of t j = 06:47 UT, almost 20 minutes +later than that of the low-energy electrons. Table 1 summarises +the results of the VDA and TSA. It shows the VDA result for +SolO only for the more reasonable VDA fit and also results of +similar TSA analyses done for STEREO-A SEPT and HET (not +shown here in a figure). +5. Discussion +In this study, we have analysed an event on 9 October 2021, +which was associated with an electron rich event in-situ and was +also observed by multiple wavelength remote sensing observa- +tions. In our analysis, we have considered observations from X- +rays, microwaves, radio waves, EUV, WL, and in-situ measure- +ments to understand the solar sources of this electron event. We +Fig. 8. Velocity dispersion analysis based on onset times of SolO EPT +and HET electron channels. The horizontal axis shows the inverse of +the average unitless speed of electrons as observed in each channel. +The vertical axis presents the determined onset time for each energy +channel. Onset times observed by EPT (HET) are marked in blue (red). +Horizontal error bars represent the width of the energy channels, and +vertical error bars represent the time resolution used to determine the +onsets (30 seconds). The orange and green lines are linear fits to the +lowest 10 and 18 energy channels of EPT, respectively. The grey lines +represent the slope corresponding to a path length matching the nominal +Parker spiral length at SolO’s radial distance. +also utilised the full capabilities of the Solar Orbiter (SolO) mis- +sion. The many features of the event are summarised with the +use of a timeline inferred from both remote sensing and in-situ +observations (Fig. 10). +The in-situ observations of energetic electrons by SolO show +distinct phases of electron energization that are indistinguishable +at other spacecraft. More specifically, the energetic electrons +observed by SolO show two increases in the electron intensity +within two hours. Additionally, the electrons’ anisotropy shows +clearly that there are two distinct phases (Sec. 4.4.1), namely, +a first anisotropic phase, which is observed immediately after +the onset of electrons in the spacecraft, and a second mostly +isotropic phase that is observed after the first peak in the in-situ +electron intensities. Throughout the Discussion, we will term +these two phases observed by SolO as "Phase 1" and "Phase 2". +A further analysis of "Phase 1" and more specifically the elec- +tron release times shows that the low- and high-energy electrons +are released at different times. +We also note that such a distinction of two phases was not +recorded for the electrons observed by STEREO-A and Wind. +The two peaks in the electron intensity that were observed at +SolO were also not seen by STEREO-A and Wind. In the fol- +lowing subsections, we will discuss the two different phases as +characterised by the anisotropy that was observed by SolO and +their possible acceleration mechanisms at or near the Sun. +5.1. Phase 1 +The first phase extends from immediately after the first electrons +arriving at SolO, until 07:14 UT, when the initially peaked elec- +tron anisotropy vanishes. During the first phase, we inferred that +the low energy electrons (≲ 142 kV) were released ∼5–10 min- +Article number, page 11 of 20 + +VDA,SOLOEPT+HETelectrons,2021-10-09 +06:55 +06:50 + (HH:mm) +Onsettime +06:45 +F +06:40 +H +06:35 +1.00 +1.25 +1.50 +1.75 +2.00 +2.25 +2.50 +2.75 +3.00 +1/β +nominal pathlength:0.736AU +EPT +fitto thelowest10 EPT channels:L=0.845AU; t inj=06:25:46 +HET +fittothelowest18EPTchannels:L=0.561AU;tinj=06:32:04A&A proofs: manuscript no. aanda +Table 1. Electron onset and inferred solar injection times. +Observer / +Energy +Onset time +Inferred injection +Path length +Method +Instrument +(keV) +at spacecraft (UT) +time at Sun (UT) +(AU) +SolO / EPT +31.2 - 62.7 +- +6:25:46 ± 0:07:56 +0.845 ± 0.380 +VDA +SolO / HET +1053 - 2410 +6:41:15 ± 0:00:30 +6:34:56 ± 0:00:30 +0.736 +TSA +STEREO-A / SEPT +45.0 - 55.0 +6:50:58 ± 0:01:00 +6:28:50 ± 0:01:00 +1.096 ± 0.1 +TSA +STEREO-A / HET +700 - 1400 +6:57:00 ± 0:01:00 +6:47:18 ± 0:01:00 +1.096 ± 0.1 +TSA +Fig. 9. Composite spectrum of all available radio observations from the Earth vantage point (9 GHz - 10 kHz) together with STIX nonthermal (25 +- 50 keV) count rates (top panel), and the in-situ electrons observed at SolO (bottom panel). The SolO electron observations from the sunward tele- +scope are time-shifted using the nominal path length and corrected for the travel time of light to make them comparable to the radio observations. +The inferred injection time of the low-energy (high-energy) electrons are marked with the dotted (dot-dashed) line. With the pink (blue) shaded +area we mark the release times using path length values larger than the nominal. For ∼0.85 AU, the earliest release of the low-energy electrons is +near the start of the first type III. The dashed line indicates the peak of the electron intensity for the ∼55 keV electron channel. +utes earlier than the more energetic electrons (Sec. 4.4.3). Given +this time difference, we can distinguish several solar features as- +sociated uniquely to each release. HXR observations show dis- +tinct pulses around the electron release and radio observations +show the escape of electrons to open field lines and the forma- +tion of multiple strong shock regions during this phase. +An anisotropic early phase of the event is not only ob- +served at SolO, but also at other observing spacecraft, namely, +STEREO-A and Wind (Fig. 6). Such a consistent increase in +anisotropy among multiple observers may indicate a common +acceleration/release process. It is unclear whether the same event +was observed at Wind, since the electrons arrive an hour after +their arrival at SolO or STEREO-A. +The intensity-time profiles of the in-situ electrons recorded +by SolO during this phase were impulsive, highly anisotropic +from the sunward direction, and exhibited a typical power- +law energy spectrum with a hard-soft spectral index (Fig. 7). +This broken power-law energy spectrum of the in-situ electrons +has been previously suggested to be due to beam propagation +through inhomogeneous plasma (Krucker et al. 2008; Kontar +& Reid 2009). Additionally, high anisotropy for this phase was +generally observed by all spacecraft within the 50◦ heliolongitu- +dinal range, which suggests that the electron injection took place +low in the corona and at regions well connected to most of the +observers. +Low-energy electrons +The inferred injection time of the low-energy electrons is +∼06:32 UT ±7:30 min based on the VDA and ∼06:34 UT using +TSA at ∼42 keV electrons (both converted to the spacecraft time +frame). A number of flare- and shock-associated features were +Article number, page 12 of 20 + +Composite Dynamic Radio Spectrum +10-1 +Frequency (MHz) +100 +101 +102 +103 +STIX: 25-50 keV +106 +EPT 0.0312 -0.0348 MeV +EPT 0.0406 - 0.0432 MeV +Electron flux +EPT 0.0533 -0.0580 MeV +EPT 0.0731 - 0.0788 MeV +103 +EPT 0.1011 -0.1109 MeV +EPT0.1423-0.1541MeV +EPT0.1995-0.2181MeV +EPT0.2817-0.3061MeV +HET0.4533 +-1.0380MeV +HET1.0530-2.4010MeV +100 +10-3 +06:00 +06:30 +07:00 +07:30 +08:00 +TimeImmanuel. C. Jebaraj et al.: Solar energetic electrons on 9 October 2021 +Fig. 10. Events during the first phase leading to the first in-situ component. (a) Timeline of the features associated with the two different electron +injections during the first phase. The exact times of the observational features are marked on the timeline bar. The gradient on the bar is red to +blue, indicating the transition between the low and high energy releases close to the Sun. The long-lasting radio features are marked below the +timeline. (b) Cartoon representation of the first phase. The coronal shock wave during the low-energy release (t=1) and high-energy release (t=2) +are represented by the black arcs. The open field lines from the periphery of the active region are represented by the green lines, while those from +the adjacent coronal hole are represented by blue lines. The closed field lines in the AR are denoted by th red coloured lines. The abundance of +energetic electrons are denoted by the small grey dots. The transparent grey rectangle is zoomed into in panel (c) and shows the shock wave’s lateral +expansion into nearby open magnetic field lines. The shock normal is denoted by the arrow and the ˆn. The perpendicular and near-perpendicular +geometry is represented by θBn ≈ 90◦ and θBn < 90◦. The electrons accelerated at the shock front on either side of the θBn ≈ 90◦ part of the shock +wave are marked by the spirals along the magnetic field lines. More details provided in the text. +observed during this time. The HXR observations presented in +Sec. 4.1 show distinct pulses around the electron release time, +which were also observed in microwaves along with a charac- +teristic diffuse gyro-resonant component indicating the presence +of trapped electrons. The analysis of the HXR photon spectrum +also shows that electrons were routinely accelerated to energies +of ∼10 – 100 keV, suggesting a flare-related contribution to the +low-energy electrons observed in-situ. +The inferred injection time of the low-energy electrons +seems to be associated with the first two HXR pulses and the +TIII-G1. The TIII-G1 was probably generated by electron beams +released from the flare site. Space-borne radio wave observations +of the TIII-G1 reveal that the source electron beam was propa- +gating at roughly -50◦ heliolongitude (Fig. B.1), that is, away +from the Parker spirals connecting SolO, STEREO-A, and Wind. +According to our analysis (Appendix B), the in-situ Langmuir +waves observed at STEREO-A and SolO were unlikely to be gen- +erated by the passing of TIII-G1 (Fig. B.1). +According to Fig. 9, another physical phenomenon was +present around the time of the inferred low-energy electron re- +lease and that was the coronal shock wave. The first signatures +of TII(1a) were observed around 06:33 UT, indicating that the +shock was able to efficiently accelerate electrons after the first +HXR pulse. Around the same time, the EUV wave was formed +and started its evolution away from the active region, mostly to +the southwest direction. According to the first-order analysis of +the shock wave (type II association presented in Appendix C), +the TII(1a) originated most likely from the extended flank re- +gions of the coronal shock wave. Furthermore, as shown in Fig. 5 +and Sec. 4.3, open magnetic field lines indicate a good magnetic +connectivity to both SolO and STEREO-A at the periphery of the +AR, where the shock is progressively connected. +Article number, page 13 of 20 + +EUV wave +18 channels +a) +EUV flare +VDA +TSA +SXR peak +HXR 4 +reflection +SXR end +06:28 +06:3 +06:34 +06:38 +06:42) +06:50 +06:53 +SXR Flare start +HXR1, TIII - G1 +HXR 2 +HXR 3 +TSA +HXR 5 +Phase 1 peak +Phase 2 peak +06:19 +06:3 +06:33 +06:3 +06:41 +06:46 +[06:52 +07:30 +LoW energy injection +HIGH energy injection +MW +6:30 +7:00 +TIII - G2 +6:37 +6:46 + 6:50 +TII (W) +6:55 +TII (1) +6:33 +7:07 +TII (2) +6:36 +7:40 +TIV +8:00 +b) +c) +t=2 +t=1 +Coronal hole +Active regionA&A proofs: manuscript no. aanda +In summary, we have found that the low-energy electrons +were accelerated mainly by a flare-related process consistent +with the first two HXR pulses. Fig. .1 shows that the flare oc- +curred in the western part of the AR. There, open field lines +are present only in the vicinity of the strong positive polarity +in the south. During the first (shown in panel a and b) and sec- +ond HXR peak, a footpoint source is located close to this area. +In contrast, in the later peaks the eastern footpoint has shifted +to a more northern location. We conclude that this is consistent +with the notion that the energetic electrons observed in-situ in +phase 1 have been accelerated (at least partly) in the flare and +were injected onto open magnetic field lines connection to IP +space. +We also found that a relative contribution from the shock +wave is difficult to be excluded, since the inferred injection and +the start of type II are very close. The shock wave possibly fa- +cilitated the low-energy electron release to the open magnetic +field lines connecting to different observers, since the EUV wave +reached the well-connected field lines (to SolO and STEREO- +A) around 06:35 UT. During this phase, the shock geometry at +the field lines is expected to be mostly quasi-perpendicular at its +flanks as it interacts with the open field lines present close to the +active region. This may allow the shock wave to accelerate elec- +trons rapidly along open magnetic field lines via the SDA mech- +anism. This scenario is illustrated in Fig. 10b where the shock +wave during this time is represented by the black arc at t=1. The +thermal and suprathermal electrons which are present in abun- +dance in the vicinity of the active region can be accelerated to +energies in the range of 40–80 keV through a fast-Fermi mech- +anism. As explained earlier, given the right coronal and shock +conditions (upstream electron density, magnetic compression ra- +tio, and shock geometry), SDA can be a potent accelerator of +energetic electrons (Leroy & Mangeney 1984; Krauss-Varban & +Wu 1989; Mann et al. 2009; Warmuth et al. 2009). +High energy electrons +The inferred injection times of electrons exceeding an energy of +∼100 keV (using VDA and TSA) show a delay of 5 to 10 min- +utes, compared to that of the low-energy electrons (Sec. 4.4.3). +This puts it amidst several observed eruptive features and poten- +tial acceleration scenarios. At the time of the electron release, the +TIII-G2, the 4th HXR pulse, and the TII(HB) were observed. We +note that there was also a restructuring of magnetic footpoints +near the flaring region minutes before the injection, during the +3rd HXR pulse (Fig. 3 and Sec. 4.1). +The TIII-G2 was observed in association with the 4th HXR +pulse, which was also one of the two lowest intensity pulses, +implying that there was a decrease in the number of available +energetic electrons (Fig. 2f). The associated TIII-G2, however, +was rather bright (dense electron beams generating the type III) +and with a prolonged duration. This often signifies an extended +period of electron release into interplanetary space. It is not clear +if parts of the TIII-G2 emanated from the type II or not. The TIII- +G2 was observed by all space-borne radio observatories and, in +particular, by STEREO-A and SolO that also observed in-situ +Langmuir waves. The results from the direction-finding analysis +of the TIII-G2 presented in Appendix B reveal that the TIII-G2 +propagated approximately in the -15◦ heliolongitude (Fig. B.1), +which was towards both SolO and STEREO-A. The radio analy- +sis suggests a good magnetic connection between the source of +the TIII-G2 and the two spacecraft. +During this phase, energetic electrons in the range of 50–100 +keV were still found in abundance in the flaring region (Sec. 4.1). +Additionally, important shock-associated phenomena were also +observed, namely, TII(2a), TII(HB), and TII(W) at 06:36 UT, +06:41 UT, and 06:50 UT, respectively. The presence of herring- +bone (HB) structures indicate that electron beams were acceler- +ated by a near-perpendicular shock wave (85◦ < θBn < 89.9◦) +via the SDA mechanism (Mann & Klassen 2005). Such a near- +perpendicular shock geometry is expected in the low corona, +which is dominated by closed magnetic field lines and a strong +lateral expansion of the shock wave (Kouloumvakos et al. 2021). +According to the analysis performed in Sec. 4.3, it is understood +that the TII(HB) and TII(W) features were observed concur- +rently with the EUV wave mirroring off a magnetic null, which +is formed in the south-west periphery of the AR close to the +south-west coronal hole. These two features mark a phase of ef- +ficient electron acceleration from the shock, which is probably +responsible for the injection of the high-energy electrons during +the first phase. Meanwhile, the TII(2a) was observed in metric +wavelengths and according to the kinematic analysis presented +in the Appendix C, the source of the TII(2a) is at the flank re- +gions of the shock wave. +A critical feature of this period was the possible interaction +between the electron beams generating the TIII-G2 and part of +the shock wave generating TII(HB) leading to a re-acceleration +of the incident electrons. Based on the interplanetary directivity +analysis of TIII-G2, we have shown that the source of the elec- +tron beams were well-connected to both SolO and STEREO-A +(Fig. B.1). Additionally, during the same period, the TIV con- +tinuum observed was probably generated by electrons trapped +within the flux rope of the expanding CME. It is possible that +some of the trapped energetic particles may have escaped during +interactions between the CME and the ambient magnetic field +lines (Petrosian 2016; Klein et al. 2022). In both circumstances, +an interaction with the shock wave could have accelerated the in- +cident electrons to higher energies. Electrons with large enough +pitch angles are readily reflected by a near-perpendicular shock +wave, gaining maximum energies of up to 13-fold their initial en- +ergy in the process (for a shock wave with magnetic compression +4, Leroy & Mangeney 1984; Ball & Melrose 2001). This mech- +anism may have increased the energy of a fraction (depends on +θBn) of the incident c/3 electrons that generated the TIII-G2 to +near-relativistic energies. Considering that electrons in the range +of 50–100 keV were also abundant during this period from, e.g., +the HXR pulse, they may be further accelerated to relativistic +energies by interacting with the coronal shock wave. +In conclusion, we find that during the release of high-energy +electrons, a number of processes were observed concurrently. +We distinguished between the different features and their asso- +ciated mechanisms and found that the acceleration most likely +happened in the corona close to the Sun. Observations suggest +that the main acceleration probably took place at the southern +flank of the shock wave, where an interaction of the shock with +a coronal streamer is observed. In these regions of interaction the +shock wave geometry can be nearly perpendicular and electrons +can accelerate to high energies efficiently. A cartoon illustrating +such a scenario is presented in Fig. 10b,c, where the shock wave +(represented by the black arc at t=2) continues to interact with +open field lines at its flanks. The presence of nonthermal elec- +trons probably eases the acceleration of the electrons to mildly- +relativistic and relativistic energies; at least for a fraction of the +incident electrons. +A contribution from a flare-related acceleration process can- +not be excluded during this high-energy electron release phase, +however, the Suprathermal Ion Spectrograph (SIS) on-board +SolO showed no particular enhancement of flare accelerating +Article number, page 14 of 20 + +Immanuel. C. Jebaraj et al.: Solar energetic electrons on 9 October 2021 +particles such as 3He ions. The presence of 3He ions would in- +dicate a direct contribution of flare accelerated particles to the +particle population measured in-situ during the event. This prob- +ably suggests that the shock wave had an important role in the +acceleration and release of this high-energy electron component. +The shock geometry which in some cases have been shown to +be predominantly quasi-perpendicular in the low corona (e.g., +Kouloumvakos et al. 2021), can enable the shock to accelerate +electrons up to 13-fold their incident energy based on their pitch +angles through an SDA mechanism (Leroy & Mangeney 1984; +Ball & Melrose 2001, and references therein). +5.2. Phase 2 +As observed in Fig. 9, a second peak was observed around +7:30 UT in the electron time profiles recorded by SolO. This +increase in intensity appears after the anisotropic first phase. In +this second phase, the electrons showed very little anisotropy +(Fig. 6), which is usually believed to be due to transport related +effects. It was not possible to determine the release of these elec- +trons, however, the peak intensities were delayed by ≈50 min- +utes after the start of the SXR flare (Fig. 10). This phase can be +characterised by the gradual increase in high-energy electron in- +tensities. An interesting aspect of this phase was the increase in +the intensities of energetic electrons exceeding 300 keV, which +was only observed by SolO and not by other spacecraft. Addi- +tionally, the high-energy electrons exhibited a harder spectrum +than during the first phase as observed by SolO (Fig. 7). A hard +spectrum for delayed and gradual electron events has previously +been attributed to transport-related effects (Strauss et al. 2020). +In the case of STEREO-A, we believe the spacecraft sampled +electrons that were accelerated mostly during the first phase and +were delayed due to transport related effects. This could explain +the anisotropy of the electrons observed by STEREO-A, which +was lower than at SolO. This would suggest that SolO was lo- +cated conveniently at a region where the electrons were largely +unaffected by transport and did not undergo strong pitch-angle +scattering. The other spacecraft located elsewhere might have +observed electrons from both the first phase and a transport- +affected second phase. +The second phase occurred after the impulsive flare phase, +that is, when the HXR flux had returned to nominal pre-flare +levels. It is therefore probable that the electrons of phase 2 were +accelerated at the CME/shock instead of during the flare. In ra- +dio wavelengths, the only features remaining in the radio spec- +trum are a hectometric type II burst and metric TIV pulsations. +At the same time, the WL shock wave reached its peak veloc- +ity of 1400 km s−1 (Fig. C.1) at a radial distance of 7 R⊙. The +presence of a long-lasting patchy type II radio burst in the hec- +tometer wavelength, namely TII(1b) and TII(2b), confirms that +shock electron acceleration took place during this phase. From +the analysis of Sec. 4.3, it is understood that the shock under- +went large-scale deformations due to the presence of coronal +magnetic structures, namely, coronal holes and streamers, and it +also continued to expand in regions confined by different stream +interactions (Sec. 4.3). An analysis of the shock kinematics (Ap- +pendix C) and the comparison with the radio TII kinematics sug- +gests that TII(2b) was most likely generated in the flank regions +of the WL shock wave observed during this second phase. +Despite the fact that SDA can be a potent accelerator of ener- +getic electrons in regions with quasi-perpendicular shock geom- +etry, it is not fully clear if the shock geometry in the high corona +is such. It is possible that small scale deformations such as cavi- +ties and ripples may provide the locally quasi-perpendicular ge- +ometry. However, in these places the shock can be corrugated, +leading to the short and efficient acceleration of electrons trapped +within them through SDA mechanism (Bale et al. 1999). On the +other hand, DSA can also be an efficient acceleration mecha- +nism if the electrons are trapped efficiently at the shock region. +In this case, pre-existing magnetic field turbulence and low fre- +quency waves generated upstream by specularly reflected ions +(about 25% of the thermal ions are reflected; Krasnoselskikh +et al. 1991; Gedalin 1996; Balikhin & Gedalin 2022) could have +played a crucial role in trapping electrons for an extended pe- +riod. Ambient magnetic field turbulence, which is ubiquitous in +interplanetary plasma, may act as magnetic scattering centres for +electrons (Tsytovich 1966). The electrons trapped within turbu- +lent fields or deformed and corrugated shock regions may di- +verge from the thermal Maxwellian (Maksimovic et al. 1997) +and provide a seed population for DSA. The energy gain of these +electrons can be in the order of 100s of keV (resonance with +whistler waves at lower-hybrid frequency; Vaisberg et al. 1983; +Galeev 1984; Krasnoselskikh et al. 1985) to several MeVs and +tens of MeVs (Alfven wave resonance; Tsytovich 1973; Bell +1978; Kirk & Dendy 2001) depending on how efficiently they +are trapped near the shock. +The fact that this second phase was only observed by SolO +indicates that the conditions for the shock scenario were pro- +vided only within a confined region of the interplanetary shock +wave. The confined region could have been formed when the +WL shock wave interacted with the streamers and propagated +close to the fast-wind region. The electrons which were acceler- +ated through a pitch-angle diffusion process under the aforemen- +tioned conditions can then escape the shock. While even small +amounts of perpendicular diffusion present in the SIR can spread +the particles to large heliolongitudes, we do not observe the sec- +ond phase in a spacecraft other than SolO. +The intensity-time profiles of the SEP event at SolO might +be influenced by the arrival of a stream interaction region (SIR) +during the SEP onset time. As discussed by Lario et al. (2022), +the high-speed stream driving the SIR was previously observed +by STEREO-A and later also by near-Earth spacecraft, where it +strongly affected the observed energetic ion intensity-time pro- +files. At SolO, the SIR and its associated magnetic compres- +sion might have acted as a magnetic mirror, reflecting some of +the electrons back towards the inner heliosphere and the CME- +driven shock wave where the electrons may subsequently accel- +erate to higher energies. +6. Conclusions +The main conclusions of this study are as follows: +• We were able to identify two distinct SEP event phases in- +situ, that show different anisotropies and are most-likely re- +lated to different acceleration phases in the solar corona. +• We also found a hardening spectra from the first to the sec- +ond phase for mildly-relativistic and relativistic electrons. +This finding further prompts the existence of two different +acceleration mechanisms. +• We have found the presence of two separate injections dur- +ing the first phase. The low-energy and high-energy electron +components were injected at different times and can be at- +tributed to different episodes of acceleration. +• We found a mix of flare and shock acceleration mechanisms +for the low energy electrons, while the high energy electron +acceleration was mostly a shock-associated phenomenon in +a highly structured corona. +Article number, page 15 of 20 + +A&A proofs: manuscript no. aanda +• We have also found that the prolonged nature of the second +phase was due to the particles being accelerated and trans- +ported diffusively within a compression region, where the +said particles are also trapped efficiently. +This study was largely possible due to the capabilities of +the EPD instrument onboard SolO, which observes in a wide +range of energy channels and also provides directional mea- +surements. Utilising multi-viewpoint remote sensing observa- +tions from widely distributed spacecraft and ground-based ob- +servations in many available wavelengths, it is possible to probe +mechanisms continuously from deep in the solar corona to inter- +planetary space. Future studies must aim to include high reso- +lution radio imaging from the LOw Frequency ARray (LOFAR; +van Haarlem et al. 2013), and the full capabilities of Solar Or- +biter and Parker Solar Probe (PSP; Fox et al. 2016) to better un- +derstand the acceleration mechanisms at the Sun and the effects +of coronal and interplanetary transport. +Acknowledgements. This work received funding from the European Union’s +Horizon 2020 research and innovation program under grant agreement No. +870405 (EUHFORIA 2.0), and No. 101004159 (SERPENTINE). I.C.J. and J.M. +acknowledge funding by the BRAIN-be project SWiM (Solar Wind Modelling +with EUHFORIA for the new heliospheric missions). A.K. acknowledges finan- +cial support from NASA’s NNN06AA01C (SO-SIS Phase-E) contract. Work in +the University of Turku was performed under the umbrella of Finnish Centre of +Excellence in Research of Sustainable Space (FORESAIL, Academy of Finland +grant no. 336809). N.D. is grateful for support by the Turku Collegium for Sci- +ence, Medicine and Technology of the University of Turku, Finland and support +of Academy of Finland (SHOCKSEE, grant no. 346902). N.W. acknowledges +support from the NASA program NNH17ZDA001N-LWS and from the Research +Foundation – Flanders (FWO – Vlaanderen, fellowship no. 1184319N) T.W. ac- +knowledges DLR grant 50 OC 2101. The work of F.S. was supported by DLR +grant No. 50 OT 1904. I.C.J. thanks Dr. Vladimir Krasnoselskikh for the produc- +tive discussions regarding the various aspects of collisionless shocks. Solar Or- +biter is a space mission of international collaboration between ESA and NASA, +operated by ESA. The STIX instrument is an international collaboration between +Switzerland, Poland, France, Czech Republic, Germany, Austria, Ireland, and +Italy. The authors also thank Dr. Milan Maksimovic for providing data products +from the Radio Plasma Waves (RPW) instrument onboard Solar Orbiter. EIT +and LASCO data have been used courtesy of the SOHO/EIT and SOHO/LASCO +consortiums, respectively. The STEREO SECCHI data are produced by a consor- +tium of RAL(UK), NRL(USA), LMSAL(USA), GSFC(USA), MPS(Germany), +CSL(Belgium), IOTA(France), and IAS(France). The Wind/WAVES instrument +was designed and built as a joint effort of the Paris-Meudon Observatory, the +University of Minnesota, and the Goddard Space Flight Center, and the data are +available at the instrument Web site. The authors are grateful to Dr. Xavier Bon- +nin for providing direction finding data from the WAVES experiment onboard +Wind. We thank the radio monitoring service at LESIA (Observatoire de Paris) +for providing value-added data that have been used for this study. HMI data are +provided courtesy of NASA/SDO science teams. +References +Agueda, N., Klein, K.-L., Vilmer, N., et al. 2014, A&A, 570, A5 +Anastasiadis, A., Lario, D., Papaioannou, A., Kouloumvakos, A., & Vourlidas, +A. 2019, Philosophical Transactions of the Royal Society of London Series +A, 377, 20180100 +Arge, C. N., Henney, C. J., Koller, J., et al. 2010, in Solar Wind 12, ed. M. Mak- +simovic, K. Issautier, N. Meyer-Vernet, M. M. Moncuquet, & F. Pantellini, +Vol. CS-1216, Am. Inst. Phys., Melville, 343–346 +Aurass, H., Hofmann, A., & Urbarz, H. W. 1998, A&A, 334, 289 +Badman, S. T., Carley, E., Cañizares, L. A., et al. 2022, ApJ, 938, 95 +Bale, S. D., Goetz, K., Harvey, P. R., et al. 2016, Space Sci. Rev., 204, 49 +Bale, S. D., Reiner, M. J., Bougeret, J. L., et al. 1999, Geophys. Res. Lett., 26, +1573 +Balikhin, M. & Gedalin, M. 2022, ApJ, 925, 90 +Ball, L. & Melrose, D. B. 2001, PASA, 18, 361 +Bastian, T. S., Benz, A. O., & Gary, D. E. 1998, ARA&A, 36, 131 +Battaglia, A. F., Saqri, J., Massa, P., et al. 2021, A&A, 656, A4 +Bell, A. R. 1978, MNRAS, 182, 147 +Benz, A. O., Monstein, C., Meyer, H., et al. 2009, Earth Moon and Planets, 104, +277 +Bougeret, J. L., Goetz, K., Kaiser, M. L., et al. 2008, Space Sci. Rev., 136, 487 +Bougeret, J.-L., Kaiser, M. L., Kellogg, P. J., et al. 1995, Space Science Reviews, +71, 231 +Brown, J. C. 1971, Sol. Phys., 18, 489 +Brueckner, G. E., Howard, R. A., Koomen, M. J., et al. 1995, Sol. Phys., 162, +357 +Cairns, I. H. 1987, Journal of Plasma Physics, 38, 169 +Cairns, I. H., Knock, S. A., Robinson, P. A., & Kuncic, Z. 2003, Space Sci. Rev., +107, 27 +Cane, H. V., Stone, R. G., Fainberg, J., et al. 1981, Geophys. Res. Lett., 8, 1285 +Domingo, V., Fleck, B., & Poland, A. I. 1995, Sol. Phys., 162, 1 +Dresing, N., Effenberger, F., Gómez-Herrero, R., et al. 2020, ApJ, 889, 143 +Dresing, N., Kouloumvakos, A., Vainio, R., & Rouillard, A. 2022, ApJ, 925, L21 +Dröge, W. 2000, ApJ, 537, 1073 +Dulk, G. A., Leblanc, Y., Bastian, T. S., & Bougeret, J.-L. 2000, J. Geophys. Res., +105, 27343 +Fárník, F., Hudson, H. S., Karlický, M., & Kosugi, T. 2003, A&A, 399, 1159 +Fox, N. J., Velli, M. C., Bale, S. D., et al. 2016, Space Sci. Rev., 204, 7 +Galeev, A. A. 1984, Soviet Journal of Experimental and Theoretical Physics, 59, +965 +Galeev, A. A., Malkov, M. A., & Völk, H. J. 1995, Journal of Plasma Physics, +54, 59 +Garcia, H. A. 1994, Sol. Phys., 154, 275 +Gedalin, M. 1996, J. Geophys. Res., 101, 4871 +Gieseler, J., Dresing, N., Palmroos, C., et al. 2022, Frontiers in Astronomy and +Space Physics, (accepted) +Gieseler, U. D. J., Jones, T. W., & Kang, H. 2000, A&A, 364, 911 +Ginzburg, V. L. & Zhelezniakov, V. V. 1958, Soviet Ast., 2, 653 +Glesener, L., Krucker, S., & Lin, R. P. 2012, ApJ, 754, 9 +Grigis, P. C. & Benz, A. O. 2004, A&A, 426, 1093 +Grigis, P. C. & Benz, A. O. 2008, ApJ, 683, 1180 +Harten, R. & Clark, K. 1995, Space Sci. Rev., 71, 23 +Howard, R. A., Moses, J. D., Vourlidas, A., et al. 2008, Space Sci. Rev., 136, 67 +Huttunen-Heikinmaa, K., Valtonen, E., & Laitinen, T. 2005, A&A, 442, 673 +Iwai, K., Kubo, Y., Ishibashi, H., et al. 2017, Earth, Planets and Space, 69, 95 +Jebaraj, I. C., Kouloumvakos, A., Magdaleni´c, J., et al. 2021, A&A, 654 +Jebaraj, I. C., Magdaleni´c, J., Krasnoselskikh, V., Krupar, V., & Poedts, S. 2022, +A&A, (accepted) +Jebaraj, I. C., Magdaleni´c, J., Podladchikova, T., et al. 2020, A&A, 639 +Kahler, S. W. & Hundhausen, A. J. 1992, J. Geophys. Res., 97, 1619 +Kaiser, M. L. 2005, Advances in Space Research, 36, 1483 +Kaiser, M. L., Kucera, T. A., Davila, J. M., et al. 2008, Space Sci. Rev., 136, 5 +Kiplinger, A. L. 1995, ApJ, 453, 973 +Kirk, J. G. & Dendy, R. O. 2001, Journal of Physics G Nuclear Physics, 27, 1589 +Klassen, A., Dresing, N., Gómez-Herrero, R., Heber, B., & Veronig, A. 2018, +A&A, 614, A61 +Klassen, A., Gómez-Herrero, R., & Heber, B. 2011a, Sol. Phys., 273, 413 +Klassen, A., Gómez-Herrero, R., Heber, B., et al. 2012, A&A, 542, A28 +Klassen, A., Gómez-Herrero, R., Müller-Mellin, R., et al. 2011b, A&A, 528, +A84 +Klein, K. L., Chupp, E. L., Trottet, G., et al. 1999, A&A, 348, 271 +Klein, K.-L. & Dalla, S. 2017, Space Sci. Rev., 212, 1107 +Klein, K.-L., Musset, S., Vilmer, N., et al. 2022, A&A, 663, A173 +Kong, X., Chen, Y., Guo, F., et al. 2016, ApJ, 821, 32 +Kong, X., Guo, F., Giacalone, J., Li, H., & Chen, Y. 2017, ApJ, 851, 38 +Kontar, E. P. & Reid, H. A. S. 2009, ApJ, 695, L140 +Kouloumvakos, A., Kwon, R. Y., Rodríguez-García, L., et al. 2022a, A&A, 660, +A84 +Kouloumvakos, A., Nindos, A., Valtonen, E., et al. 2015, A&A, 580, A80 +Kouloumvakos, A., Rodríguez-García, L., Gieseler, J., et al. 2022b, Frontiers, 0, +0 +Kouloumvakos, A., Rouillard, A., Warmuth, A., et al. 2021, ApJ, 913, 99 +Kouloumvakos, A., Rouillard, A. P., Wu, Y., et al. 2019, ApJ, 876, 80 +Krasnoselskikh, V., Voshchepynets, A., & Maksimovic, M. 2019, ApJ, 879, 51 +Krasnoselskikh, V. V., Balikhin, M. A., Alleyne, H. S. C., et al. 1991, Advances +in Space Research, 11, 15 +Krasnoselskikh, V. V., Kruchina, E. N., Volokitin, A. S., & Thejappa, G. 1985, +A&A, 149, 323 +Krauss-Varban, D. & Wu, C. S. 1989, J. Geophys. Res., 94, 15367 +Krucker, S., Hurford, G. J., Grimm, O., et al. 2020, A&A, 642, A15 +Krucker, S., Kontar, E. P., Christe, S., Glesener, L., & Lin, R. P. 2011, ApJ, 742, +82 +Krucker, S., Kontar, E. P., Christe, S., & Lin, R. P. 2007, ApJ, 663, L109 +Krucker, S., Saint-Hilaire, P., Christe, S., et al. 2008, ApJ, 681, 644 +Krupar, V., Eastwood, J. P., Kruparova, O., et al. 2016, ApJ, 823 +Krupar, V., Maksimovic, M., Santolik, O., et al. 2014, Sol. Phys., 289, 3121 +Krupar, V., Szabo, A., Maksimovic, M., et al. 2020, ApJS, 246, 57 +Lario, D., Wijsen, N., Kwon, R. Y., et al. 2022, ApJ, 934, 55 +Lemen, J. R., Title, A. M., Akin, D. J., et al. 2012, Sol. Phys., 275, 17 +Leroy, M. M. & Mangeney, A. 1984, Annales Geophysicae, 2, 449 +Article number, page 16 of 20 + +Immanuel. C. Jebaraj et al.: Solar energetic electrons on 9 October 2021 +Lin, R. P., Anderson, K. A., Ashford, S., et al. 1995, Space Sci. Rev., 71, 125 +Long, D. M., Bloomfield, D. S., Chen, P. F., et al. 2017, Sol. Phys., 292, 7 +Lucas, J. M. 1985, Technometrics, 27, 129 +Magdaleni´c, J., Marqué, C., Krupar, V., et al. 2014, ApJ, 791, 115 +Maksimovic, M., Bale, S. D., Chust, T., et al. 2020, A&A, 642, A12 +Maksimovic, M., Pierrard, V., & Lemaire, J. F. 1997, A&A, 324, 725 +Mann, G. & Klassen, A. 2005, A&A, 441, 319 +Mann, G., Warmuth, A., & Aurass, H. 2009, A&A, 494, 669 +Massa, P., Battaglia, A. F., Volpara, A., et al. 2022, Sol. Phys., 297, 93 +Massa, P., Schwartz, R., Tolbert, A. K., et al. 2020, ApJ, 894, 46 +Masson, S., Antiochos, S. K., & DeVore, C. R. 2019, ApJ, 884, 143 +McClements, K. G., Dendy, R. O., Bingham, R., Kirk, J. G., & Drury, L. O. +1997, MNRAS, 291, 241 +McLean, D. J. & Labrum, N. R. 1985, Solar radiophysics : studies of emission +from the sun at metre wavelengths +Melrose, D. B. 2017, Reviews of Modern Plasma Physics, 1, 5 +Müller, D., St. Cyr, O. C., Zouganelis, I., et al. 2020, A&A, 642, A1 +Müller-Mellin, R., Böttcher, S., Falenski, J., et al. 2008, Space Sci. Rev., 136, +363 +Musset, S., Jeunon, M., & Glesener, L. 2020, ApJ, 889, 183 +Newkirk, G. J. 1961, The Astrophysical Journal, 133, 983 +Nindos, A. 2020, Frontiers in Astronomy and Space Sciences, 7, 57 +Page, E. S. 1954, Biometrika, 41, 100 +Papaioannou, A., Sandberg, I., Anastasiadis, A., et al. 2016, Journal of Space +Weather and Space Climate, 6, A42 +Pesnell, W. D., Thompson, B. J., & Chamberlin, P. C. 2012, Sol. Phys., 275, 3 +Petrosian, V. 2016, ApJ, 830, 28 +Pohjolainen, S. 2008, A&A, 483, 297 +Pulupa, M., Bale, S. D., Bonnell, J. W., et al. 2017, Journal of Geophysical Re- +search (Space Physics), 122, 2836 +Ramesh, R., Kathiravan, C., & Chellasamy, E. E. 2022, ApJ, 932, 48 +Reames, D. V. 2021, Solar Energetic Particles. A Modern Primer on Understand- +ing Sources, Acceleration and Propagation, Vol. 978 +Reid, H. A. S. & Ratcliffe, H. 2014, Research in Astronomy and Astrophysics, +14, 773 +Rodríguez-Pacheco, J., Wimmer-Schweingruber, R. F., Mason, G. M., et al. +2020, A&A, 642, A7 +Saito, K. 1970, Annals of the Tokyo Astronomical Observatory, 12, 51 +Schmidt, J. M. & Cairns, I. H. 2016, Geophys. Res. Lett., 43, 50 +Schrijver, C. J. & De Rosa, M. L. 2003, Sol. Phys., 212, 165 +Stewart, R. T. & Magun, A. 1980, PASA, 4, 53 +Strauss, R. D., Dresing, N., Kollhoff, A., & Brüdern, M. 2020, ApJ, 897, 24 +Thejappa, G. 1987, Sol. Phys., 111, 45 +Tkachenko, A., Krasnoselskikh, V., & Voshchepynets, A. 2021, ApJ, 908, 126 +Treumann, R. A. & Jaroschek, C. H. 2008, arXiv e-prints, arXiv:0805.2579 +Tsytovich, V. N. 1966, Soviet Physics Uspekhi, 9, 370 +Tsytovich, V. N. 1973, ARA&A, 11, 363 +Uchida, Y., Altschuler, M. D., & Newkirk, Gordon, J. 1973, Sol. Phys., 28, 495 +Vainio, R., Valtonen, E., Heber, B., et al. 2013, Journal of Space Weather and +Space Climate, 3, A12 +Vaisberg, D. L., Galeev, A. A., Zastenker, G. N., et al. 1983, Soviet Journal of +Experimental and Theoretical Physics, 58, 716 +van Haarlem, M. P., Wise, M. W., Gunst, A. W., et al. 2013, A&A, 556, A2 +Vlahos, L., Anastasiadis, A., Papaioannou, A., Kouloumvakos, A., & Isliker, H. +2019, Philosophical Transactions of the Royal Society of London Series A, +377, 20180095 +von Rosenvinge, T. T., Reames, D. V., Baker, R., et al. 2008, Space Sci. Rev., +136, 391 +Vršnak, B. & Luli´c, S. 2000a, Sol. Phys., 196, 157 +Vršnak, B. & Luli´c, S. 2000b, Sol. Phys., 196, 181 +Warmuth, A. 2015, Living Reviews in Solar Physics, 12, 3 +Warmuth, A. & Mann, G. 2005, A&A, 435, 1123 +Warmuth, A. & Mann, G. 2020, A&A, 644, A172 +Warmuth, A., Mann, G., & Aurass, H. 2009, A&A, 494, 677 +Warmuth, A., Önel, H., Mann, G., et al. 2020, Sol. Phys., 295, 90 +Wiegelmann, T., Inhester, B., & Sakurai, T. 2006, Sol. Phys., 233, 215 +Wiegelmann, T., Thalmann, J. K., Inhester, B., et al. 2012, Sol. Phys., 281, 37 +Zhang, Y., Musset, S., Glesener, L., Panesar, N., & Fleishman, G. 2022, arXiv +e-prints, arXiv:2207.05668 +Article number, page 17 of 20 + +A&A proofs: manuscript no. aanda +Appendix A: NLFFF extrapolations of AR magnetic +field topology +In order to understand the magnetic topology of eruptive event, +we extrapolate photospheric vector magnetograms of the source +AR obtained from SDO/HMI into the solar corona under the +nonlinear force-free field (NLFFF) assumption with the help of +an optimization code, as described in Wiegelmann et al. (2012). +The photospheric magnetic vector field measurements have been +preprocessed to obtain consistent boundary data for the NLFFF- +model (see Wiegelmann et al. 2006 for details). +Figure .1 depicts a sample of the extrapolated field lines, +which are plotted over the vertical photospheric magnetic field +shown in greyscale. Additionally, we overplot the nonthermal +STIX sources as coloured contours. Note that we have rotated +the STIX images to conform with the vantage point of SDO, +which is exactly the inverse process as used for Fig. 3 where we +have rotated SDO/AIA images to the SolO view. We can do per- +form this rotation only for the nonthermal STIX sources since +they originate in the chromosphere, while the thermal source is +an extended 3D structure in the corona for which a simple rota- +tion would cause severe projection effects. +We show this comparison for three of the nonthermal STIX +peaks for which nearly co-temporal HMI data was available +(times are UT at 1 AU): HMI at 06:24 and the STIX peak +around 06:31 UT (panels a +b ), HMI at 06:36 and STIX around +06:37 UT (panels c +d ), and finally HMI at 06:48 UT and STIX +at 06:46 UT (panels e + f). The top row of panels (a, c, e) shows +closed loops where both footpoints of the magnetic field line are +anchored in the photosphere. The bottom panels (b, d, f) contain +open magnetic field lines, where open means that the field lines +have only one footpoint in the photosphere and the upper end +reaches the top boundary of the computational domain. These +are the field lines which potentially offer access to interplanetary +space. +Appendix B: Directivity of type III bursts +The +interplanetary +radio +emissions +associated +with +the +event were observed by all spacecraft: STEREO-A/WAVES, +Wind/WAVES, SolO/RPW, and PSP/FIELDS. A combined dy- +namic radio spectrum from the three spacecraft excluding WIND +is presented in Fig. B.1. The figure also includes the linear po- +larisation measurements from PSP and STEREO-A, which also +provides goniopolarimetric measurements. +Locating the source and the propagation of the type III ra- +dio sources in the corona and interplanetary space without in- +terferometric imaging (e.g., LOw Frequency ARray (LOFAR); +van Haarlem et al. 2013) or radio triangulation (e.g., Magdaleni´c +et al. 2014; Krupar et al. 2016; Jebaraj et al. 2020) is challenging. +However, we make use of the direction finding observations from +STEREO-A and Wind observations (azimuth and co-latitude an- +gles of the radio emission, Krupar et al. 2014). +In the case of the directivity analysis, we have compared cal- +ibrated radio fluxes measured by Parker Solar Probe, Solar Or- +biter, STEREO A and Wind at six frequency channels. We as- +sumed that the radio emission pattern S as a function of helio- +centric longitude λ can be described by the von Mises distribu- +tion (also known as the circular normal distribution) as: +S (λ) = exp(κ cos(λ − λ0)) +2πI0(κ) +, +(B.1) +where λ0 is a direction corresponding to a peak radio flux, +κ is a measure of concentration, and I0 is the modified Bessel +function of the first kind of order 0, with this scaling constant +chosen so that the distribution sums to unity. +Fig B.1 panels e & f show the propagation direction of TIII- +G1 and TIII-G2 sources obtained using the radio triangulation +technique (short black arrows, for more details see Krupar et al. +2014) and a directivity analysis (long black arrows, radio emis- +sion pattern is described by the von Mises distribution). Both +methods employ cross-calibrated measurements from multiple +spacecraft, all four spacecraft in the case of the directivity anal- +ysis, Wind and STEREO-A in the case of radio triangulation. We +perform both analyses to better constrain the direction of source +propagation and eliminate intrinsic errors which may arise from +each method. Even so, we shall put more weight on the radio tri- +angulation results as they do not depend on free parameters such +as a coronal electron density model. +The radio triangulation results of TIII-G1 between the fre- +quencies 1000 kHz and 600 kHz indicate that the source prop- +agated in the -58◦ heliolongitude (possible error of ±6◦). The +directivity analysis indicated that the source propagated in -43◦ +heliolongitude (possible error of ±2◦). Both results indicate that +TIII-G1 most likely propagated westward of STEREO-A close to +the reference line (black dashed line) plotted in Fig. 1. The clos- +est spacecraft to the propagation of TIII-G1 was PSP, however, +no in-situ Langmuir waves were recorded by PSP. Therefore, we +can conclude that it was highly unlikely for TIII-G1 to be associ- +ated with the in-situ electrons recorded by SolO, and STEREO-A. +Lario et al. (2022) found a very impulsive rise of particle flux at +PSP during the early stages of the event, which is to be expected +when there was direct connectivity in the direction rather close +to PSP. +For TIII-G2, we performed a similar analysis (Fig. B.1e) and +found, according to radio triangulation, the source propagated in +the -15◦ heliolongitude with a possible error of ±2◦. Using the +directivity analysis, we found the propagation to be in the heli- +olongitudes -30◦ with possibly a ±3◦ error. The larger spread in +propagation direction between radio triangulation and the direc- +tivity analysis could be due to errors arising from the propagation +of radio waves in the presence of large-scale density fluctuations +(Krupar et al. 2020). Alternatively, the large spread may also +likely be due to multiple electron beams generating type III ra- +dio bursts within TIII-G2. The electron beams that generated the +different type III bursts may propagate in slightly different solar +wind due to their time-varying injections at the source - causing +a large spread in their directivity. However, it should be noted +that TIII-G2 propagated mostly in the direction of SolO and +STEREO-A, which is further corroborated by the in-situ Lang- +muir waves. Therefore, from our analysis, TIII-G2 may be asso- +ciated with the energetic electrons recorded in-situ at both SolO +and STEREO-A. +Appendix C: Shock wave kinematics +We reconstructed the 3D structure of the shock wave using +PyThea, a software package to reconstruct the 3D structure of +CMEs and shock waves (Kouloumvakos et al. 2022b), written +in Python language and available online4. We took advantage +of the two viewpoints provided by STEREO-A and near-Earth +spacecraft (i.e., SOHO and SDO), and we fitted an ellipsoid +model to EUV and WL observations of the shock wave. We +adjusted the free parameters of the geometrical ellipsoid model +to achieve the best visual fit to near-simultaneous observations +for the two available viewpoints. From the 3D reconstruction, +4 https://doi.org/10.5281/zenodo.5713659 +Article number, page 18 of 20 + +Immanuel. C. Jebaraj et al.: Solar energetic electrons on 9 October 2021 +Fig. .1. Nonlinear force-free field extrapolations based on SDO/HMI vector magnetograms and STIX overplotted (see text for details). +Fig. B.1. Space-borne radio dynamic spectrograms from: (a) PSP, (b) SolO, (c) STEREO-A, and (d) Wind. Panels (e) and (f) show the direction +finding analysis of TIII-G1 and TIII-G2. +we determined the position and kinematics of the shock in the +corona. +Fig. C.1 suggests that the wave propagated quite rapidly at +the apex and reached a maximum possible speed of 1400 km s−1 +at around 07:25 UT. The shock apex was at a radial distance of +7 R⊙ at this time. The flank regions of the shock expanded slower +and reached their peak speeds of 900 km s−1 a little earlier at +07:15 UT when they were at a radial distance of 3 R⊙. According +to the ellipsoidal fit, both flanks expanded at a similar rate till +their respective peaks after which their velocities were slightly +different. +According to the analysis of the EUV wave and the WL +shock wave, the apex of the shock expanded considerably faster +than the flank regions. Comparing these with the kinematics of +the type II radio burst, we may get a first-order understanding +of their association. To do so, we converted the spectral drift +rate of the type II radio bursts into km s−1 using radial coronal +electron density models. We applied a 5-fold Newkirk (1961) +model to the metric and decametric type II bursts, and a 3.5- +fold Saito (1970) model for the hectometer bursts. These coro- +nal electron density models were chosen based on the projec- +tion analysis of type II radio sources at shock flanks and for +non-radial shock propagation done by Jebaraj et al. (2020). We +Article number, page 19 of 20 + +a) +C +e10 +a) +0.4 +PSP/RFS +e) +TIII-G1 +1 +0.2 +0.0 F +0.1 +-0.2 +HEEQ( +10 +-0.4 E +b) +SolO/RPW +-0.6 上 +106 +(MHz) + sfu +0.1 +F 682 kHz, 675 kHz, 675 kHz, 708 kHz +-1.0 +105 +-0.5 +0.0 +0.5 +1.0 +0.01 +XHEEQ (au) +Frequency ( +10 +STEREO A +104 +/WAVES +0.4 +f) +TIII-G2 +0.2 +103 +0.1 +0.0 E +Langmuir:waves +(au) +0.01 +-0.2 +10 +HEEQ +d) +-0.4 +TIII-G2 +/WAVES +Wind +-0.6 +1 +1013kHz,1025kHz,1025kHz,1040kHz +-0.8Fo +855 kHz, 875 kHz, 875 kHz, 916 kHz +764 kHz, 775 kHz, 775 kHz, 804 kHz +0.1 +-1.0 +TII-G1 +-0.5 +0.0 +0.5 +1.0 +00:90 +06:30 +07:00 +07:30 +08:00 +08:30 +):60 +XHEEQ (au) +Time (UT)A&A proofs: manuscript no. aanda +Fig. C.1. The kinematics of the reconstructed shock wave. The three axes of an ellipsoidal fit, namely, the shock apex (red), and the two flank axes +(r-axis 1 - blue and r-axis 2 - green) are plotted together and represented with different colours. The left and the right panel of the figure show the +height-time evolution of all the three shock axes, and its first derivative (velocity), respectively. The possible errors are in each axis and indicated +by the shaded regions. The "+" and "x" markings on the curves are the constraints provided by the observations. +obtained 1100±30 km s−1 for TII(1a) and TII(1b), which sug- +gests that the emission may have been from regions between the +flanks and the apex. Meanwhile, for TII(2a) and TII(2b), using +a similar approach, we obtained a drift speed of 700±50 km s−1 +associating it with the flank regions of the shock wave. The ± +values here have been calculated from the varying bandwidth +of the respective type II bursts over time. TII(2b) on the other +hand was treated separately due to its prolonged emission pe- +riod in the hectometer wavelength. We obtained a drift speed +of 750±50 km s−1, which corresponds well to the flank regions +of the shock wave during this time. Meanwhile, TII(HB) and +TII(W) are not as straightforward to analyse using a simple ra- +dial density profile such as the one applied here. The two struc- +tured type II bursts were most likely due to shock wave propaga- +tion in regions of enhanced density and therefore exhibit spectral +deformities. +Article number, page 20 of 20 + +Event: FLM1.6|2021-10-09T06:38:00 I spline3 (0.2) +Event: FLM1.6|2021-10-09T06:38:00 I spline3 (0.2) +20.0 - +1600 +17.5 - +1400 +15.0 - +[Rsun] +1200 +12.5 +Speed [km/s] +Length +1000 +10.0 +Height or l +800 +7.5 - +600 +5.0 +400 +h-apex +h-apex +2.5 - +200 - +r-axisl +r-axis1 +r-axis2 +r-axis2 +X +0.0 +-0 +2021 +2021 +2021 +2021 +2021 +2021 +2021 +2021 +2021 +2021 +2021 +2021 +2021 +2021 +Oct-09 +Oct-09 +Oct-09 +Oct-09 +Oct-09 +Oct-09 +Oct-09 +Oct-09 +Oct-09 +Oct-09 +Oct-09 +Oct-09 +Oct-09 +Oct-09 +06:30 +07:00 +07:30 +08:00 +08:30 +09:00 +09:30 +06:30 +07:00 +07:30 +08:00 +08:30 +09:00 +09:30 +Time [UT] +Time [UT] \ No newline at end of file diff --git a/ctE2T4oBgHgl3EQfFwaR/content/tmp_files/load_file.txt b/ctE2T4oBgHgl3EQfFwaR/content/tmp_files/load_file.txt new file mode 100644 index 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+page_content=' Vainio4, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Krupar7, 8, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Magdalenic1, 2, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Wiegelmann9, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Schuller5, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Battaglia10, 11, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Fedeli4 1 Center for mathematical Plasma Astrophysics-CmPA, Department of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2 Solar–Terrestrial Centre of Excellence—SIDC, Royal Observatory of Belgium, 1180 Brussels, Belgium 3 The Johns Hopkins University Applied Physics Laboratory, 11101 Johns Hopkins Road, Laurel, MD 20723, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4 Department of Physics and Astronomy, University of Turku, Finland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 5 Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 6 Department of Astronomy, University of Maryland, College Park, MD 20742, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 7 Heliospheric Physics Laboratory, Heliophysics Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 8 Goddard Planetary Heliophysics Institute, University of Maryland, Baltimore County, Baltimore, MD 21250, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 9 Max-Planck-Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 10 Institute for Data Science (I4DS), University of Applied Sciences and Arts Northwestern Switzerland, Bahnhofstrasse 6, 5210 Windisch, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 11 Institute for Particle Physics and Astrophysics (IPA), Swiss Federal Institute of Technology in Zurich (ETHZ), Wolfgang-Pauli- Strasse 27, 8039 Zurich, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We study the solar energetic particle (SEP) event observed on 9 October 2021, by multiple spacecraft including Solar Orbiter (SolO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The event was associated with an M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='6 flare, a coronal mass ejection (CME) and a shock wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' During the event, high-energy protons and electrons were recorded by multiple instruments located within a narrow longitudinal cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' An interesting aspect of the event was the multi-stage particle energization during the flare impulsive phase and also what appears to be a separate phase of electron acceleration detected at SolO after the flare maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We aim to investigate and identify the multiple sources of energetic electron acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We utilize SEP electron observations from the Energetic Particle Detector (EPD) and hard X-ray (HXR) observations from the Spectrometer/Telescope for Imaging X-rays (STIX) on-board SolO, in combination with radio observations at a broad frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We focus on establishing an association between the energetic electrons and the different HXR and radio emissions associated with the multiple acceleration episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We have found that the flare was able to accelerate electrons for at least 20 minutes during the nonthermal phase observed in the form of five discrete HXR pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We also show evidence that the shock wave has contributed to the electron acceleration during and after the impulsive flare phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The detailed analysis of EPD electron data shows that there was a time difference in the release of low- and high-energy electrons, with the high-energy release delayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Also, the observed electron anisotropy characteristics suggest different connectivity during the two phases of acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' type III radio emission;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Energetic electrons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Solar flares 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Introduction Acceleration of solar energetic particles (SEPs) during eruptive events may be associated with several different physical phe- nomena and mechanisms, such as solar jets and flares, CMEs, or shock waves (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' see Klein & Dalla 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Anastasiadis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Vlahos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Reames 2021), and they can accelerate particles to energies ranging from a few tens of keV to several GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The origins of SEPs measured in-situ have been a long- standing debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This is because it is difficult to distinguish be- tween several possible processes and interpretations using in-situ observations only near 1 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Previous studies suggest that both flare- and shock-related physical processes can contribute to the acceleration of SEPs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Kouloumvakos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Papaioan- nou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' however, it remains an open issue to quantify the contribution of each process to each species and a broad en- ergy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Energetic electrons are thought to be primarily accel- erated in the low corona, and the physical mechanisms respon- sible for the acceleration of electrons may then be constrained to reconnection, for example at solar jets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' see Krucker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Glesener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Musset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2022) or at current sheets forming at the wake of CMEs (Kahler & Hundhausen 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Klein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' It is not yet well under- stood if shock waves can have any role in the acceleration at high energies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' >1 MeV) through a drift acceleration at shock fronts (Ball & Melrose 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' While it is commonly believed that both protons and elec- trons can be accelerated to very high energies at magnetic recon- nection sites in the low corona, it is still not clear which condi- tions can facilitate the escape of the accelerated particles from the acceleration site to the interplanetary space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The direct injec- tion and escape of the energised electrons can be described in three scenarios, namely, the electrons released 1) by propagating shock fronts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Kouloumvakos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2022a), 2) CME interac- Article number, page 1 of 20 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='03650v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='SR] 9 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' aanda tion with the ambient magnetic field lines, 3) by open magnetic field lines rooted directly to the active region that gives direct access to interplanetary space (Masson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Depending on the proximity of the acceleration site to open magnetic field lines, the electrons can escape directly into the interplanetary space via open magnetic field lines and usually manifest as type III radio bursts (see, Reid & Ratcliffe 2014, for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Type III radio emission is produced when beams of ener- getic electrons resonantly generate Langmuir/slow-electrostatic waves which are linearly (Krasnoselskikh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Tkachenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Jebaraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2022) or non-linearly (Ginzburg & Zhelezniakov 1958;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Cairns 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Melrose 2017) transformed into electromagnetic waves at the plasma frequency and/or its harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Type III radio bursts are known to be associated with flares, jets, and other solar eruptive phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' There is also a very good connection of energetic electron events with type III bursts at energies below 300 keV (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Krucker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2007, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Klassen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2011a,b, 2012, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' On the other hand, efficient shock acceleration of electrons, while possible, may be constrained to the shock drift acceler- ation mechanism (SDA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Ball & Melrose 2001, and references therein) in regions where a quasi-perpendicular shock geome- try is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' However, the efficiency of energy gain from the mechanism itself is rather limited since most electrons are trans- mitted through the shock transition regardless of electron energy when the shock geometry tends towards perpendicular to the up- stream magnetic field (θBn ≈ 90◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Evidence of shock waves ac- celerating beams of energetic electrons is common during solar eruptions and can be seen manifesting as drifting type II radio emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Numerous studies of type II radio bursts have shown that the emission most likely arises from the upstream regions of the shock wave, implying an active acceleration process at the shock wave (Krasnoselskikh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Thejappa 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Jebaraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Kouloumvakos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2021) at multiple acceleration sites in some events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Aurass et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' (1998) have suggested that coronal shock waves are able to accelerate electrons to energies considerably higher than the background thermal population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A recent statistical study (Dresing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2022) has suggested that the acceleration of mildly-relativistic and relativistic elec- trons correlate rather well with critical shock parameters close to the Sun, namely, the shock strength (fast-magnetosonic Mach number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Similarly, good correlations have been found for high- energy protons as well (Kouloumvakos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This good correlation for electrons also suggests that efficient acceleration at shock regions with oblique and quasi-parallel shock geometry may also be present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' However, the mechanism of electron accel- eration in oblique shocks follows a diffusive shock acceleration mechanism (DSA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Bell 1978, and references therein), which is highly dependent on the shock wave’s ability to accelerate ions (for review;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Treumann & Jaroschek 2008) and the subsequent generation of upstream wave turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The electrons may then be trapped by the upstream waves and be accelerated through a Fermi acceleration mechanism similar to that of the protons (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Tsytovich 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Vaisberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Galeev 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Galeev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' McClements et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Gieseler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In this study, we investigate an SEP event observed on 9 Oc- tober 2021, by Solar Orbiter (SolO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Müller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The event was associated with a CME/shock wave and an M-class flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' During the event, high-energy protons and electrons were observed by multiple observers, such as, near-Earth spacecraft and Parker Solar Probe (PSP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Fox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Our motivation is to analyse what appears to be a second phase of electron accel- eration as suggested by SolO SEP electron observations, in com- bination with the complex radio observations that show multiple stages of particle energization during and after the flare impul- sive phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We find that this is an interesting aspect of the event, hence, our analysis is focused on establishing an association be- tween the energetic electrons and the different radio emissions during the impulsive phase of the flare and understanding the origin of the apparent second phase of electron energization af- ter the flare maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Specifics on the energetic proton obser- vations for this event can be found in Lario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This multi-faceted study is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We start with a brief introduction to the space-based and ground-based instrumentation and an overview of the event in Section 2 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In Section 4, we show observations of the so- lar event with a special emphasis on the X-rays (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1), the radio waves (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='2), and the solar energetic electrons (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In this section we also analyse the different obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The results of our analysis and our conclusions about the two apparent phases of electron energization are provided in Section 5 and 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Instrumentation For this study, we utilize data from instruments on-board SolO, PSP, Solar TErrestrial RElations Observatory Ahead (STEREO- A Kaiser 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Kaiser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2008), SOlar and Heliospheric Ob- servatory (SOHO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Domingo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1995), Solar Dynamics Obser- vatory (SDO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Pesnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2012), and the Wind (Harten & Clark 1995) spacecraft, as well as measurements from ground-based instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Below we give a summary of the data used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Energetic Particle observations: From SolO, we utilize en- ergetic particle measurements from different sensors of the Energetic Particle Detector (EPD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Rodríguez-Pacheco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020) instrument suite, namely Electron Proton Telescope (EPT), High Energy Telescope (HET), and the Suprathermal Electrons and Protons (STEP), in the energy range from a few keV to a few MeV for electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' From STEREO-A, we utilize SEP measurements from the High Energy Telescope (HET;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' von Rosenvinge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2008) and the Solar Electron and Proton Telescope (SEPT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Müller-Mellin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In addition, electron measurements from the 3DP (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1995) instrument on-board Wind have been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Hard and soft X-rays observations: We utilize hard X-ray (HXR) spectra and images from the Spectrometer/Telescope for Imaging X-rays (STIX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Krucker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020) on-board SolO and soft X-ray (SXR) observations from the Geosta- tionary Operational Environmental Satellite (GOES;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Garcia 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Radio observations: We utilize radio observations from both, ground-based and space-borne instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' For the inter- planetary part of the dynamic radio spectrum we utilize ob- servations from the Radio and Plasma Waves instrument (RPW;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Maksimovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020) on-board SolO and from the Radio Frequency Spectrometer (RFS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Pulupa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2017) part of the FIELDS electric antennas (Bale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2016) on- board PSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Observations from legacy instrumentation such as the SWAVES (Bougeret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2008) instrument on-board STEREO-A and the WAVES experiment on-board the Wind spacecraft (Bougeret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1995) are also employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We also utilize ground-based radio observations from the Yamagawa radio spectrograph (9 GHz - 70 MHz, Iwai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2017) and the e-Callisto network of radio telescopes (in particular, ASSA (Astronomical Society of South Australia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 80 MHz 16 MHz), Benz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' These observations cover a Article number, page 2 of 20 Immanuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Jebaraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' : Solar energetic electrons on 9 October 2021 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The right panel shows a view of the heliographic equatorial plane from the north, in the Stonyhurst coordinate system, and the spacecraft constellation on 9 October 2021 at 06:30 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The different coloured squares indicate the location of all observing spacecraft, namely, L1 (Earth, green), STEREO-A (red), SolO (blue), and PSP (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The nominal Parker spiral (curved line) and line of sight (dotted lines) from the Sun to each spacecraft is shown for each observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The small back arrow and the dashed black line indicate the flare location and the reference Parker spiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The large black arrows help locate the in-situ omni-directional electron recordings at SolO and STEREO-A, which are presented in the two rectangular panels on the left (note that STEREO-A/HET data is not omni-directional, but pointing almost perpendicular to the nominal Parker spiral direction in the ecliptic plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' broad range of the radio wavelengths, from the millimetric to the decametric domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Extreme ultra violet observations (EUV): We utilize observa- tions from the Atmospheric Imaging Assembly (AIA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Lemen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2012) on-board SDO and the Extreme Ultra Violet Im- agers (EUVI), part of the Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2008) in- strument suite on-board STEREO-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Coronagraphs white light observations (WL): We utilize coronagraphic observations in WL, from the Large An- gle and Spectroscopic Coronagraphs (C2 & C3) (LASCO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Brueckner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1995) on-board SOHO and the two coron- agraphs (COR1 & COR2), part of the SECCHI instrument suite on-board STEREO-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Event overview The event on 9 October 2021, was one of the first major erup- tions of Solar Cycle 25 and was associated with an M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='6 flare in GOES classification, a filament eruption, and a halo CME originating from the NOAA AR 12882.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The active region had a βγ configuration of its photospheric magnetic field 1 and was located at the central meridian (N20W01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Soft X-ray (SXR) observations from GOES showed that the solar flare started at 06:19 UT, peaked at 06:38 UT, and decayed to flux levels between the maximum and the pre-flare background level at 06:53 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The flare was also observed in hard X-rays (HXR) by the STIX instrument on-board SolO, which at that time was at a heliocentric distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='68 AU and was separated from the 1 based on the classification provided in: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' solarmonitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='org/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='date=20211009®ion=12882 Sun-Earth line by 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='2◦ to the east (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Hence, the flare was close to the disk centre for both SolO and near-Earth assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Radio observations were also very rich during the event, showing several complex spectral features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We provide details about the X-ray and radio observations in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Figure 1 shows an overview of the energetic electron obser- vations from SolO (EPD-EPT) and STEREO-A (SEPT) that ob- served the SEP event on 9 October 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' All the spacecraft, and in particular SolO and STEREO-A, clearly observed an intensity increase of electrons at energies >1 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The heliospheric view on the right panel of Figure 1 shows the heliographic equato- rial plane from solar north and illustrates the spacecraft loca- tions close to the start of the solar event, on 9 October 2021 at 06:30 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The new solar mission spacecraft (SolO and PSP), STEREO-A, and near-Earth spacecraft were closely positioned in a quadrant and covered a narrow range of heliolongitudes of about 50◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' During the SEP event, SolO (located at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='68 AU), and STEREO-A (located at 1 AU) were both trailing Earth for about 15◦ and ∼40◦, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In the heliospheric view of Figure 1, we also show the Parker spiral field lines connecting each space- craft with the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' For the illustration of the spacecraft location, we used the Solar-MACH tool2 (Gieseler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' For each spacecraft, we used a solar wind speed of about 350 km/s to cal- culate the Parker spiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We see that theinterplanetary magnetic field (IMF) lines connecting STEREO-A and SolO with the Sun are spatially close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We estimated the longitudinal difference of the magnetic footpoints at the solar surface to be around 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='9◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' From the electron recordings (left panel of Figure 1, we see that at SolO there is a rapid rise of the electron flux that lasts for 2 https://solar-mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='io Article number, page 3 of 20 180° OMNI-DIRECTIONAL ELECTRONS 105- r s MeV) 225° 135° 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' SOLO EPD Flux / (cm² sr EPT 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='2-45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='9 keV EPT 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='7-67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='3 keV 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' EPT 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4-101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1 keV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0 EPT-142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='3-154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' EPT 218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1-237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1 keV 10-1 HET 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='45-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='04 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='5 HET 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='05-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4 MeV 270° .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='06 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2021-10-09 (cm² sr s MeV) 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 06:30/UT STEREO-A SEPT 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0-55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0 keV STEREO A 101 SEPT 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0-125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0 keV Earth ) /xn SEPT 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0-255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0 keV BepiColombo Parker Solar Probe SEPT 375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0-425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0 keVL tu 315° Solar Orbiter HET 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='7-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4 MeVM 10-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' field line connecting to HET 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='8 MeV ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' (vsw=350 km/s) 06:00 07:00 08:00 00:60 10:00 11:00 reference long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Time [UT] of 2021-10-09 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' aanda Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' (a) GOES soft X-ray fluxes shows the thermal emission of the M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='6 flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' (b) STIX hard X-ray count rates in five broad energy bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The STIX times have been shifted to be consistent with the GOES ob- servations from 1 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Results of the spectral fitting of STIX spectra show: (c) the background-subtracted STIX count rate in the 25–50 keV range, (d) the spectral index of the injected electrons, δ, (e) the low- energy cutoff, ELC, and (f) the total injected electron flux, as well as the fluxes above 50 and 100 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The five major nonthermal peaks are indicated by red dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' about 15 minutes until the maximum, and then there is a second more gradual increase after the maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This second increase, however, was not observed by the closely connected STEREO-A spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Observations and data analysis 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' X-ray observations Figure 2 (panels (a) and (b)) shows the GOES soft X-ray fluxes in comparison with the STIX hard X-ray count rates in five wide energy bands (note that 160 s have been added to the STIX times to account for the light travel time difference between SolO and GOES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' While the count rates below 15 keV show the smooth time evolution characteristics for the thermal bremsstrahlung produced by hot plasma, the multiple spikes visible at higher energies suggest several episodes of electron acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' To in- vestigate this quantitatively, we have forward-fitted a time se- ries of observed STIX count spectra with the combination of an isothermal model and a thick-target nonthermal component (Brown 1971), using the OSPEX spectral analysis tool3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The fit results for the nonthermal component are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2 (panels (c), (d), (e), and (f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Similarly to panels (a) and (b) of the same figure, the times have been shifted to be con- sistent with a viewpoint at 1 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The background-subtracted count rates in the range of 25–50 keV (panel (c)), show at least five major peaks, indicated by dashed vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The spec- tral index δ of the injected electron flux, in panel (d), shows a clear anti-correlation with the nonthermal count rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This is the well-known soft-hard-soft behaviour (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Grigis & Benz 2004), and it is indeed present in each peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Overall, the spec- trum gets harder over time (soft-hard-harder evolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This be- haviour was observed in some solar flares (Grigis & Benz 2008), particularly in those associated with interplanetary proton events (Kiplinger 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Next, panel (e) shows the low-energy cutoff ELC, which lies mostly in the range of 12–16 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This quantity is rather ardous to constrain (note the large uncertainties), and we can only determine the highest ELC that is consistent with the data because the true cutoff is usually masked by the thermal emission (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Warmuth & Mann 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Adopting this, panel (f) shows the total injected electron flux, which reflects a lower estimate for this quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Additionally, we show the fluxes above 50 keV and 100 keV in red and blue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Note that the HXR count rate peaks are indeed associated with higher electron fluxes, which are most clearly seen at higher energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' To investigate electron acceleration in more detail, we per- formed image reconstruction of the hard X-ray sources using the STIX pixelated science data (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Krucker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Battaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Massa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Figure 3 shows the evolution of the thermal (6–10 keV, red contours) and nonthermal X- ray sources (22–70 keV, blue contours) reconstructed with the MEM_GE imaging algorithm (Massa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Countour lev- els start at 20% of the maximum intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The image times cor- respond to the five major nonthermal peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' While the integra- tion time was one minute for the thermal images, it varied be- tween one and four minutes for the nonthermal peaks in order to optimize counting statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The sources are overplotted on AIA 1600 Å images that have been rotated to conform with the STIX viewpoint (see Battaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The inverted colour table shows the flaring ribbons and kernels in dark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The UV images show that the flare had an unusually complex morphology with multiple ribbons and kernels, lying partly at right angles to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The X-ray sources were co-aligned with the rotated AIA im- ages using the pointing information provided by the STIX As- pect System (SAS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Warmuth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020), which has a nominal positioning accuracy of ±4".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We see that during the first two nonthermal peaks, two HXR footpoints (FPs) are located at the southern end of the eastern flaring ribbon and the western flaring kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' At the second peak, an additional weaker FP appears near the central part of the eastern ribbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' At the third nonthermal peak, the southeastern FP has faded away, and we have again the classical two-footpoint configuration, which is also maintained 3 http://hesperia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='gov/ssw/packages/spex/doc/ Article number, page 4 of 20 10-4 a) 1-8 A 10~5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='5-4 A / m"] M 10*6 flux GOES 10° 10 10 105 4-10 keV 10-15 keV 104 15-25keV count rate 25-50keV 103 50-84 keV 102 XIIS 101 L 10° 06:30 06:40 06:50 07:00 07:10 07:20 200 25-50 keV c) 150 rate 100 E count 50 0 8 d) 7 5盒 3 e) [keV] 10 ELc 5日 0 total 1035 >50 keV 1034 >100 keV flux 1033 electron 1032 1031 06:30 06:36 06:42 06:48 06:54 9-Oct-2021 [UT]Immanuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Jebaraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' : Solar energetic electrons on 9 October 2021 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Flare evolution as seen in a series of SDO/AIA 1600 Å images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Due to the inverted colour table, the flare ribbons and kernels are dark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The AIA frames have been rotated to conform to the view from SolO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The provided times represent the recording times of the AIA frames (UT at Earth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' STIX images showing the thermal and nonthermal X-ray sources are overlaid as red and blue contours, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' during the subsequent two peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' While the western FP remains stationary, the eastern FP moves along the flare ribbon to the northeast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Note that the eastern FP has nearly faded away in the fifth peak, so that in contrast to the other panels, the nonthermal contour lines shown here start already at the 5% level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In an effort to visualise the complex STIX observations with respect to the magnetic field topology of the active region, we have performed a non-linear force free field extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The details of the analysis are provided in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The HXR sources and their movement correspond well with the western part of the AR and the dominant footpoint rooted in the positive polarity region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The shifting of the HXR footpoints is seen in the south-eastern parts of the AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In summary, we conclude that the flare was able to accelerate electrons from at least 06:28 to 06:54 UT, mostly in the form of five discrete pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A change of magnetic topology appears to have taken place between the first and third nonthermal burst, shown by a significant shift of the nonthermal emission in the eastern part of the flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Radio observations In Figure 4, we show a composite dynamic radio spectrum constructed from the observations of several ground-based and space-borne instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Such a composite spectrum provides uninterrupted high time and frequency resolution observations, starting from microwave wavelengths that are generated at alti- tudes very close to the chromosphere and up to the kilometric wavelengths, correspond to the interplanetary space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The com- bined observations indicate a wide variety of coronal processes related to the acceleration and propagation of nonthermal elec- trons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The solar radio event is rich with different types of radio emissions i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', type II, III, radio bursts, IV continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In mi- crowave wavelengths (9 GHz – 1 GHz) we observe a diffuse ra- dio emission which is most-likely gyro-resonant in nature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Bastian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1998) and is emitted by near-relativistic electrons (≈100 keV) trapped in the flaring loops (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Nindos 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The low-decimetric to decametric wavelengths (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', from ∼1 GHz to ∼20 MHz) are mostly dominated by plasma emission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', type II, III, IV radio emissions) produced by fast electron beams that are associated with flares, propagating shock waves, and elec- trons trapped within flare loops or in flux rope CMEs (McLean & Labrum 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' From the composite dynamic radio spectrum of these wavelengths (Figure 4), we distinguish different type III radio bursts, some of them associated with the HXR pulses, multiple parts of type II burst that exhibit various fine structures, and a type IV continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' At hecto-kilometric wavelengths, the interplanetary radio emissions associated with the event were ob- served by all spacecraft namely, PSP, SolO, STEREO-A, Wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We observe two main type III radio burst groups and two patchy parts of type II radio burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The event of 9 October 2021, was first observed in radio wavelengths with the emission starting at around 06:30 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The diffuse radio emission observed in microwave wavelengths lasted for the entire duration of the flare and corresponds well with the HXR pulses as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2, from 06:30 to 06:54 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' At the beginning of the event, we observe the first type III ra- dio burst (TIII-G1), which was associated with the first peak in HXR during the flare impulsive phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The first properly dis- tinguishable signature (although faint) of G1 was observed at 06:31:40 UT, starting at ∼40 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Apart from the TIII-G1 and the microwave continuum, near the very beginning of the event, we observe structured and narrowband spike-like emission at around 350 MHz, at 06:30 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Similar type of emission was also observed at higher frequencies, around 1000 MHz, at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The morphology and the apparent drift of the fea- tures suggest that they could be precursors (Fárník et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Pohjolainen 2008) to the type II radio burst observed a few min- utes later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The first part of the type II (TII(1a)) radio burst was observed at 300 MHz around 06:33 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This 2nd harmonic emission lane (2fpe) shows rather patchy morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The fundamental lane ( fpe) of the same type II was observed later (06:34:20 UT) start- ing at about 90 MHz, and it was considerably patchier than the second harmonic and consisted of distinguishable fine structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The nature of the narrow-band patchy features indicates rather localized source regions with rapidly changing plasma condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Such morphological characteristics are common in the case of metric-decametric type II bursts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Cairns et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Kouloumvakos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Jebaraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The harmonic component was observed at ∼250 MHz meaning that a shock wave was formed at ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4 R⊙ (applying a 2-fold Newkirk coronal electron density model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Newkirk 1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The coronal electron density is much higher in the corona close to the flaring region and at regions that are probably dominated by closed magnetic field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The radio emission ended at around 06:42 UT, and at 23 MHz and 50 MHz for the fundamental and harmonic lanes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Recently, Ramesh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' (2022) imaged the 80 MHz harmonic component of TII(1a) at ≈ 06:38 UT, which corresponds to the end of the inferred injection time of the low-energy electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The source of TII(1a) according to this study was located close to the southwest periphery of the flaring active region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This po- sition roughly coincides with the direction of the strong EUV wave expansion, toward the southwest direction from the source region, where the open magnetic field lines were connected to SolO and STEREO-A (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Simultaneously with the TII(1a), we traced also a second, more diffuse and broadband radio emission possibly also type II Article number, page 5 of 20 AIA 1600 A - 9-Oct-2021 06:31:02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='130 AIA 1600 A - 9-Oct-2021 06:33:02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='120 AIA 1600 A - 9-Oct-2021 06:37:02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='130 AIA 1600 A - 9-Oct-2021 06:40:38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='130 AIA 1600 A - 9-Oct-2021 06:45:50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='130 500 500 500 500 500 450 450 450 450 450 400 Y 350 (0 350 350 350 350 6-10 keV 22-70 keV 300 300 300 L 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 X (arcsec) X (arcsec) X (arcsec) X (arcsec) X (arcsec)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' aanda Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The radio event associated with the flare/CME on 9 October 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Panel (a) shows the full radio spectrum from the Earth vantage point including both ground-based and space-borne observations in the range of 10 GHz to 20 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The different types of radio bursts are marked on the spectra with their respective abbreviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Panel (b) presents a zoomed in view of the decimetric-metric-decametric observations that showcase different structured radio emission, such as herringbones and a stationary flare continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The details of the structured radio emission, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' type II herringbones(TII(HB)) and very narrow-band wavy part of the type II (TII(W)), are presented in panels (c1) and (c2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' burst (TII(2a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This emission lane had a comparable drift rate to the one of TII(1a), and it was observed starting at high fre- quency ∼600 MHz, at around 06:36 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Approximately at the same time, also a continuum emission was observed, starting at ∼1 GHz (at around 06:31 UT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The continuum emission seems to be superposed with the TII(2a) burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Such a synchronous appearance makes the separation of the two different types of radio emission very difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A continuation of the broad-band continuum was also observed after the cessation of TII(2a), but with variable intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The intense part of the broadband contin- uum with fine structures (marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4a,b as TIV) started at 06:47 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The stationary TIV extended from ∼600 MHz to 200 MHz and consisted of intense broadband pulsation, extending till about 08:00 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The second and most intense group of TIII-G2 radio burst was observed after the flare impulsive phase at around 06:37 UT, starting at ∼70 MHz and corresponding well to one of the HXR pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We find no clear evidence to confirm that the type III bursts emanate from the type II burst, so called type II-associated bursts (Cane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Dulk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' However, concur- rently with the appearance of the TIII-G2 radio burst a clear brightening of a type II was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The distinguishable her- ringbone features (TII(HB)) at the decametric wavelengths (40 MHz fundamental emission) can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' TII(HB) was ob- served together with TIII-G2 starting from 06:41 UT and marks a period of very efficient electron acceleration at the shock wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' It is possible that some of the electron beams generating TIII-G2 were accelerated by the shock wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We show the details of the corresponding dynamic spectrum in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='(c1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The fast drift- ing herringbone bursts originating from both sides of the type II backbone are generated by the fast electron beams accelerated in the upstream region of the shock wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The geometry of the shock wave in these regions is close to θBn ≈ 90◦, which is evi- dent by the lack of a bright backbone (Stewart & Magun 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The drift rate of the herringbone features and the irregular mor- phology of the backbone may reflect the physical characteristics of either the shock wave or the variations in the local plasma frequency close to the shock wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The spectral irregularities of the backbone structure are somewhat to be expected when the shock wave interacts with an electron-rich environment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', a streamer) and therefore a reverse drift into higher plasma fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A few minutes (∼5 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=') after the end of the TII(HB), we ob- served a similar (wavy) type II-like burst (TII(W)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The TII(W) burst starts at 06:50 UT, and as it is unclear if it is the decamet- ric continuation of TII(2a) ,we distinguish it just as TII(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A zoomed-in view of TII(W) is presented in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' (c2), which shows that the second harmonic of the emission was brighter than the fundamental, which is opposite to the fundamental brightening in the case of TII(HB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' While TII(W) also appears to have some herringbone-like features, they are not as clear as those observed in TII(HB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Another important characteristic of TII(W) is also the presence of a bright backbone together with the herringbones, indicating a shock wave that is probably not as perpendicular as TII(HB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The two main parts of the type II radio burst, namely TII(1a) and TII(2a), observed in the metric wavelengths were also ob- served in the hectometer wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The hectometric counter- Article number, page 6 of 20 Composite radio dynamic spectra a) Ic1) b) C2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1 MHz TII(W) TII(HB) 1 TII(G1) TII(G2) TII(2b) (1b) TIV Tll(2a) b 1000 6:35 6:40 6:45 6:50 6:55 7:00 100 Il(1al c1) c2) Pulsations AII 1000 4 HB H STIX25-50keV 8 6:42 6:44 6:46 6:51 6:53 6:55 6:30 7:00 7:30 8:00 Time (UT)Immanuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Jebaraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' : Solar energetic electrons on 9 October 2021 part of the two parts of the type II burst were also noticeably patchier and bursty compared to their decametric counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The continuation of the two parts of the metric type II radio burst was recorded by the space-based instruments and it is marked as TII(1b) and TII(2b) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The TII(2b) was observed in the hectometer range starting at 7:08 UT, which was about 12 minutes after the appearance of the TII(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The relative bandwidth of the hectometric type II burst was comparable to that of the decametric, suggesting that the source of the type II bursts may have been located in a rela- tively small region of the shock wave (Schmidt & Cairns 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Kouloumvakos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Jebaraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Both bursts, namely, TII(1b) and TII(2b) come to a cessation at 07:07 UT (6 MHz) and 07:40 UT (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='5 MHz), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Around the same time that the type II radio emission ap- peared in the hectometric wavelengths, we continued to observe also the TIV with sporadic broad-band radio pulsations in the metric wavelengths centred at 150 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This emission was most likely produced by electrons trapped in post-flare loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Hecto-kilometric observations of the two type III groups, namely, TIII-G1 and TIII-G2 were recorded by all available space-borne observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' At kilometric wavelengths, both TIII-G1 and G2 were also observed together with Langmuir waves at SolO and STEREO-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Langmuir waves are fundamental plasma waves that are generated by the electron beam, which can be subsequently converted into type III radio emission observed at large angles (Jebaraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In-situ Langmuir waves are an indication of the electron beams propagating at close prox- imity to the observing spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Despite the fact that all the spacecraft observed the different type III bursts, only STEREO- A and SolO observe Langmuir waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This confirms the passage of the type III generating electron beams through the position of the spacecraft and indicates that the magnetic connectivity of the two spacecraft with the electron beam was most likely simi- lar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' On the other hand, PSP and Wind did not observe Langmuir waves, and this indicates that the electron beams did not prop- agate in their vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' An in-depth analysis of the directivity of both TIII groups is presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Remote sensing observations in EUV and WL The solar event on 9 October 2021, was associated with a CME and a large-scale propagating coronal wave that was observed in both EUV and WL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' While there has been an extended debate on the physical nature of such large-scale coronal disturbances (for a review, see Warmuth 2015), they are now generally interpreted as signature of fast-mode waves or shocks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The expansion of the EUV wave was first observed by SDO/AIA at 06:30 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Figure 5 presents SDO/AIA running difference im- ages at 193 Å, from different times of the EUV wave evolu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Due to the presence of a small coronal hole and a streamer south of the flaring active region, the evolution of the wave was rather constrained and deformed in the low corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The pres- ence of pre-existing magnetic structures such as coronal holes and streamers can affect the propagation and characteristics of large amplitude waves (Vršnak & Luli´c 2000a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This mainly occurs because, in these regions, there is a rapid change in the characteristic speed of the medium (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', the fast-magnetosonic speed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The evolution of the low-coronal EUV wave shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 5 was observed to be rather strong in the southwest quad- rant away from the eruptive source region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Such an intense evo- lution of the EUV wave in this direction was also accompanied by a propagating pressure wave at higher altitudes, which was observed by the coronagraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We utilise the information from radio observations in tandem with EUV images to understand the evolution of the pressure wave and the formation of a shock wave in the low corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The presence of multiple type II radio bursts (described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='2) suggests that a shock wave was present very early during the event at multiple locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' TII(1a) was observed at a lower fre- quency, while TII(2a) was observed at a much higher frequency 3 minutes later (≈ 06:36 UT, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Such a disparity in start- ing frequencies suggests that the source of TII(1a) was possibly closer to the leading edge of the pressure wave, while the source region of TII(2a) could have been located closer to the flanks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Near the start of both TII(1a) and TII(2a), the propagation and interaction of the wave with closed field magnetic structures at the periphery of the active region most likely led to favourable conditions for electron acceleration and the subsequent genera- tion of type II radio emission in regions close to the apex and the flanks of the wave (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2016, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Kouloumvakos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' After 06:41 UT, the wave evolution was rather strong in the southwest direction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Close to the start of the TIII-G2 and also TII(HB), we observed an interaction of the EUV wave’s southwest flank with open field lines that are probably part of a streamer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 5a, we show the open magnetic field lines that connect to STEREO-A, SolO, and Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' These field lines were derived using the Potential Field Source Surface (PFSS) model (Schrijver & De Rosa 2003) and Air Force Data Assim- ilative Photospheric Flux Transport (ADAPT) magnetograms (Arge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The interaction of the pressure wave with the magnetic structures in this region was most likely an important catalyst for efficient electron acceleration that we observed in radio wavelengths (TIII-G2 and TII(HB) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4c1) and in-situ energetic electrons observed by SolO and STEREO-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Further- more, during the time of these interactions, both spacecraft were connected to the EUV wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' When the TII(HB) first appears in the radio spectrum at 06:41 UT, we observe that the EUV wave propagated past the streamer towards the boundaries of a coronal hole southwest of the active region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This area is marked roughly by the red ar- rows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 5b at 06:43 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' When the wave interacted with this coronal hole, part of the wave was reflected by the coro- nal hole boundary at ∼06:50 UT (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 5c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A faint part of the wave transmitted across the coronal hole boundary and also re- fracted into regions where it was more favourable for the shock to exist, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', higher in the middle corona where the local speed of the medium decreases (Uchida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Warmuth & Mann 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Due to the low density of the coronal hole region and the high magnetic field strength (increased characteristic speed of the medium), the large amplitude wave may propagate faster and also dampen at an increased rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' During and well after this part of the interaction at the coronal hole boundary, we observed the TII(W) presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In WL, the pressure wave was first observed by SOHO/LASCO C2 coronagraph at 07:12 UT as a halo-shaped coronal wave propagating higher in the middle solar corona along the position angle (PA) 263◦ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', in the south-west quad- rant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This part of the WL shock wave corroborates well with the fastest component of the EUV wave as we discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The observations also validate the aforementioned refraction of the pressure wave higher in the corona, where there may exist more favourable conditions for the formation of a shock wave (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', lower Alfven speed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' As the event originated close to the central meridian, the pressure wave was observed as a halo event Article number, page 7 of 20 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' aanda Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The evolution of the EUV wave as observed by SDO/AIA 193 Å at three time instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Panel (a) shows the EUV wave close to the SXR peak time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The open magnetic field lines connecting to both SolO, STEREO-A, and Wind are drawn in blue, red, and green, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The red line across the Sun face represents the heliospheric current sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Panels (b) and (c) show the EUV wave expansion at 06:43 UT and 06:51 UT roughly corresponding to TII(HB) and TII(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The red arrows indicate the region of interest where the wave also undergoes reflection at the coronal hole boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The blue, red, and green markings in (b) and (c) represent the spacecraft connections shown in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' encircling the solar disk from the Earth’s point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Due to projection effects, tracking the shock wave becomes increasingly difficult at later times in both LASCO C2 and C3 coronagraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' At STEREO-A, the event was observed ∼40◦ closer to the west limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The pressure wave was first observed in the COR1 coronagraph at 06:46:18 UT and in the COR2 coronagraph at 07:08:45 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' At the time when the pressure wave enters the COR1 field of view, all the observed radio emission was in the decametric wavelengths (mid-coronal plasma) and the TII(1a), TII(HB), and TIII-G2 were near their cessation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This shows that the most efficient electron acceleration took place when the shock wave was still located low in the middle corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The pres- sure wave evolved fastest along PA 266◦ which suggests that this was almost the same leading-edge identified in the LASCO C2 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The wave was highly deformed by the presence of two streamers at PA 230◦ and at PA 280◦ further suggest- ing the interaction between the pressure wave and the various density structures on the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' These interactions may also have contributed to the patchy and bursty yet continuous observations of the TII(1b) and TII(2b) at hectometric wavelengths until their cessation at 07:07 UT and 07:40 UT, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The kinemat- ics of the shock wave and their association with TII(1a), TII(ab), TII(2a), and TII(2b) are presented in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Energetic electrons 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Pitch-angle distributions Figure 6a shows the electron event observed by SolO/EPT in the 40–50 keV energy channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The top panel, which shows the intensities as measured by the four viewing directions of EPT, reveals a strongly anisotropic event starting at ∼6:40 UT fol- lowed by an isotropic phase starting around 7:15 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The second panel shows the pitch-angle coverage of EPT’s four viewing di- rections, which is ideal and exceptionally stable over the course of the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The good pitch-angle coverage allows not only to determine the strong first order parallel anisotropy (shown in the bottom panel) in the early phase of the event, but also to verify that the later isotropic phase is real and not caused by poor-pitch angle coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Figure 6b shows the electron event at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4 MeV measured by SolO/HET, which provides the same four viewing directions, allowing us to determine the pitch-angle distribution also at MeV energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The initial phase of the MeV-electron event shows a significantly smaller anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' However, the time series still reveals a two-component picture with the later, almost isotropic phase starting like the one observed by EPT around 7:15 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A smaller anisotropy for higher electron energies is expected be- cause of the energy-dependence on pitch-angle scattering, which is stronger for higher energy electrons than for the lower en- ergy ones (Dröge 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Agueda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Strauss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' However, in contrast to the low energy electrons (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 6a), the intensity level of the two components is rather similar at MeV energies, with the later component being even slightly more in- tense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' At 40–50 keV (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 6a), the first, anisotropic component is much more intense than the second, isotropic component, in- dicating a softer spectrum of the first component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This is con- firmed by analysing the peak spectra of both components (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Figure 6c shows the 45–55 keV electron event observed in the four viewing directions of STEREO-A/SEPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Because the spacecraft was put upside down after its superior solar conjunc- tion in 2014, the SEPT sun and anti-sun sectors do not point anymore along a nominal Parker spiral field line but perpendicu- lar to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This causes often non-ideal pitch-angle coverage, which is, however, not too bad during the electron event of 9 October 2021, as shown by the 2nd and 3rd panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 6c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The onset of the event is observed by the sun sector of SEPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' However, during the rise phase of the event, the magnetic field direction changes at ∼7 UT, so that the anti-sunward propagating beam is then observed best in the anti-sun telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We note that the pointing of the SEPT instrument has changed after the superior conjunction in 2015, since when the spacecraft was put upside down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Therefore, the north and south telescopes are swapped and the sun and anti-sun telescopes point perpendicular to the nominal Parker spiral (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 8 in Badman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2022) At the time of the onset the pitch-angle coverage is ideal showing a smaller anisotropy than at SolO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The rise phase of the event also shows a smaller anisotropy than observed by SolO, with no in- dication of two distinct components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' However, this could be po- Article number, page 8 of 20 AIA 193 A 2021-10-09 06:37:04 AIA 193 A 2021-10-09 06:43:04 AIA 193 A 2021-10-09 06:51:04 1000" 1000" b) 1000" a C Helioprojective Latitude (Solar-Y) 500" 500" 500" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0 500" 500" 500" 1000" 1000" 1000" 1000" 500" 0" 500" 1000" 1000" 500" 0" 500" 1000" 1000" 500" 0" 500" 1000" Helioprojective Longitude (Solar-X) Helioprojective Longitude (Solar-X) Helioprojective Longitude (Solar-X)Immanuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Jebaraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' : Solar energetic electrons on 9 October 2021 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In-situ electron recordings by multiple observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Energetic electron event observed by SolO/EPT (a) and HET (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The top panels of (a) and (b) show the intensities of 40–50 keV and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4 MeV electrons as observed by the four different sensors of the EPT and HET instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The second panels show the corresponding pitch-angle coverage, and the third panels show the pitch-angle dependent intensities with intensity level marked by colour coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The bottom panels show the first order anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Energetic electron event observed by STEREO-A/SEPT in the 45–55 keV energy channel (c) and Wind/3DP (d) in the 50–82 keV channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The panels have the same content as for (a) and (b), but (d) shows a longer time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' tentially masked by the non-ideal pitch-angle coverage during the rising phase of the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Unfortunately, STEREO-A/HET provides only one viewing direction, which is aligned with the pointing of SEPT-sun so that we cannot infer the anisotropy at MeV energies for STEREO-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Sectored electron observations by Wind/3DP are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 6d with a longer time period than shown in the other pan- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The event is much more gradual and less intense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' How- ever, the initial phase shows some anisotropy, followed by a sec- ond step with less anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' At a first glance, this might ap- pear like the same two-component picture as observed by SolO, but the time periods are significantly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' While the first, anisotropic component lasts about 30 min at SolO, it has a du- ration of about five hours at Wind so that a one-to-one corre- spondence is unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The second increase coincides with the crossing of a magnetic sector boundary and is therefore likely related with a change of magnetic connectivity rather than with a new injection at the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Energy spectra observed by SolO SolO is the only spacecraft that clearly observes a two compo- nent electron event as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Figure 7 shows peak electron spectra observed by the three EPD instruments STEP (orange), EPT (red), and HET (brown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' For EPT and HET, we use the sunward-looking telescopes that align with the viewing direction of STEP and cover the usual incoming direction of SEPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The figure shows a spectrum for each of the two compo- nents, with circles (squares) denoting the first (second) compo- Article number, page 9 of 20 a) b) Solar Orbiter (R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='68 au) Solar Orbiter (R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='68 au) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0406 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0497 MeV EPT Electrons MeV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0530-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4010MeV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='HET Electrons ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='20:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='00:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='Date / Time in year 2021 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='Date / Time in year 2021A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' aanda Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Electron peak intensity energy spectra determined for the two SEP components of the electron event observed by SolO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' nent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The light-grey points represent the pre-event background spectrum that has been subtracted from each of the two spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Points with dark-grey colour represent energy bins that did not show a significant increase above the background level (HET) or which were contaminated by ions (STEP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Both spectra have been fit with a broken power law using the methods described in Dresing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Strauss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The resulting spectral indices below and above the break as well as the break energy are provided in the figure legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We only use part of the STEP en- ergy channels in the fit to avoid fitting the upturn in the very low energy range observed by STEP, which could be caused by mix- ing with another low-energy event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We also exclude the first en- ergy channel of HET from the fit because of instrumental effects that lead to too low intensity measurement, which has been fixed in a patch uploaded to the spacecraft only after this event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Com- pared to the first component, which has a spectral index above the break energy of (δ2 = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='64±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='18), the spectrum of the sec- ond component is clearly harder (δ2 = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='67±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' At energies ≳ 200 keV, the spectra even intersect, meaning that the second component is more intense than the first one at these energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This suggests that the second component is formed by a distinct particle injection, rather than being a part of the first compo- nent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Furthermore, the significantly harder spectrum of the sec- ond component suggests a more efficient acceleration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Release times The electron event is well observed at SolO, especially in the near-relativistic range covered by EPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We determine the onset time at the spacecraft of each energy channel using a statisti- cal quality control scheme that is designed to decide when the quality of the monitored variable changed from being in con- trol to out of control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' There are many different quality control schemes, and cumulative sum (CUSUM) schemes are used in many industries due to their capability to give an early warn- ing of changes in the monitored process (Page 1954).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A tradi- tional CUSUM method assumes that the variable in question is normally distributed, and in the case of the monitored vari- able having a Poisson distribution, the quality control scheme used should be the Poisson-CUSUM (Lucas 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Huttunen- Heikinmaa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2005), which we apply here for the determi- nation of the onset times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' For the parameters of the method we use the mean and stan- dard deviation of intensity measurements in the chosen pre-event background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We average the original measurements from 1 sec- ond resolution to 30 second resolution, in order to minimise the effects of counting noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Furthermore, to avert false positives in the determination of onset times, we demand that at least 60 con- secutive out-of-control signals must be counted by the method before identifying the onset of the event as the first of these sig- nals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This limit corresponds to 30 minutes of intensity measure- ments exceeding the threshold of Poisson-CUSUM method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Using these onset times, we perform a velocity dispersion analysis (VDA, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Vainio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2013) to infer the common so- lar injection time of the observed electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Figure 8 shows the onset times as a function of the inverse unit-less speed (1/β = c/v) of the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Blue symbols correspond to the onset times observed in the different energy channels of EPT, which is capa- ble of observing electrons in the energy range 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='2–471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4 keV, and red symbols mark HET observations, which measures elec- trons in the MeV range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' For some channels the onset could not be resolved, which is why there are fewer data points in the plot than there are total channels in EPT and HET combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The VDA method assumes that particles of all energies were injected at the Sun into the interplanetary space at the same time and that their propagation path length is identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Consequently, the onset times as a function of inverse beta should align them- selves to a line with a rising slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This slope then allows to de- termine the length of the path travelled by the electrons, and the intersection with the vertical time axis tells the time of the com- mon particle injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The grey lines in the background of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 8 represent slopes that one would get with the nominal Parker spi- ral path length of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='736 AU, assuming a solar wind speed of vsw = 360 km/s and taking into account SolO’s heliocentric dis- Article number, page 10 of 20 61=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='22 62 = -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='64±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='18 α= 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='48 Eb = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0 106 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0 keV 61=-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='14±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='09 62 = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='67±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='19 α= 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='17 oreg Eb = 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='5 104 ±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0 keV sr MeV) STEP 1st comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' EPT sun 1st comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' HET sun 1st comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' + not significant cm² 102 申 STEP 2nd comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 串 EPT sun 2nd comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 串 HET sun 2nd comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 申 not significant 100 10-2 10-2 10-1 100 101 Energy (MeV)Immanuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Jebaraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' : Solar energetic electrons on 9 October 2021 tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 8, the onset times do not follow this trend, nor do they form a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Instead, the higher energy channels of EPT show more and more delayed onsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Above an energy of about 142 keV, it becomes unreasonable to include further data points into the VDA, since the general trend from that energy channel upward are more and more de- layed onsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We therefore perform a fit only to the first 18 EPT lowest energy channels (shown by the green line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' However, this results in a path length of L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='56 AU, which is nonphysical (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', too short) since the spacecraft’s radial distance is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='68 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We therefore apply another fit only to the 10 lowest energy chan- nels (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='2–62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='7 keV), which is shown by the orange line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This fit yields a reasonable path length of L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='845 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='380 AU and an inferred injection time at the Sun for electrons of these energies at 6:25:46 UT ± 476 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Like for the high energy channels of EPT, the onset times of MeV electrons as observed by SolO/HET (red points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 8) are also systematically delayed and do not fit the velocity dis- persion trend as indicated by either fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This suggests a delayed solar injection with respect to the lower energy electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We apply a time shift analysis (TSA, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Vainio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2013) to the onset time of the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='05–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='41 MeV energy channel using a nomi- nal Parker spiral length of L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='736 AU and a solar wind speed of 360 km/s in order to infer the solar injection time of these electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This channel was selected because it has the clear- est and most resolvable onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We obtain an injection time of 06:35:56 UT at the Sun, which is about 10 minutes later than that for the ∼30–60 keV energy electrons using TSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' For same low energy channel, using the path length determined by the VDA (L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='845 AU), we only obtain a one minute earlier injection time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This strongly suggests the delayed solar injection of the high-energy electrons compared to the low-energy electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' An application of the TSA is also presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 9 for a number of energy channels assuming a path length larger than normal (L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='85 AU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We also performed a VDA analysis for STEREO-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' From the onset times determined from the STEREO-A/SEPT electron channels, a trend similar to SolO/EPT arises: The onset times in the highest channels, namely those above 195 keV, are con- sistently more delayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' On the other hand, a fit to the lower en- ergy channels in the range 45–195 keV yields a nonphysical path length of L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='53 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' If we instead choose the onset time as the moment when the measured electron intensity reaches 1% of the event peak intensity, then a fit to the same energy range yields a longer, but still nonphysical path length of L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='72 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' If we instead use TSA on onset time of the 45–55 keV channel to infer the solar injection time, we get t j = 06:28:50, assuming the nom- inal Parker spiral arm for a solar wind speed of 390 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Simi- larly, for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='7–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4 MeV electrons measured by STEREO/HET we obtain a solar release time of t j = 06:47 UT, almost 20 minutes later than that of the low-energy electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Table 1 summarises the results of the VDA and TSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' It shows the VDA result for SolO only for the more reasonable VDA fit and also results of similar TSA analyses done for STEREO-A SEPT and HET (not shown here in a figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Discussion In this study, we have analysed an event on 9 October 2021, which was associated with an electron rich event in-situ and was also observed by multiple wavelength remote sensing observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In our analysis, we have considered observations from X- rays, microwaves, radio waves, EUV, WL, and in-situ measure- ments to understand the solar sources of this electron event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Velocity dispersion analysis based on onset times of SolO EPT and HET electron channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The horizontal axis shows the inverse of the average unitless speed of electrons as observed in each channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The vertical axis presents the determined onset time for each energy channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Onset times observed by EPT (HET) are marked in blue (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Horizontal error bars represent the width of the energy channels, and vertical error bars represent the time resolution used to determine the onsets (30 seconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The orange and green lines are linear fits to the lowest 10 and 18 energy channels of EPT, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The grey lines represent the slope corresponding to a path length matching the nominal Parker spiral length at SolO’s radial distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' also utilised the full capabilities of the Solar Orbiter (SolO) mis- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The many features of the event are summarised with the use of a timeline inferred from both remote sensing and in-situ observations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The in-situ observations of energetic electrons by SolO show distinct phases of electron energization that are indistinguishable at other spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' More specifically, the energetic electrons observed by SolO show two increases in the electron intensity within two hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Additionally, the electrons’ anisotropy shows clearly that there are two distinct phases (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1), namely, a first anisotropic phase, which is observed immediately after the onset of electrons in the spacecraft, and a second mostly isotropic phase that is observed after the first peak in the in-situ electron intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Throughout the Discussion, we will term these two phases observed by SolO as "Phase 1" and "Phase 2".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A further analysis of "Phase 1" and more specifically the elec- tron release times shows that the low- and high-energy electrons are released at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We also note that such a distinction of two phases was not recorded for the electrons observed by STEREO-A and Wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The two peaks in the electron intensity that were observed at SolO were also not seen by STEREO-A and Wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In the fol- lowing subsections, we will discuss the two different phases as characterised by the anisotropy that was observed by SolO and their possible acceleration mechanisms at or near the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Phase 1 The first phase extends from immediately after the first electrons arriving at SolO, until 07:14 UT, when the initially peaked elec- tron anisotropy vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' During the first phase, we inferred that the low energy electrons (≲ 142 kV) were released ∼5–10 min- Article number, page 11 of 20 VDA,SOLOEPT+HETelectrons,2021-10-09 06:55 06:50 (HH:mm) Onsettime 06:45 F 06:40 H 06:35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='00 1/β nominal pathlength:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='736AU EPT fitto thelowest10 EPT channels:L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='845AU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' t inj=06:25:46 HET fittothelowest18EPTchannels:L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='561AU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='tinj=06:32:04A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' aanda Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Electron onset and inferred solar injection times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Observer / Energy Onset time Inferred injection Path length Method Instrument (keV) at spacecraft (UT) time at Sun (UT) (AU) SolO / EPT 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='2 - 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='7 6:25:46 ± 0:07:56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='845 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='380 VDA SolO / HET 1053 - 2410 6:41:15 ± 0:00:30 6:34:56 ± 0:00:30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='736 TSA STEREO-A / SEPT 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0 - 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0 6:50:58 ± 0:01:00 6:28:50 ± 0:01:00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='096 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1 TSA STEREO-A / HET 700 - 1400 6:57:00 ± 0:01:00 6:47:18 ± 0:01:00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='096 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1 TSA Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Composite spectrum of all available radio observations from the Earth vantage point (9 GHz - 10 kHz) together with STIX nonthermal (25 50 keV) count rates (top panel), and the in-situ electrons observed at SolO (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The SolO electron observations from the sunward tele- scope are time-shifted using the nominal path length and corrected for the travel time of light to make them comparable to the radio observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The inferred injection time of the low-energy (high-energy) electrons are marked with the dotted (dot-dashed) line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' With the pink (blue) shaded area we mark the release times using path length values larger than the nominal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' For ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='85 AU, the earliest release of the low-energy electrons is near the start of the first type III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The dashed line indicates the peak of the electron intensity for the ∼55 keV electron channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' utes earlier than the more energetic electrons (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Given this time difference, we can distinguish several solar features as- sociated uniquely to each release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' HXR observations show dis- tinct pulses around the electron release and radio observations show the escape of electrons to open field lines and the forma- tion of multiple strong shock regions during this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' An anisotropic early phase of the event is not only ob- served at SolO, but also at other observing spacecraft, namely, STEREO-A and Wind (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Such a consistent increase in anisotropy among multiple observers may indicate a common acceleration/release process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' It is unclear whether the same event was observed at Wind, since the electrons arrive an hour after their arrival at SolO or STEREO-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The intensity-time profiles of the in-situ electrons recorded by SolO during this phase were impulsive, highly anisotropic from the sunward direction, and exhibited a typical power- law energy spectrum with a hard-soft spectral index (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This broken power-law energy spectrum of the in-situ electrons has been previously suggested to be due to beam propagation through inhomogeneous plasma (Krucker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Kontar & Reid 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Additionally, high anisotropy for this phase was generally observed by all spacecraft within the 50◦ heliolongitu- dinal range, which suggests that the electron injection took place low in the corona and at regions well connected to most of the observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Low-energy electrons The inferred injection time of the low-energy electrons is ∼06:32 UT ±7:30 min based on the VDA and ∼06:34 UT using TSA at ∼42 keV electrons (both converted to the spacecraft time frame).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A number of flare- and shock-associated features were Article number, page 12 of 20 Composite Dynamic Radio Spectrum 10-1 Frequency (MHz) 100 101 102 103 STIX: 25-50 keV 106 EPT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0312 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0348 MeV EPT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0406 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0432 MeV Electron flux EPT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0533 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0580 MeV EPT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0731 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0788 MeV 103 EPT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1011 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1109 MeV EPT0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1423-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1541MeV EPT0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1995-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='2181MeV EPT0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='2817-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='3061MeV HET0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4533 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0380MeV HET1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0530-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4010MeV 100 10-3 06:00 06:30 07:00 07:30 08:00 TimeImmanuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Jebaraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' : Solar energetic electrons on 9 October 2021 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Events during the first phase leading to the first in-situ component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' (a) Timeline of the features associated with the two different electron injections during the first phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The exact times of the observational features are marked on the timeline bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The gradient on the bar is red to blue, indicating the transition between the low and high energy releases close to the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The long-lasting radio features are marked below the timeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' (b) Cartoon representation of the first phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The coronal shock wave during the low-energy release (t=1) and high-energy release (t=2) are represented by the black arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The open field lines from the periphery of the active region are represented by the green lines, while those from the adjacent coronal hole are represented by blue lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The closed field lines in the AR are denoted by th red coloured lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The abundance of energetic electrons are denoted by the small grey dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The transparent grey rectangle is zoomed into in panel (c) and shows the shock wave’s lateral expansion into nearby open magnetic field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The shock normal is denoted by the arrow and the ˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The perpendicular and near-perpendicular geometry is represented by θBn ≈ 90◦ and θBn < 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The electrons accelerated at the shock front on either side of the θBn ≈ 90◦ part of the shock wave are marked by the spirals along the magnetic field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' More details provided in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' observed during this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The HXR observations presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1 show distinct pulses around the electron release time, which were also observed in microwaves along with a charac- teristic diffuse gyro-resonant component indicating the presence of trapped electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The analysis of the HXR photon spectrum also shows that electrons were routinely accelerated to energies of ∼10 – 100 keV, suggesting a flare-related contribution to the low-energy electrons observed in-situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The inferred injection time of the low-energy electrons seems to be associated with the first two HXR pulses and the TIII-G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The TIII-G1 was probably generated by electron beams released from the flare site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Space-borne radio wave observations of the TIII-G1 reveal that the source electron beam was propa- gating at roughly -50◦ heliolongitude (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1), that is, away from the Parker spirals connecting SolO, STEREO-A, and Wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' According to our analysis (Appendix B), the in-situ Langmuir waves observed at STEREO-A and SolO were unlikely to be gen- erated by the passing of TIII-G1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 9, another physical phenomenon was present around the time of the inferred low-energy electron re- lease and that was the coronal shock wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The first signatures of TII(1a) were observed around 06:33 UT, indicating that the shock was able to efficiently accelerate electrons after the first HXR pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Around the same time, the EUV wave was formed and started its evolution away from the active region, mostly to the southwest direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' According to the first-order analysis of the shock wave (type II association presented in Appendix C), the TII(1a) originated most likely from the extended flank re- gions of the coronal shock wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Furthermore, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 5 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='3, open magnetic field lines indicate a good magnetic connectivity to both SolO and STEREO-A at the periphery of the AR, where the shock is progressively connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Article number,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' page 13 of 20 EUV wave 18 channels a) EUV flare VDA TSA SXR peak HXR 4 reflection SXR end 06:28 06:3 06:34 06:38 06:42) 06:50 06:53 SXR Flare start HXR1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' TIII - G1 HXR 2 HXR 3 TSA HXR 5 Phase 1 peak Phase 2 peak 06:19 06:3 06:33 06:3 06:41 06:46 [06:52 07:30 LoW energy injection HIGH energy injection MW 6:30 7:00 TIII - G2 6:37 6:46 6:50 TII (W) 6:55 TII (1) 6:33 7:07 TII (2) 6:36 7:40 TIV 8:00 b) c) t=2 t=1 Coronal hole Active regionA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' aanda In summary, we have found that the low-energy electrons were accelerated mainly by a flare-related process consistent with the first two HXR pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1 shows that the flare oc- curred in the western part of the AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' There, open field lines are present only in the vicinity of the strong positive polarity in the south.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' During the first (shown in panel a and b) and sec- ond HXR peak, a footpoint source is located close to this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In contrast, in the later peaks the eastern footpoint has shifted to a more northern location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We conclude that this is consistent with the notion that the energetic electrons observed in-situ in phase 1 have been accelerated (at least partly) in the flare and were injected onto open magnetic field lines connection to IP space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We also found that a relative contribution from the shock wave is difficult to be excluded, since the inferred injection and the start of type II are very close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The shock wave possibly fa- cilitated the low-energy electron release to the open magnetic field lines connecting to different observers, since the EUV wave reached the well-connected field lines (to SolO and STEREO- A) around 06:35 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' During this phase, the shock geometry at the field lines is expected to be mostly quasi-perpendicular at its flanks as it interacts with the open field lines present close to the active region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This may allow the shock wave to accelerate elec- trons rapidly along open magnetic field lines via the SDA mech- anism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This scenario is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 10b where the shock wave during this time is represented by the black arc at t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The thermal and suprathermal electrons which are present in abun- dance in the vicinity of the active region can be accelerated to energies in the range of 40–80 keV through a fast-Fermi mech- anism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' As explained earlier, given the right coronal and shock conditions (upstream electron density, magnetic compression ra- tio, and shock geometry), SDA can be a potent accelerator of energetic electrons (Leroy & Mangeney 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Krauss-Varban & Wu 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Warmuth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' High energy electrons The inferred injection times of electrons exceeding an energy of ∼100 keV (using VDA and TSA) show a delay of 5 to 10 min- utes, compared to that of the low-energy electrons (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This puts it amidst several observed eruptive features and poten- tial acceleration scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' At the time of the electron release, the TIII-G2, the 4th HXR pulse, and the TII(HB) were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We note that there was also a restructuring of magnetic footpoints near the flaring region minutes before the injection, during the 3rd HXR pulse (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 3 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The TIII-G2 was observed in association with the 4th HXR pulse, which was also one of the two lowest intensity pulses, implying that there was a decrease in the number of available energetic electrons (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The associated TIII-G2, however, was rather bright (dense electron beams generating the type III) and with a prolonged duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This often signifies an extended period of electron release into interplanetary space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' It is not clear if parts of the TIII-G2 emanated from the type II or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The TIII- G2 was observed by all space-borne radio observatories and, in particular, by STEREO-A and SolO that also observed in-situ Langmuir waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The results from the direction-finding analysis of the TIII-G2 presented in Appendix B reveal that the TIII-G2 propagated approximately in the -15◦ heliolongitude (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1), which was towards both SolO and STEREO-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The radio analy- sis suggests a good magnetic connection between the source of the TIII-G2 and the two spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' During this phase, energetic electrons in the range of 50–100 keV were still found in abundance in the flaring region (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Additionally, important shock-associated phenomena were also observed, namely, TII(2a), TII(HB), and TII(W) at 06:36 UT, 06:41 UT, and 06:50 UT, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The presence of herring- bone (HB) structures indicate that electron beams were acceler- ated by a near-perpendicular shock wave (85◦ < θBn < 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='9◦) via the SDA mechanism (Mann & Klassen 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Such a near- perpendicular shock geometry is expected in the low corona, which is dominated by closed magnetic field lines and a strong lateral expansion of the shock wave (Kouloumvakos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' According to the analysis performed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='3, it is understood that the TII(HB) and TII(W) features were observed concur- rently with the EUV wave mirroring off a magnetic null, which is formed in the south-west periphery of the AR close to the south-west coronal hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' These two features mark a phase of ef- ficient electron acceleration from the shock, which is probably responsible for the injection of the high-energy electrons during the first phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Meanwhile, the TII(2a) was observed in metric wavelengths and according to the kinematic analysis presented in the Appendix C, the source of the TII(2a) is at the flank re- gions of the shock wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A critical feature of this period was the possible interaction between the electron beams generating the TIII-G2 and part of the shock wave generating TII(HB) leading to a re-acceleration of the incident electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Based on the interplanetary directivity analysis of TIII-G2, we have shown that the source of the elec- tron beams were well-connected to both SolO and STEREO-A (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Additionally, during the same period, the TIV con- tinuum observed was probably generated by electrons trapped within the flux rope of the expanding CME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' It is possible that some of the trapped energetic particles may have escaped during interactions between the CME and the ambient magnetic field lines (Petrosian 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Klein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In both circumstances, an interaction with the shock wave could have accelerated the in- cident electrons to higher energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Electrons with large enough pitch angles are readily reflected by a near-perpendicular shock wave, gaining maximum energies of up to 13-fold their initial en- ergy in the process (for a shock wave with magnetic compression 4, Leroy & Mangeney 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Ball & Melrose 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This mech- anism may have increased the energy of a fraction (depends on θBn) of the incident c/3 electrons that generated the TIII-G2 to near-relativistic energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Considering that electrons in the range of 50–100 keV were also abundant during this period from, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', the HXR pulse, they may be further accelerated to relativistic energies by interacting with the coronal shock wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In conclusion, we find that during the release of high-energy electrons, a number of processes were observed concurrently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We distinguished between the different features and their asso- ciated mechanisms and found that the acceleration most likely happened in the corona close to the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Observations suggest that the main acceleration probably took place at the southern flank of the shock wave, where an interaction of the shock with a coronal streamer is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In these regions of interaction the shock wave geometry can be nearly perpendicular and electrons can accelerate to high energies efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A cartoon illustrating such a scenario is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 10b,c, where the shock wave (represented by the black arc at t=2) continues to interact with open field lines at its flanks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The presence of nonthermal elec- trons probably eases the acceleration of the electrons to mildly- relativistic and relativistic energies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' at least for a fraction of the incident electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A contribution from a flare-related acceleration process can- not be excluded during this high-energy electron release phase, however, the Suprathermal Ion Spectrograph (SIS) on-board SolO showed no particular enhancement of flare accelerating Article number, page 14 of 20 Immanuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Jebaraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' : Solar energetic electrons on 9 October 2021 particles such as 3He ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The presence of 3He ions would in- dicate a direct contribution of flare accelerated particles to the particle population measured in-situ during the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This prob- ably suggests that the shock wave had an important role in the acceleration and release of this high-energy electron component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The shock geometry which in some cases have been shown to be predominantly quasi-perpendicular in the low corona (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Kouloumvakos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2021), can enable the shock to accelerate electrons up to 13-fold their incident energy based on their pitch angles through an SDA mechanism (Leroy & Mangeney 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Ball & Melrose 2001, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Phase 2 As observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 9, a second peak was observed around 7:30 UT in the electron time profiles recorded by SolO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This increase in intensity appears after the anisotropic first phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In this second phase, the electrons showed very little anisotropy (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 6), which is usually believed to be due to transport related effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' It was not possible to determine the release of these elec- trons, however, the peak intensities were delayed by ≈50 min- utes after the start of the SXR flare (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This phase can be characterised by the gradual increase in high-energy electron in- tensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' An interesting aspect of this phase was the increase in the intensities of energetic electrons exceeding 300 keV, which was only observed by SolO and not by other spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Addi- tionally, the high-energy electrons exhibited a harder spectrum than during the first phase as observed by SolO (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A hard spectrum for delayed and gradual electron events has previously been attributed to transport-related effects (Strauss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In the case of STEREO-A, we believe the spacecraft sampled electrons that were accelerated mostly during the first phase and were delayed due to transport related effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This could explain the anisotropy of the electrons observed by STEREO-A, which was lower than at SolO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This would suggest that SolO was lo- cated conveniently at a region where the electrons were largely unaffected by transport and did not undergo strong pitch-angle scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The other spacecraft located elsewhere might have observed electrons from both the first phase and a transport- affected second phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The second phase occurred after the impulsive flare phase, that is, when the HXR flux had returned to nominal pre-flare levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' It is therefore probable that the electrons of phase 2 were accelerated at the CME/shock instead of during the flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In ra- dio wavelengths, the only features remaining in the radio spec- trum are a hectometric type II burst and metric TIV pulsations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' At the same time, the WL shock wave reached its peak veloc- ity of 1400 km s−1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1) at a radial distance of 7 R⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The presence of a long-lasting patchy type II radio burst in the hec- tometer wavelength, namely TII(1b) and TII(2b), confirms that shock electron acceleration took place during this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' From the analysis of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='3, it is understood that the shock under- went large-scale deformations due to the presence of coronal magnetic structures, namely, coronal holes and streamers, and it also continued to expand in regions confined by different stream interactions (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' An analysis of the shock kinematics (Ap- pendix C) and the comparison with the radio TII kinematics sug- gests that TII(2b) was most likely generated in the flank regions of the WL shock wave observed during this second phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Despite the fact that SDA can be a potent accelerator of ener- getic electrons in regions with quasi-perpendicular shock geom- etry, it is not fully clear if the shock geometry in the high corona is such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' It is possible that small scale deformations such as cavi- ties and ripples may provide the locally quasi-perpendicular ge- ometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' However, in these places the shock can be corrugated, leading to the short and efficient acceleration of electrons trapped within them through SDA mechanism (Bale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' On the other hand, DSA can also be an efficient acceleration mecha- nism if the electrons are trapped efficiently at the shock region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In this case, pre-existing magnetic field turbulence and low fre- quency waves generated upstream by specularly reflected ions (about 25% of the thermal ions are reflected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Krasnoselskikh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Gedalin 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Balikhin & Gedalin 2022) could have played a crucial role in trapping electrons for an extended pe- riod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Ambient magnetic field turbulence, which is ubiquitous in interplanetary plasma, may act as magnetic scattering centres for electrons (Tsytovich 1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The electrons trapped within turbu- lent fields or deformed and corrugated shock regions may di- verge from the thermal Maxwellian (Maksimovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1997) and provide a seed population for DSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The energy gain of these electrons can be in the order of 100s of keV (resonance with whistler waves at lower-hybrid frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Vaisberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Galeev 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Krasnoselskikh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1985) to several MeVs and tens of MeVs (Alfven wave resonance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Tsytovich 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Bell 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Kirk & Dendy 2001) depending on how efficiently they are trapped near the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The fact that this second phase was only observed by SolO indicates that the conditions for the shock scenario were pro- vided only within a confined region of the interplanetary shock wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The confined region could have been formed when the WL shock wave interacted with the streamers and propagated close to the fast-wind region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The electrons which were acceler- ated through a pitch-angle diffusion process under the aforemen- tioned conditions can then escape the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' While even small amounts of perpendicular diffusion present in the SIR can spread the particles to large heliolongitudes, we do not observe the sec- ond phase in a spacecraft other than SolO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The intensity-time profiles of the SEP event at SolO might be influenced by the arrival of a stream interaction region (SIR) during the SEP onset time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' As discussed by Lario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' (2022), the high-speed stream driving the SIR was previously observed by STEREO-A and later also by near-Earth spacecraft, where it strongly affected the observed energetic ion intensity-time pro- files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' At SolO, the SIR and its associated magnetic compres- sion might have acted as a magnetic mirror, reflecting some of the electrons back towards the inner heliosphere and the CME- driven shock wave where the electrons may subsequently accel- erate to higher energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Conclusions The main conclusions of this study are as follows: We were able to identify two distinct SEP event phases in- situ, that show different anisotropies and are most-likely re- lated to different acceleration phases in the solar corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We also found a hardening spectra from the first to the sec- ond phase for mildly-relativistic and relativistic electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This finding further prompts the existence of two different acceleration mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We have found the presence of two separate injections dur- ing the first phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The low-energy and high-energy electron components were injected at different times and can be at- tributed to different episodes of acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We found a mix of flare and shock acceleration mechanisms for the low energy electrons, while the high energy electron acceleration was mostly a shock-associated phenomenon in a highly structured corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Article number, page 15 of 20 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' aanda We have also found that the prolonged nature of the second phase was due to the particles being accelerated and trans- ported diffusively within a compression region, where the said particles are also trapped efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This study was largely possible due to the capabilities of the EPD instrument onboard SolO, which observes in a wide range of energy channels and also provides directional mea- surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Utilising multi-viewpoint remote sensing observa- tions from widely distributed spacecraft and ground-based ob- servations in many available wavelengths, it is possible to probe mechanisms continuously from deep in the solar corona to inter- planetary space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Future studies must aim to include high reso- lution radio imaging from the LOw Frequency ARray (LOFAR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' van Haarlem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2013), and the full capabilities of Solar Or- biter and Parker Solar Probe (PSP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Fox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2016) to better un- derstand the acceleration mechanisms at the Sun and the effects of coronal and interplanetary transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' This work received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 870405 (EUHFORIA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0), and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 101004159 (SERPENTINE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' acknowledge funding by the BRAIN-be project SWiM (Solar Wind Modelling with EUHFORIA for the new heliospheric missions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' acknowledges finan- cial support from NASA’s NNN06AA01C (SO-SIS Phase-E) contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Work in the University of Turku was performed under the umbrella of Finnish Centre of Excellence in Research of Sustainable Space (FORESAIL, Academy of Finland grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 336809).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' is grateful for support by the Turku Collegium for Sci- ence, Medicine and Technology of the University of Turku, Finland and support of Academy of Finland (SHOCKSEE, grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 346902).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' acknowledges support from the NASA program NNH17ZDA001N-LWS and from the Research Foundation – Flanders (FWO – Vlaanderen, fellowship no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1184319N) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' ac- knowledges DLR grant 50 OC 2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The work of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' was supported by DLR grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 50 OT 1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' thanks Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Vladimir Krasnoselskikh for the produc- tive discussions regarding the various aspects of collisionless shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Solar Or- biter is a space mission of international collaboration between ESA and NASA, operated by ESA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The STIX instrument is an international collaboration between Switzerland, Poland, France, Czech Republic, Germany, Austria, Ireland, and Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The authors also thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Milan Maksimovic for providing data products from the Radio Plasma Waves (RPW) instrument onboard Solar Orbiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' EIT and LASCO data have been used courtesy of the SOHO/EIT and SOHO/LASCO consortiums, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The STEREO SECCHI data are produced by a consor- tium of RAL(UK), NRL(USA), LMSAL(USA), GSFC(USA), MPS(Germany), CSL(Belgium), IOTA(France), and IAS(France).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The Wind/WAVES instrument was designed and built as a joint effort of the Paris-Meudon Observatory, the University of Minnesota, and the Goddard Space Flight Center, and the data are available at the instrument Web site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The authors are grateful to Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Xavier Bon- nin for providing direction finding data from the WAVES experiment onboard Wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We thank the radio monitoring service at LESIA (Observatoire de Paris) for providing value-added data that have been used for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' HMI data are provided courtesy of NASA/SDO science teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' References Agueda, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Klein, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Vilmer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2014, A&A, 570, A5 Anastasiadis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Lario, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Papaioannou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Kouloumvakos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Vourlidas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2019, Philosophical Transactions of the Royal Society of London Series A, 377, 20180100 Arge, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Henney, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Koller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2010, in Solar Wind 12, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Mak- simovic, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Issautier, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Meyer-Vernet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Moncuquet, & F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Pantellini, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' CS-1216, Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Melville, 343–346 Aurass, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Hofmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Urbarz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1998, A&A, 334, 289 Badman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Carley, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Cañizares, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2022, ApJ, 938, 95 Bale, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Goetz, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Harvey, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2016, Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 204, 49 Bale, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Reiner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Bougeret, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1999, Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 26, 1573 Balikhin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & Gedalin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2022, ApJ, 925, 90 Ball, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & Melrose, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2001, PASA, 18, 361 Bastian, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Benz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Gary, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1998, ARA&A, 36, 131 Battaglia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Saqri, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Massa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2021, A&A, 656, A4 Bell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1978, MNRAS, 182, 147 Benz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Monstein, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Meyer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2009, Earth Moon and Planets, 104, 277 Bougeret, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Goetz, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Kaiser, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2008, Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 136, 487 Bougeret, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Kaiser, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Kellogg, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1995, Space Science Reviews, 71, 231 Brown, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1971, Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 18, 489 Brueckner, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Howard, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Koomen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1995, Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 162, 357 Cairns, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1987, Journal of Plasma Physics, 38, 169 Cairns, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Knock, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Robinson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Kuncic, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2003, Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 107, 27 Cane, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Stone, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Fainberg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1981, Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 8, 1285 Domingo, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Fleck, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Poland, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1995, Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 162, 1 Dresing, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Effenberger, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Gómez-Herrero, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020, ApJ, 889, 143 Dresing, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Kouloumvakos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Vainio, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Rouillard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2022, ApJ, 925, L21 Dröge, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2000, ApJ, 537, 1073 Dulk, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Leblanc, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Bastian, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Bougeret, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2000, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 105, 27343 Fárník, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Hudson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Karlický, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Kosugi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2003, A&A, 399, 1159 Fox, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Velli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Bale, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2016, Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 204, 7 Galeev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1984, Soviet Journal of Experimental and Theoretical Physics, 59, 965 Galeev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Malkov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Völk, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1995, Journal of Plasma Physics, 54, 59 Garcia, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1994, Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 154, 275 Gedalin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1996, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 101, 4871 Gieseler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Dresing, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Palmroos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2022, Frontiers in Astronomy and Space Physics, (accepted) Gieseler, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Jones, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Kang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2000, A&A, 364, 911 Ginzburg, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & Zhelezniakov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1958, Soviet Ast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 2, 653 Glesener, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Krucker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Lin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2012, ApJ, 754, 9 Grigis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & Benz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2004, A&A, 426, 1093 Grigis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & Benz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2008, ApJ, 683, 1180 Harten, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & Clark, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1995, Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 71, 23 Howard, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Moses, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Vourlidas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2008, Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 136, 67 Huttunen-Heikinmaa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Valtonen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Laitinen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2005, A&A, 442, 673 Iwai, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Kubo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Ishibashi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2017, Earth, Planets and Space, 69, 95 Jebaraj, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Kouloumvakos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Magdaleni´c, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2021, A&A, 654 Jebaraj, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Magdaleni´c, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Krasnoselskikh, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Krupar, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Poedts, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2022, A&A, (accepted) Jebaraj, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Magdaleni´c, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Podladchikova, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020, A&A, 639 Kahler, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & Hundhausen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1992, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 97, 1619 Kaiser, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2005, Advances in Space Research, 36, 1483 Kaiser, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Kucera, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Davila, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2008, Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 136, 5 Kiplinger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1995, ApJ, 453, 973 Kirk, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & Dendy, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2001, Journal of Physics G Nuclear Physics, 27, 1589 Klassen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Dresing, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Gómez-Herrero, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Heber, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Veronig, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2018, A&A, 614, A61 Klassen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Gómez-Herrero, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Heber, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2011a, Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 273, 413 Klassen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Gómez-Herrero, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Heber, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2012, A&A, 542, A28 Klassen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Gómez-Herrero, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Müller-Mellin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2011b, A&A, 528, A84 Klein, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Chupp, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Trottet, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1999, A&A, 348, 271 Klein, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & Dalla, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2017, Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 212, 1107 Klein, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Musset, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Vilmer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2022, A&A, 663, A173 Kong, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Guo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2016, ApJ, 821, 32 Kong, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Guo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Giacalone, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2017, ApJ, 851, 38 Kontar, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & Reid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2009, ApJ, 695, L140 Kouloumvakos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Kwon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Rodríguez-García, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2022a, A&A, 660, A84 Kouloumvakos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Nindos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Valtonen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2015, A&A, 580, A80 Kouloumvakos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Rodríguez-García, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Gieseler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2022b, Frontiers, 0, 0 Kouloumvakos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Rouillard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Warmuth, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2021, ApJ, 913, 99 Kouloumvakos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Rouillard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2019, ApJ, 876, 80 Krasnoselskikh, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Voshchepynets, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Maksimovic, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2019, ApJ, 879, 51 Krasnoselskikh, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Balikhin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Alleyne, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1991, Advances in Space Research, 11, 15 Krasnoselskikh, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Kruchina, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Volokitin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Thejappa, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1985, A&A, 149, 323 Krauss-Varban, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1989, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 94, 15367 Krucker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Hurford, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Grimm, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020, A&A, 642, A15 Krucker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Kontar, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Christe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Glesener, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Lin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2011, ApJ, 742, 82 Krucker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Kontar, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Christe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Lin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2007, ApJ, 663, L109 Krucker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Saint-Hilaire, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Christe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2008, ApJ, 681, 644 Krupar, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Eastwood, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Kruparova, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2016, ApJ, 823 Krupar, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Maksimovic, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Santolik, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2014, Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 289, 3121 Krupar, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Szabo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Maksimovic, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020, ApJS, 246, 57 Lario, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Wijsen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Kwon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2022, ApJ, 934, 55 Lemen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Title, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Akin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2012, Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 275, 17 Leroy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & Mangeney, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1984, Annales Geophysicae, 2, 449 Article number, page 16 of 20 Immanuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Jebaraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' : Solar energetic electrons on 9 October 2021 Lin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Anderson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Ashford, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1995, Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 71, 125 Long, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Bloomfield, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Chen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2017, Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 292, 7 Lucas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1985, Technometrics, 27, 129 Magdaleni´c, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Marqué, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Krupar, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2014, ApJ, 791, 115 Maksimovic, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Bale, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Chust, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020, A&A, 642, A12 Maksimovic, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Pierrard, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Lemaire, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1997, A&A, 324, 725 Mann, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & Klassen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2005, A&A, 441, 319 Mann, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Warmuth, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Aurass, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2009, A&A, 494, 669 Massa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Battaglia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Volpara, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2022, Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 297, 93 Massa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Schwartz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Tolbert, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020, ApJ, 894, 46 Masson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Antiochos, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & DeVore, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2019, ApJ, 884, 143 McClements, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Dendy, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Bingham, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Kirk, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Drury, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1997, MNRAS, 291, 241 McLean, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & Labrum, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1985, Solar radiophysics : studies of emission from the sun at metre wavelengths Melrose, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2017, Reviews of Modern Plasma Physics, 1, 5 Müller, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Cyr, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Zouganelis, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020, A&A, 642, A1 Müller-Mellin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Böttcher, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Falenski, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2008, Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 136, 363 Musset, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Jeunon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Glesener, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020, ApJ, 889, 183 Newkirk, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1961, The Astrophysical Journal, 133, 983 Nindos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020, Frontiers in Astronomy and Space Sciences, 7, 57 Page, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1954, Biometrika, 41, 100 Papaioannou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Sandberg, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Anastasiadis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2016, Journal of Space Weather and Space Climate, 6, A42 Pesnell, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Thompson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Chamberlin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2012, Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 275, 3 Petrosian, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2016, ApJ, 830, 28 Pohjolainen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2008, A&A, 483, 297 Pulupa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Bale, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Bonnell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2017, Journal of Geophysical Re- search (Space Physics), 122, 2836 Ramesh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Kathiravan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Chellasamy, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2022, ApJ, 932, 48 Reames, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2021, Solar Energetic Particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A Modern Primer on Understand- ing Sources, Acceleration and Propagation, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 978 Reid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & Ratcliffe, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2014, Research in Astronomy and Astrophysics, 14, 773 Rodríguez-Pacheco, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Wimmer-Schweingruber, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Mason, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020, A&A, 642, A7 Saito, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1970, Annals of the Tokyo Astronomical Observatory, 12, 51 Schmidt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & Cairns, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2016, Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 43, 50 Schrijver, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & De Rosa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2003, Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 212, 165 Stewart, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & Magun, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1980, PASA, 4, 53 Strauss, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Dresing, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Kollhoff, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Brüdern, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020, ApJ, 897, 24 Thejappa, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1987, Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 111, 45 Tkachenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Krasnoselskikh, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Voshchepynets, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2021, ApJ, 908, 126 Treumann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & Jaroschek, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2008, arXiv e-prints, arXiv:0805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='2579 Tsytovich, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1966, Soviet Physics Uspekhi, 9, 370 Tsytovich, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1973, ARA&A, 11, 363 Uchida, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Altschuler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Newkirk, Gordon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1973, Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 28, 495 Vainio, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Valtonen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Heber, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2013, Journal of Space Weather and Space Climate, 3, A12 Vaisberg, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Galeev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Zastenker, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1983, Soviet Journal of Experimental and Theoretical Physics, 58, 716 van Haarlem, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Wise, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Gunst, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2013, A&A, 556, A2 Vlahos, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Anastasiadis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Papaioannou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Kouloumvakos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Isliker, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2019, Philosophical Transactions of the Royal Society of London Series A, 377, 20180095 von Rosenvinge, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Reames, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Baker, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2008, Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 136, 391 Vršnak, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & Luli´c, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2000a, Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 196, 157 Vršnak, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & Luli´c, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2000b, Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 196, 181 Warmuth, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2015, Living Reviews in Solar Physics, 12, 3 Warmuth, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & Mann, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2005, A&A, 435, 1123 Warmuth, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' & Mann, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020, A&A, 644, A172 Warmuth, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Mann, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Aurass, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2009, A&A, 494, 677 Warmuth, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Önel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Mann, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020, Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 295, 90 Wiegelmann, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Inhester, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Sakurai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2006, Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 233, 215 Wiegelmann, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Thalmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Inhester, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2012, Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', 281, 37 Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Musset, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Glesener, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Panesar, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', & Fleishman, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='05668 Article number, page 17 of 20 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' aanda Appendix A: NLFFF extrapolations of AR magnetic field topology In order to understand the magnetic topology of eruptive event, we extrapolate photospheric vector magnetograms of the source AR obtained from SDO/HMI into the solar corona under the nonlinear force-free field (NLFFF) assumption with the help of an optimization code, as described in Wiegelmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The photospheric magnetic vector field measurements have been preprocessed to obtain consistent boundary data for the NLFFF- model (see Wiegelmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2006 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Figure .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1 depicts a sample of the extrapolated field lines, which are plotted over the vertical photospheric magnetic field shown in greyscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Additionally, we overplot the nonthermal STIX sources as coloured contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Note that we have rotated the STIX images to conform with the vantage point of SDO, which is exactly the inverse process as used for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 3 where we have rotated SDO/AIA images to the SolO view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We can do per- form this rotation only for the nonthermal STIX sources since they originate in the chromosphere, while the thermal source is an extended 3D structure in the corona for which a simple rota- tion would cause severe projection effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We show this comparison for three of the nonthermal STIX peaks for which nearly co-temporal HMI data was available (times are UT at 1 AU): HMI at 06:24 and the STIX peak around 06:31 UT (panels a +b ), HMI at 06:36 and STIX around 06:37 UT (panels c +d ), and finally HMI at 06:48 UT and STIX at 06:46 UT (panels e + f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The top row of panels (a, c, e) shows closed loops where both footpoints of the magnetic field line are anchored in the photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The bottom panels (b, d, f) contain open magnetic field lines, where open means that the field lines have only one footpoint in the photosphere and the upper end reaches the top boundary of the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' These are the field lines which potentially offer access to interplanetary space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Appendix B: Directivity of type III bursts The interplanetary radio emissions associated with the event were observed by all spacecraft: STEREO-A/WAVES, Wind/WAVES, SolO/RPW, and PSP/FIELDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' A combined dy- namic radio spectrum from the three spacecraft excluding WIND is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The figure also includes the linear po- larisation measurements from PSP and STEREO-A, which also provides goniopolarimetric measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Locating the source and the propagation of the type III ra- dio sources in the corona and interplanetary space without in- terferometric imaging (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', LOw Frequency ARray (LOFAR);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' van Haarlem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2013) or radio triangulation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', Magdaleni´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Krupar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Jebaraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020) is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' However, we make use of the direction finding observations from STEREO-A and Wind observations (azimuth and co-latitude an- gles of the radio emission, Krupar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' In the case of the directivity analysis, we have compared cal- ibrated radio fluxes measured by Parker Solar Probe, Solar Or- biter, STEREO A and Wind at six frequency channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We as- sumed that the radio emission pattern S as a function of helio- centric longitude λ can be described by the von Mises distribu- tion (also known as the circular normal distribution) as: S (λ) = exp(κ cos(λ − λ0)) 2πI0(κ) , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1) where λ0 is a direction corresponding to a peak radio flux, κ is a measure of concentration, and I0 is the modified Bessel function of the first kind of order 0, with this scaling constant chosen so that the distribution sums to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Fig B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1 panels e & f show the propagation direction of TIII- G1 and TIII-G2 sources obtained using the radio triangulation technique (short black arrows, for more details see Krupar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2014) and a directivity analysis (long black arrows, radio emis- sion pattern is described by the von Mises distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Both methods employ cross-calibrated measurements from multiple spacecraft, all four spacecraft in the case of the directivity anal- ysis, Wind and STEREO-A in the case of radio triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We perform both analyses to better constrain the direction of source propagation and eliminate intrinsic errors which may arise from each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Even so, we shall put more weight on the radio tri- angulation results as they do not depend on free parameters such as a coronal electron density model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The radio triangulation results of TIII-G1 between the fre- quencies 1000 kHz and 600 kHz indicate that the source prop- agated in the -58◦ heliolongitude (possible error of ±6◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The directivity analysis indicated that the source propagated in -43◦ heliolongitude (possible error of ±2◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Both results indicate that TIII-G1 most likely propagated westward of STEREO-A close to the reference line (black dashed line) plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The clos- est spacecraft to the propagation of TIII-G1 was PSP, however, no in-situ Langmuir waves were recorded by PSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Therefore, we can conclude that it was highly unlikely for TIII-G1 to be associ- ated with the in-situ electrons recorded by SolO, and STEREO-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Lario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' (2022) found a very impulsive rise of particle flux at PSP during the early stages of the event, which is to be expected when there was direct connectivity in the direction rather close to PSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' For TIII-G2, we performed a similar analysis (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1e) and found, according to radio triangulation, the source propagated in the -15◦ heliolongitude with a possible error of ±2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Using the directivity analysis, we found the propagation to be in the heli- olongitudes -30◦ with possibly a ±3◦ error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The larger spread in propagation direction between radio triangulation and the direc- tivity analysis could be due to errors arising from the propagation of radio waves in the presence of large-scale density fluctuations (Krupar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Alternatively, the large spread may also likely be due to multiple electron beams generating type III ra- dio bursts within TIII-G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The electron beams that generated the different type III bursts may propagate in slightly different solar wind due to their time-varying injections at the source - causing a large spread in their directivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' However, it should be noted that TIII-G2 propagated mostly in the direction of SolO and STEREO-A, which is further corroborated by the in-situ Lang- muir waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Therefore, from our analysis, TIII-G2 may be asso- ciated with the energetic electrons recorded in-situ at both SolO and STEREO-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Appendix C: Shock wave kinematics We reconstructed the 3D structure of the shock wave using PyThea, a software package to reconstruct the 3D structure of CMEs and shock waves (Kouloumvakos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' 2022b), written in Python language and available online4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We took advantage of the two viewpoints provided by STEREO-A and near-Earth spacecraft (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=', SOHO and SDO), and we fitted an ellipsoid model to EUV and WL observations of the shock wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We adjusted the free parameters of the geometrical ellipsoid model to achieve the best visual fit to near-simultaneous observations for the two available viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' From the 3D reconstruction, 4 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='5713659 Article number, page 18 of 20 Immanuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Jebaraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' : Solar energetic electrons on 9 October 2021 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Nonlinear force-free field extrapolations based on SDO/HMI vector magnetograms and STIX overplotted (see text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Space-borne radio dynamic spectrograms from: (a) PSP, (b) SolO, (c) STEREO-A, and (d) Wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Panels (e) and (f) show the direction finding analysis of TIII-G1 and TIII-G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' we determined the position and kinematics of the shock in the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1 suggests that the wave propagated quite rapidly at the apex and reached a maximum possible speed of 1400 km s−1 at around 07:25 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The shock apex was at a radial distance of 7 R⊙ at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The flank regions of the shock expanded slower and reached their peak speeds of 900 km s−1 a little earlier at 07:15 UT when they were at a radial distance of 3 R⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' According to the ellipsoidal fit, both flanks expanded at a similar rate till their respective peaks after which their velocities were slightly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' According to the analysis of the EUV wave and the WL shock wave, the apex of the shock expanded considerably faster than the flank regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Comparing these with the kinematics of the type II radio burst, we may get a first-order understanding of their association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' To do so, we converted the spectral drift rate of the type II radio bursts into km s−1 using radial coronal electron density models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We applied a 5-fold Newkirk (1961) model to the metric and decametric type II bursts, and a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='5- fold Saito (1970) model for the hectometer bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' These coro- nal electron density models were chosen based on the projec- tion analysis of type II radio sources at shock flanks and for non-radial shock propagation done by Jebaraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We Article number, page 19 of 20 a) C e10 a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4 PSP/RFS e) TIII-G1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='2 HEEQ( 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4 E b) SolO/RPW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='6 上 106 (MHz) sfu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1 F 682 kHz, 675 kHz, 675 kHz, 708 kHz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='01 XHEEQ (au) Frequency ( 10 STEREO A 104 /WAVES 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4 f) TIII-G2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='2 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0 E Langmuir:waves (au) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='2 10 HEEQ d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='4 TIII-G2 /WAVES Wind 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='6 1 1013kHz,1025kHz,1025kHz,1040kHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='8Fo 855 kHz, 875 kHz, 875 kHz, 916 kHz 764 kHz, 775 kHz, 775 kHz, 804 kHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0 TII-G1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0 00:90 06:30 07:00 07:30 08:00 08:30 ):60 XHEEQ (au) Time (UT)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' aanda Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The kinematics of the reconstructed shock wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The three axes of an ellipsoidal fit, namely, the shock apex (red), and the two flank axes (r-axis 1 - blue and r-axis 2 - green) are plotted together and represented with different colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The left and the right panel of the figure show the height-time evolution of all the three shock axes, and its first derivative (velocity), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The possible errors are in each axis and indicated by the shaded regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The "+" and "x" markings on the curves are the constraints provided by the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' obtained 1100±30 km s−1 for TII(1a) and TII(1b), which sug- gests that the emission may have been from regions between the flanks and the apex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Meanwhile, for TII(2a) and TII(2b), using a similar approach, we obtained a drift speed of 700±50 km s−1 associating it with the flank regions of the shock wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The ± values here have been calculated from the varying bandwidth of the respective type II bursts over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' TII(2b) on the other hand was treated separately due to its prolonged emission pe- riod in the hectometer wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' We obtained a drift speed of 750±50 km s−1, which corresponds well to the flank regions of the shock wave during this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Meanwhile, TII(HB) and TII(W) are not as straightforward to analyse using a simple ra- dial density profile such as the one applied here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' The two struc- tured type II bursts were most likely due to shock wave propaga- tion in regions of enhanced density and therefore exhibit spectral deformities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content=' Article number, page 20 of 20 Event: FLM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='6|2021-10-09T06:38:00 I spline3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='2) Event: FLM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='6|2021-10-09T06:38:00 I spline3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='2) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0 - 1600 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='5 - 1400 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0 - [Rsun] 1200 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='5 Speed [km/s] Length 1000 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='0 Height or l 800 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE2T4oBgHgl3EQfFwaR/content/2301.03650v1.pdf'} +page_content='5 - 600 5.' metadata={'source': 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Chatterjeea,∗, S. Dalla Torrea, C. Dilksb, A. Kiseleve, H.Klestf, R. Preghenellac, A. Vossenb +aINFN Trieste, Italy +bDuke University, USA +cINFN Bologna, Italy +dINFN Roma 1, Italy +eBrookheaven National Laboratory, USA +fStonyBrook University, USA +Abstract +The Electron-Ion collider (EIC) will be the ultimate facility to study the dynamics played by the colored quarks and gluons to +the emergence of the global phenomenology of the nucleons and nuclei as described by Quantum Chromodynamics. The physics +programs will greatly rely on efficient particle identification (PID) in both the forward and the backward regions. The forward +and the backward RICHes of the EIC have to be able to cover wide acceptance and momentum ranges; in the forward region a +dual radiator RICH (dRICH) is foreseen and in the backward region a proximity-focusing RICH can be foreseen to be employed. +The geometry and the performance studies of the dRICH have been performed as prescribed in the EIC Yellow Report using the +ATHENA software framework. This part of our work reports the effort following the call for EIC detector proposal the studies +related to the forward and the backward RICHes performance. In the forward region, dRICH performance showed a pion- kaon +separation from around 1 GeV/c to 50 GeV/c at a three sigma level; the proximity focusing RICH (pfRICH) foreseen for the +backward region can reach three sigma separation up to 3 GeV/c for e/π and up to 10 GeV/c for π/K mass hypothesis. +1. Requirements of particle identification in EIC +EIC at Brookhaven National Laboratory(USA)[1] is ex- +pected to start data-taking in the early 2030s to answer crit- +ical questions related to Quantum ChromoDynamics (QCD). +A state-of-the-art machine and cutting-edge detector technol- +ogy are fundamental for its success. Highly polarized elec- +trons (∼70-80%) will be collided with highly polarized nucle- +ons and light nuclei (∼70-80%) and also with heavy ions. The +high luminosity collisions (∼1034 electron proton cm−2s−1) will +take place over a wide center of mass energy (20-141 GeV) +with a possibility of more than one interaction points. Many +of EIC physics require excellent particle identifications over +a wide phase space. The requirements are documented as a +common effort of the entire EIC community; afterwards, we +will refer to this report as Yellow Report (YR) [1]. +In re- +sponse to the call for detector proposal submission, three proto- +collaborations ATHENA [2], ECCE [3] and CORE[4] had been +formed. Our report mainly focuses on the simulation studies +made using ATHENA software framework for studying the for- +ward and backward RICH detectors’ performances. +2. PID performance requirements for EIC physics +The YR gives indicative requirements for the hadron PID in +the electron-going endcap: better than 3σ π/K up to 10 GeV/c +(table 3.1 of ref.[1]). In the following part of the article, we will +consider this as our reference. The forward dRICH is aimed to +perform 3σ π/K separation up to 50 GeV/c and e/π separation +up to 15 GeV/c. As mentioned in the YR, the required accep- +tance for the dRICH is 1.0 ≤ η ≤ 3.5. These reference numbers +were taken as guidance for the ATHENA implementation. Fig- +∗Corresponding author +Email address: chandradoy.chatterjee@ts.infn.it (C. Chatterjee) +IP +pfRICH +dRICH +Aerogel +Aerogel +Sensors +Services +Sensors +Mirrors +z +x +Figure 1: Schematic of the forward dual radiator RICH (dRICH) and backward +proximity focusing (pfRICH) with respect to the interaction point (IP). +ure 1 shows the location of the two RICH detectors with respect +to the interaction point (IP). +3. pfRICH geometry +The ATHENA design assumed that the proximity-focusing +RICH will occupy space from -150 cm to -210 cm from the +nominal IP (available space between the central tracker and +crystal calorimeter). In our implementation, the vessel has a +cylindrical shape, with an outer radius of 93 cm, and a cutaway +at small radii as determined by the design of the beam pipe. +3 cm thick aerogel with an average refractive index ⟨n⟩ = 1.019 +was used in the simulations as the main Cherenkov radiator. +The properties of the aerogel (refractive index variation, ab- +sorption, and Rayleigh scattering parameterizations as a func- +Preprint submitted to Nuclear Instruments & Methods in Physics Research, Section A. +January 23, 2023 +arXiv:2301.08334v1 [physics.ins-det] 17 Jan 2023 + +tion of wavelength) were taken from the available CLAS12 +data [5]. A 40 cm long expansion volume was assumed to be +filled with C4F10 to provide an additional e/π separation capa- +bility in a threshold mode below ∼ 2.9 GeV/c, with parameter- +izations taken from [6]. Hamamatsu S13361-3050AE-08 8x8 +SiPM panels [7] are anticipated as a reference photosensor type +(3 mm single SiPM size). SiPM Photon Detection Efficiency +(PDE) as well as the geometric fill factor is taken according +to the Hamamatsu specifications [8]. About 15 cm of space be- +hind the SiPM plane is reserved inside the vessel for the readout +electronics and services. We applied an additional safety factor +of 0.7 on top of this, namely, we assume that only 70% of pho- +tons that pass the PDE and the geometric fill-factor selection are +actually detected and used in the Cherenkov angle evaluation. +4. Forward dual radiator RICH (dRICH) +In general, the dRICH configuration is very similar to the +one from the YR. A substantial effort was made though in order +to accommodate such an apparatus in the overall tight space +available for the ATHENA detector in RHIC IP6 Interaction +Region and to quantify its expected performance. +4.1. dRICH location and vessel boundaries +During the optics tuning it was realized that the originally al- +located space of ∼ 120 cm along the beamline is not sufficient +to contain the focal plane inside the vessel, due to a very large +polar angular acceptance. It became clear that to shift the ves- +sel further away from the IP in order to minimize the adverse +effects of the solenoid fringe field, even though the coil and +the return flux configuration of the new magnet were carefully +tuned to observe the so-called projectivity requirement, namely +to minimize the overall bending of the charged secondary par- +ticles originated from the IP. As a consequence of these studies, +in the final ATHENA configuration, which is presented in the +Proposal, the dRICH vessel was shifted by ∼30 cm away from +the IP, and at the same time the solenoid coils were moved by +25 cm towards the electron-going endcap (see figure 2). +Figure 2: +Example of the effect of the magnetic field in single photon +Cherenkov angle RMS. Black points represent dRICH located at nominal po- +sitions prior to finalization, red points represent the dRICH additionally shifted +by 30 cm away from IP and blue points represent dRICH is shifted like the +red points additionally the solenoid coil is shifted by 25 cm towards electron +endcap. +4.2. dRICH geometry +The dRICH used the same aerogel type and the same SiPM +sensors as pfRICH. The safety factor of 70% was also included. +The vessel length along the beam line is 140 cm, and it occupies +the range from +190 cm to +330 cm from the IP. The detector +will have six 60◦ sectors, each equipped with its own spherical +mirror segment and a sensor plane. Aerogel thickness is 4 cm. +We have used C2F6 gas as a radiator. This allows achieving a +comfortable overlap with the aerogel, and a 3σ π/K separation +is obtained up to ∼50 GeV/c.C2F6 refractive index parameteri- +zation is taken from [9, 10], and a conservative value of 10 m +absorption length is taken for the simulations. The sensors are +positioned on a sphere, with a square tiling algorithm. Three +variables parameterize the spherical mirrors: the z position of +the backplane, which is the maximum z the spherical mirror will +reach, along with two focus tune parameters fx and fz. Point-to- +point focusing of the IP on the centre of the sensor sphere cor- +responds to focus tunes fx = fz = 0, which represents a starting +point for the focus tuning; because the IP is far from the opti- +cal axis of the spherical mirror, spherical aberrations cause the +proper point-to-point focal region to be significantly blurred. In +order to focus Cherenkov rings on the sensors, parallel-to-point +focusing is used. By changing the values of fx and fz it is pos- +sible to steer the parallel-to-point focal region to be as close to +the sensor surfaces as possible. +5. Simulation studies for the ATHENA proto collaboration +A DD4Hep-based [11] framework was used for the +ATHENA proto-collaboration. The reconstruction was based +on the Juggler framework [12]. +A newly written Inverse +Ray Tracing code (IRT), equally applicable in a standalone +GEANT4 environment and in the ATHENA software frame- +work. The code represents itself a substantial generalization of +the IRT algorithm initially developed for HERMES dual radia- +tor RICH [13]. It allows one to perform ray tracing between the +detected photon location in 3D space and the expected emission +point range along the charged particle trajectory on a predefined +sequence of refractive and reflective boundaries, using a 2D it- +erative Newton-Gauss minimization procedure. It is fully con- +figurable and has a persistent model, which allows one to ex- +port and import ROOT files with the actual optics description. +The code was originally developed to quantify the performance +of the LHCB-like RICH#1 configuration for the ATHENA for- +ward RICH, with spherical and flat mirrors in a sequence, which +cannot be easily handled by a simple 2D IRT algorithm. How- +ever, it is equally applicable to a simpler pfRICH geometry, +where both absorption and Rayleigh scattering in the aerogel +and refraction on the aerogel-gas boundary still play a role in +the unbiased Cherenkov angle evaluation. The IRT library [14] +is available in the ATHENA software repository, together with +the standalone GEANT4 stepping code and the so-called Jug- +gler plugin in the ATHENA reconstruction environment. IRT +algorithm implementation in the ATHENA Juggler PID plugin +is complemented by a sophisticated logic, performing sampling +along the charged particle trajectory, which allows one to es- +timate the average Cherenkov angle in the same way for both +straight tracks and in presence of a relatively strong bending +component of the ATHENA solenoid magnetic field. A prop- +erly weighted ensemble of the Cherenkov photon angle esti- +mates is then checked against e/π/K/p mass hypotheses, and a +probability of each hypothesis is provided as an output. +6. Performance Studies +6.1. pFRICH performance and consistency checks +To check the consistency of the software stack the perfor- +mance studies of the pfRICH were done before, thanks to its +simple geometry. Reconstructed Cherenkov angles as a func- +tion of momentum have been studied for different mass hy- +potheses to check the consistency (see figure 3,4 from pfRICH +2 + +n = 1.5 or 0, = 0.44 rad +n = 2.0 or 0, = 0.27 rad +RMS (mrad) +Rich: 30 cm shifted; Map: Solenoid +Rich: 30 cm shifted;: Map: Solenoid +No shit; Map: Solenoid +od +No shift; Map: Solenoid +Rich: 30 cm shifted; Map: New Shifted Solednoid +Rich: 30 cm shifted; Map: New Shifted Solednoid +3 +50 +p (GeV) +3 +p (GeV) +n = 2.5 or @ = 0.16 rad +n = 3.0 or 0 = 0.10 rad +Rich: 30 cm shifted; Map: Solenoid +Rich: 30 cm shifted; Map: Solenoid +No shift; Map: Solenoid +8 +No shift; Map: Solenoid +Rich: 30 cm shifted; Map: New Shifted Solednoid +Rich: 30 cm shifted; Map: New Shifted Solednoid +2+ +2 +30 +35 +40 +45 +50 +p (GeV) +20 +25 +3 +p (GeV)section of [15]) and also for dRICH (see figure 5,7 from dRICH +section of [15]). Single particles (e/π/K) were shot to estimate +the Nσ separation as a function of momentum and pseudora- +pidity (η). It has been demonstrated that the Yellow Report re- +quirements can be achieved using the simple pfRICH geometry +and the reconstructed Cherenkov angles provide a satisfactory +kaon rejection factor without diluting the pion identification ef- +ficiency. The acceptance plot shows that between η from 1.6 to +3 the number of detected photons is constant. We have defined +50% of the maximum number of the detected photon as the +working acceptance. This demonstrates the pfRICH can cover +a region larger than eta η 3.5 in the backward region (see figure +4 b). For three different η regions the Nσ had been studied. One +can reach three sigma separation up to 3 GeV/c for e/π and up +to 10 GeV/c for π/K mass hypothesis (see figure 4 c). Using +an equal mixture of pion and kaon samples for particles at sat- +uration a pion rejection factor as a function of kaon detection +efficiency is also computed. That shows that without diluting +the kaon detection efficiency high level of pion rejection can be +obtained (figure 4 d). +Figure 3: Number of detected photons as in pfRICH rings (aerogel) as a func- +tion of pseudorapidity. The sharp drop at the edges is due to the containment of +the partial ring in the sensors due to its acceptance. 50% of the ring contained +in the sensor plane is defined as the acceptance limit of the detector +Figure 4: pfRICH Nσ separation power as a function of momentum. Using +only aerogel information. +6.2. dRICH performance +The forward dRICH also showed performance as prescribed +by the YR. Both for the aerogel and gas we have observed the +acceptance is from η 1.2 to around 3.5 (figure 5). For the sepa- +ration power, it is evident that using the aerogel and gas infor- +mation once can reach from some hundreds of MeV/c up to 50 +GeV/c without the presence of any hole for π/K separation (fig- +ure 6). +As mentioned the forward dRICH should be able to +Figure 5: Number of detected photons as in dRICH rings (aerogel and C2F6) +as a function of pseudorapidity. The sharp drop at the edges is due to the con- +tainment of the partial ring in the sensors due to its acceptance. 50% of the ring +contained in the sensor plane is defined as the acceptance limit of the detector +Figure 6: Nσ separation as a function of momentum of positively identified π/K +using aerogel and C2F6 information from dRICH +Figure 7: Nσ separation of e/π using aerogel and C2F6 information from dRICH +reject electrons, it has been demonstrated that up to 15 GeV/c +the separation can be done from mid to high η (figure 7). At +low η the separation power is diluted. At very low η the effect +of the solenoidal field plays a critical role in the e/π separation. +Instead of using mirrors with a single radius of curvature, each +sector can be equipped with a mirror divided into two sections +with slightly different radii of curvature in order to obtain better +resolution over the region. Standalone studies with dual mirrors +3 + +8 +Acceptance +6 +pfRICH +8-GeV.pions +O.7.safety factor.included +3.5 +-3 +-2.5 +-2 +1.5 +nON +20 +元-K(red) e-π(Blue) +18 +n=-1.5 +16 + n=-2.3 +n=-3.2 +14 +Separation Power +12 +10 +8 +6 +4 +2 +0 +2 +4 +6 +8 +10 +12 +momentum (GeVlc)ad +dRICHacceptance +25 +CF. +Gas50GeV pions +Aerogel +20 +15 +Aerogel12 GeV pions +10 +5 +1.5 +2 +2.5 +3 +3.5 +4 +Pseudorapidity( n(元-K) +30 +Aerogel(Blue) C,Fc(Red) +No +- n= 1.5 +25 + n= 2.4 +- n= 3.2 +20 +15 +10 +5 +10 +20 +30 +40 +50 +60 +momentum (GeV/c)(1-a) +20 +C,F6 +ON +18 +n = 1.5 +16 +n = 2.4 +n= 3.2 +14 +Aerogel +n = 1.5 +12 +n = 2.4 +10 +n= 3.2 +8 +6 +2 +4 +6 +8 +10 +12 +14 +16 +momentum (GeV/c)Figure 8: Pion rejection factor as a function of kaon identification efficiency +for pfRICH (left) and dRICH (right). The corner plots in both panels show +the reconstructed Cherenkov angles of a sample of an equal fraction of the +pion/kaon mixture. +have shown promising results. Similar to the pfRICH the pion +rejection factor as a function of kaon detection efficiency shows +for saturated pions and kaons that dRICH is able to reject pi- +ons without any dilution of the kaon detection efficiency (see +figure 8). We have also demonstrated that the reconstructed +mass of the particles as a function of particle momentum ob- +tained from the reconstructed Cherenkov angle, reconstructed +refractive index (using pion mass hypothesis) and known par- +ticle momentum are physical for both aerogel and C2F6. The +central bands are straight lines and the spread are consistent +with the Cherenkov angle resolution (see figure 6 and 8 from +dRICH section of [15]). +7. Conclusions +In the ATHENA software framework, we have studied and +demonstrated that the PID requirements can be achieved us- +ing two RICH detectors in the forward and backward endcaps +by exploiting the emitted Cherenkov photons and their depen- +dency on a threshold momentum (see figure 1 and 2 of [15]). +The obtained Nσ separation from simulation as a function of +the particle momentum and pseudorapidity had been translated +into detection efficiencies to use in Delphes [16] framework +for physics simulation. +The reconstruction software can be +more sophisticated in the future, and the implementation of +likelihood-based PID algorithms and machine learning can be +options for higher-level physics studies to perform PID. Never- +theless, the studies performed in the ATHENA framework are +already promising. Independent of ATHENA software, ECCE +proto-collaboration has also demonstrated similar results us- +ing a modular RICH (mRICH) in the backward endcap and a +dRICH with different geometrical parameters in the forward di- +rection [3]. Following the detector committee advisory panel, +the 1.5T magnet configuration has been chosen as the base- +line configuration, it has been mentioned that both ECCE and +ATHENA proto collaborations are capable of delivering the +entire EIC physics program. Hence, the new EPIC collabo- +ration has been formed taking advantage of lessons learnt by +the both ECCE and ATHENA collaborations, within EPIC a re- +optimization of the forward RICH is ongoing. The choice of +the backward RICH technology is currently under discussion. +Nevertheless, the software framework is almost identical to that +used in the ATHENA framework. Keeping the IRT software as +a baseline, optimization of the dRICH, pfRICH is already ongo- +ing. For the tuning and characterization studies, single photon +resolution, and the number of detected photons per particle will +be used also for characterizing the RICH detectors of the EPIC +collaboration [15]. +8. Acknowledgement +• One of the authors (C.Chattejee) is supported by the Eu- +ropean Union’s Horizon 2020 Research and Innovation +Programme under Grant Agreement AIDAinnova - No +101004761 +• The work of C. Dilks and A. Vossen is supported by the +U.S. Department of Energy, Office of Science, Office of +Nuclear Physics. +References +[1] R. A. Khalek, et al., Science requirements and detector concepts for the +electron-ion collider: Eic yellow report (2021). doi:10.48550/ARXIV. +2103.05419. +URL https://arxiv.org/abs/2103.05419 +[2] J. Adam, et al., Athena detector proposal – a totally hermetic electron +nucleus apparatus proposed for ip6 at the electron-ion collider, JINST 17 +(2022) 10 P10019. +[3] J. K. Adkins, et al, Design of the ecce detector for the electron ion collider +(2022). doi:10.48550/ARXIV.2209.02580. +URL https://arxiv.org/abs/2209.02580 +[4] R. Alarcon, et al, Core – a compact detector for the eic (2022). doi: +10.48550/ARXIV.2209.00496. +URL https://arxiv.org/abs/2209.00496 +[5] M. Contalbrigo, et al., Aerogel mass production for the clas12 rich: Novel +characterization methods and optical performance, NIM A 876 (2017) +168–172. +[6] E. Albrecht, et al., Vuv absorbing vapours in n-perfluorocarbons, +NIMA A 510 (3) (2003) 262–272. doi:https://doi.org/10.1016/ +S0168-9002(03)01867-9. +URL https://www.sciencedirect.com/science/article/pii/ +S0168900203018679 +[7] H. Photonics, S13361-3050ae-08. +URL +https://www.hamamatsu.com/us/en/product/type/ +S13361-3050AE-08/index.html +[8] H. Photonics, Mppc arrays, s13361-3050 series. +URL +https://www.hamamatsu.com/resources/pdf/ssd/ +s13361-3050_series_kapd1054e.pdf +[9] R. Abjean, et al., Refractive index of hexafluoroethane (c2f6) in the 300- +150 nm wavelength range, NIMA 354 (2) (1995) 417–418. doi:https: +//doi.org/10.1016/0168-9002(94)01006-4. +URL https://www.sciencedirect.com/science/article/pii/ +0168900294010064 +[10] A. Burner, W. Goad, Measurement of the specific refractivities of cf4 and +c2f6, The Journal of Chemical Physics 73 (2) (1980) 990. +[11] M. Frank, et. al, Dd4hep: A detector description toolkit for high en- +ergy physics experiments, Journal of Physics: Conference Series 513 (2) +(2014) 022010. doi:10.1088/1742-6596/513/2/022010. +URL https://dx.doi.org/10.1088/1742-6596/513/2/022010 +[12] Juggler webpage. +URL https://eicweb.phy.anl.gov/EIC/juggler +[13] N.Akopov, et al., The hermes dual-radiator ring imaging cherenkov de- +tector, NIMA 479 (2002) 511–530. +[14] A. Kiselev, Inverse ray tracing library. +URL https://eicweb.phy.anl.gov/EIC/irt +[15] Pid supplementary materials wiki page. +URL +https://wiki.bnl.gov/athena/index.php/Particle_ +Identification +[16] M. Arratia, S. Sekula, A delphes card for the eic yellow-report detector +(2021). doi:10.5281/ZENODO.4592887. +URL https://zenodo.org/record/4592887 +4 + +200 +150 +100 +50 +0.6 +0.7 +0.8 +60 +Kaon detection efficiency, [0..1]150 +100 +50 +80 +185 +190 +195 +200 +Reconstructed Cherenkov angle, [mrad]140 +uo +120 +ectio +0100 +uol +80 +60 +40 +20 +0.6 +0.7 +0.8 +0.9 +Kaon detection efficiency, [0..1]400 +Number +200 +100 +36 +37 +38 +39 +40 +41 +Reconstructed Cherenkov angle, [mrad] \ No newline at end of file diff --git a/eNE_T4oBgHgl3EQf1Ryp/content/tmp_files/load_file.txt b/eNE_T4oBgHgl3EQf1Ryp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0cdf59374e3f5797daf072762339f8f9d6f5a7e6 --- /dev/null +++ b/eNE_T4oBgHgl3EQf1Ryp/content/tmp_files/load_file.txt @@ -0,0 +1,304 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf,len=303 +page_content='Simulation studies related to the particle identification by the forward and backward RICH detectors at Electron Ion Collider D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='Bhattacharyaa, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='Cisbanid, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Chatterjeea,∗, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Dalla Torrea, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Dilksb, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Kiseleve, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='Klestf, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Preghenellac, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Vossenb aINFN Trieste, Italy bDuke University, USA cINFN Bologna, Italy dINFN Roma 1, Italy eBrookheaven National Laboratory, USA fStonyBrook University, USA Abstract The Electron-Ion collider (EIC) will be the ultimate facility to study the dynamics played by the colored quarks and gluons to the emergence of the global phenomenology of the nucleons and nuclei as described by Quantum Chromodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The physics programs will greatly rely on efficient particle identification (PID) in both the forward and the backward regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The forward and the backward RICHes of the EIC have to be able to cover wide acceptance and momentum ranges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' in the forward region a dual radiator RICH (dRICH) is foreseen and in the backward region a proximity-focusing RICH can be foreseen to be employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The geometry and the performance studies of the dRICH have been performed as prescribed in the EIC Yellow Report using the ATHENA software framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' This part of our work reports the effort following the call for EIC detector proposal the studies related to the forward and the backward RICHes performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' In the forward region, dRICH performance showed a pion- kaon separation from around 1 GeV/c to 50 GeV/c at a three sigma level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' the proximity focusing RICH (pfRICH) foreseen for the backward region can reach three sigma separation up to 3 GeV/c for e/π and up to 10 GeV/c for π/K mass hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Requirements of particle identification in EIC EIC at Brookhaven National Laboratory(USA)[1] is ex- pected to start data-taking in the early 2030s to answer crit- ical questions related to Quantum ChromoDynamics (QCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' A state-of-the-art machine and cutting-edge detector technol- ogy are fundamental for its success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Highly polarized elec- trons (∼70-80%) will be collided with highly polarized nucle- ons and light nuclei (∼70-80%) and also with heavy ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The high luminosity collisions (∼1034 electron proton cm−2s−1) will take place over a wide center of mass energy (20-141 GeV) with a possibility of more than one interaction points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Many of EIC physics require excellent particle identifications over a wide phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The requirements are documented as a common effort of the entire EIC community;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' afterwards, we will refer to this report as Yellow Report (YR) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' In re- sponse to the call for detector proposal submission, three proto- collaborations ATHENA [2], ECCE [3] and CORE[4] had been formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Our report mainly focuses on the simulation studies made using ATHENA software framework for studying the for- ward and backward RICH detectors’ performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' PID performance requirements for EIC physics The YR gives indicative requirements for the hadron PID in the electron-going endcap: better than 3σ π/K up to 10 GeV/c (table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='1 of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='[1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' In the following part of the article, we will consider this as our reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The forward dRICH is aimed to perform 3σ π/K separation up to 50 GeV/c and e/π separation up to 15 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' As mentioned in the YR, the required accep- tance for the dRICH is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='0 ≤ η ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' These reference numbers were taken as guidance for the ATHENA implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Fig- ∗Corresponding author Email address: chandradoy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='chatterjee@ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='it (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Chatterjee) IP pfRICH dRICH Aerogel Aerogel Sensors Services Sensors Mirrors z x Figure 1: Schematic of the forward dual radiator RICH (dRICH) and backward proximity focusing (pfRICH) with respect to the interaction point (IP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' ure 1 shows the location of the two RICH detectors with respect to the interaction point (IP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' pfRICH geometry The ATHENA design assumed that the proximity-focusing RICH will occupy space from -150 cm to -210 cm from the nominal IP (available space between the central tracker and crystal calorimeter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' In our implementation, the vessel has a cylindrical shape, with an outer radius of 93 cm, and a cutaway at small radii as determined by the design of the beam pipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' 3 cm thick aerogel with an average refractive index ⟨n⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='019 was used in the simulations as the main Cherenkov radiator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The properties of the aerogel (refractive index variation, ab- sorption, and Rayleigh scattering parameterizations as a func- Preprint submitted to Nuclear Instruments & Methods in Physics Research, Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' January 23, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='08334v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='ins-det] 17 Jan 2023 tion of wavelength) were taken from the available CLAS12 data [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' A 40 cm long expansion volume was assumed to be filled with C4F10 to provide an additional e/π separation capa- bility in a threshold mode below ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='9 GeV/c, with parameter- izations taken from [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Hamamatsu S13361-3050AE-08 8x8 SiPM panels [7] are anticipated as a reference photosensor type (3 mm single SiPM size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' SiPM Photon Detection Efficiency (PDE) as well as the geometric fill factor is taken according to the Hamamatsu specifications [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' About 15 cm of space be- hind the SiPM plane is reserved inside the vessel for the readout electronics and services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' We applied an additional safety factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='7 on top of this, namely, we assume that only 70% of pho- tons that pass the PDE and the geometric fill-factor selection are actually detected and used in the Cherenkov angle evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Forward dual radiator RICH (dRICH) In general, the dRICH configuration is very similar to the one from the YR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' A substantial effort was made though in order to accommodate such an apparatus in the overall tight space available for the ATHENA detector in RHIC IP6 Interaction Region and to quantify its expected performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' dRICH location and vessel boundaries During the optics tuning it was realized that the originally al- located space of ∼ 120 cm along the beamline is not sufficient to contain the focal plane inside the vessel, due to a very large polar angular acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' It became clear that to shift the ves- sel further away from the IP in order to minimize the adverse effects of the solenoid fringe field, even though the coil and the return flux configuration of the new magnet were carefully tuned to observe the so-called projectivity requirement, namely to minimize the overall bending of the charged secondary par- ticles originated from the IP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' As a consequence of these studies, in the final ATHENA configuration, which is presented in the Proposal, the dRICH vessel was shifted by ∼30 cm away from the IP, and at the same time the solenoid coils were moved by 25 cm towards the electron-going endcap (see figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Figure 2: Example of the effect of the magnetic field in single photon Cherenkov angle RMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Black points represent dRICH located at nominal po- sitions prior to finalization, red points represent the dRICH additionally shifted by 30 cm away from IP and blue points represent dRICH is shifted like the red points additionally the solenoid coil is shifted by 25 cm towards electron endcap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' dRICH geometry The dRICH used the same aerogel type and the same SiPM sensors as pfRICH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The safety factor of 70% was also included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The vessel length along the beam line is 140 cm, and it occupies the range from +190 cm to +330 cm from the IP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The detector will have six 60◦ sectors, each equipped with its own spherical mirror segment and a sensor plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Aerogel thickness is 4 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' We have used C2F6 gas as a radiator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' This allows achieving a comfortable overlap with the aerogel, and a 3σ π/K separation is obtained up to ∼50 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='C2F6 refractive index parameteri- zation is taken from [9, 10], and a conservative value of 10 m absorption length is taken for the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The sensors are positioned on a sphere, with a square tiling algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Three variables parameterize the spherical mirrors: the z position of the backplane, which is the maximum z the spherical mirror will reach, along with two focus tune parameters fx and fz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Point-to- point focusing of the IP on the centre of the sensor sphere cor- responds to focus tunes fx = fz = 0, which represents a starting point for the focus tuning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' because the IP is far from the opti- cal axis of the spherical mirror, spherical aberrations cause the proper point-to-point focal region to be significantly blurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' In order to focus Cherenkov rings on the sensors, parallel-to-point focusing is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' By changing the values of fx and fz it is pos- sible to steer the parallel-to-point focal region to be as close to the sensor surfaces as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Simulation studies for the ATHENA proto collaboration A DD4Hep-based [11] framework was used for the ATHENA proto-collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The reconstruction was based on the Juggler framework [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' A newly written Inverse Ray Tracing code (IRT), equally applicable in a standalone GEANT4 environment and in the ATHENA software frame- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The code represents itself a substantial generalization of the IRT algorithm initially developed for HERMES dual radia- tor RICH [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' It allows one to perform ray tracing between the detected photon location in 3D space and the expected emission point range along the charged particle trajectory on a predefined sequence of refractive and reflective boundaries, using a 2D it- erative Newton-Gauss minimization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' It is fully con- figurable and has a persistent model, which allows one to ex- port and import ROOT files with the actual optics description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The code was originally developed to quantify the performance of the LHCB-like RICH#1 configuration for the ATHENA for- ward RICH, with spherical and flat mirrors in a sequence, which cannot be easily handled by a simple 2D IRT algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' How- ever, it is equally applicable to a simpler pfRICH geometry, where both absorption and Rayleigh scattering in the aerogel and refraction on the aerogel-gas boundary still play a role in the unbiased Cherenkov angle evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The IRT library [14] is available in the ATHENA software repository, together with the standalone GEANT4 stepping code and the so-called Jug- gler plugin in the ATHENA reconstruction environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' IRT algorithm implementation in the ATHENA Juggler PID plugin is complemented by a sophisticated logic, performing sampling along the charged particle trajectory, which allows one to es- timate the average Cherenkov angle in the same way for both straight tracks and in presence of a relatively strong bending component of the ATHENA solenoid magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' A prop- erly weighted ensemble of the Cherenkov photon angle esti- mates is then checked against e/π/K/p mass hypotheses, and a probability of each hypothesis is provided as an output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Performance Studies 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' pFRICH performance and consistency checks To check the consistency of the software stack the perfor- mance studies of the pfRICH were done before, thanks to its simple geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Reconstructed Cherenkov angles as a func- tion of momentum have been studied for different mass hy- potheses to check the consistency (see figure 3,4 from pfRICH 2 n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='5 or 0, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='44 rad n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='0 or 0, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='27 rad RMS (mrad) Rich: 30 cm shifted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Map: Solenoid Rich: 30 cm shifted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=': Map: Solenoid No shit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Map: Solenoid od No shift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Map: Solenoid Rich: 30 cm shifted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Map: New Shifted Solednoid Rich: 30 cm shifted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Map: New Shifted Solednoid 3 50 p (GeV) 3 p (GeV) n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='5 or @ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='16 rad n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='0 or 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='10 rad Rich: 30 cm shifted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Map: Solenoid Rich: 30 cm shifted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Map: Solenoid No shift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Map: Solenoid 8 No shift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Map: Solenoid Rich: 30 cm shifted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Map: New Shifted Solednoid Rich: 30 cm shifted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Map: New Shifted Solednoid 2+ 2 30 35 40 45 50 p (GeV) 20 25 3 p (GeV)section of [15]) and also for dRICH (see figure 5,7 from dRICH section of [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Single particles (e/π/K) were shot to estimate the Nσ separation as a function of momentum and pseudora- pidity (η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' It has been demonstrated that the Yellow Report re- quirements can be achieved using the simple pfRICH geometry and the reconstructed Cherenkov angles provide a satisfactory kaon rejection factor without diluting the pion identification ef- ficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The acceptance plot shows that between η from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='6 to 3 the number of detected photons is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' We have defined 50% of the maximum number of the detected photon as the working acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' This demonstrates the pfRICH can cover a region larger than eta η 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='5 in the backward region (see figure 4 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' For three different η regions the Nσ had been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' One can reach three sigma separation up to 3 GeV/c for e/π and up to 10 GeV/c for π/K mass hypothesis (see figure 4 c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Using an equal mixture of pion and kaon samples for particles at sat- uration a pion rejection factor as a function of kaon detection efficiency is also computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' That shows that without diluting the kaon detection efficiency high level of pion rejection can be obtained (figure 4 d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Figure 3: Number of detected photons as in pfRICH rings (aerogel) as a func- tion of pseudorapidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The sharp drop at the edges is due to the containment of the partial ring in the sensors due to its acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' 50% of the ring contained in the sensor plane is defined as the acceptance limit of the detector Figure 4: pfRICH Nσ separation power as a function of momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Using only aerogel information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' dRICH performance The forward dRICH also showed performance as prescribed by the YR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Both for the aerogel and gas we have observed the acceptance is from η 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='2 to around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='5 (figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' For the sepa- ration power, it is evident that using the aerogel and gas infor- mation once can reach from some hundreds of MeV/c up to 50 GeV/c without the presence of any hole for π/K separation (fig- ure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' As mentioned the forward dRICH should be able to Figure 5: Number of detected photons as in dRICH rings (aerogel and C2F6) as a function of pseudorapidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The sharp drop at the edges is due to the con- tainment of the partial ring in the sensors due to its acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' 50% of the ring contained in the sensor plane is defined as the acceptance limit of the detector Figure 6: Nσ separation as a function of momentum of positively identified π/K using aerogel and C2F6 information from dRICH Figure 7: Nσ separation of e/π using aerogel and C2F6 information from dRICH reject electrons, it has been demonstrated that up to 15 GeV/c the separation can be done from mid to high η (figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' At low η the separation power is diluted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' At very low η the effect of the solenoidal field plays a critical role in the e/π separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Instead of using mirrors with a single radius of curvature, each sector can be equipped with a mirror divided into two sections with slightly different radii of curvature in order to obtain better resolution over the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Standalone studies with dual mirrors 3 8 Acceptance 6 pfRICH 8-GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='pions O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='safety factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='included 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='5 nON 20 元-K(red) e-π(Blue) 18 n=-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='5 16 n=-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='3 n=-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='2 14 Separation Power 12 10 8 6 4 2 0 2 4 6 8 10 12 momentum (GeVlc)ad dRICHacceptance 25 CF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Gas50GeV pions Aerogel 20 15 Aerogel12 GeV pions 10 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='5 4 Pseudorapidity( n(元-K) 30 Aerogel(Blue) C,Fc(Red) No n= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='5 25 n= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='4 n= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='2 20 15 10 5 10 20 30 40 50 60 momentum (GeV/c)(1-a) 20 C,F6 ON 18 n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='5 16 n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='4 n= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='2 14 Aerogel n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='5 12 n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='4 10 n= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='2 8 6 2 4 6 8 10 12 14 16 momentum (GeV/c)Figure 8: Pion rejection factor as a function of kaon identification efficiency for pfRICH (left) and dRICH (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The corner plots in both panels show the reconstructed Cherenkov angles of a sample of an equal fraction of the pion/kaon mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' have shown promising results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Similar to the pfRICH the pion rejection factor as a function of kaon detection efficiency shows for saturated pions and kaons that dRICH is able to reject pi- ons without any dilution of the kaon detection efficiency (see figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' We have also demonstrated that the reconstructed mass of the particles as a function of particle momentum ob- tained from the reconstructed Cherenkov angle, reconstructed refractive index (using pion mass hypothesis) and known par- ticle momentum are physical for both aerogel and C2F6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The central bands are straight lines and the spread are consistent with the Cherenkov angle resolution (see figure 6 and 8 from dRICH section of [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Conclusions In the ATHENA software framework, we have studied and demonstrated that the PID requirements can be achieved us- ing two RICH detectors in the forward and backward endcaps by exploiting the emitted Cherenkov photons and their depen- dency on a threshold momentum (see figure 1 and 2 of [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The obtained Nσ separation from simulation as a function of the particle momentum and pseudorapidity had been translated into detection efficiencies to use in Delphes [16] framework for physics simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The reconstruction software can be more sophisticated in the future, and the implementation of likelihood-based PID algorithms and machine learning can be options for higher-level physics studies to perform PID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Never- theless, the studies performed in the ATHENA framework are already promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Independent of ATHENA software, ECCE proto-collaboration has also demonstrated similar results us- ing a modular RICH (mRICH) in the backward endcap and a dRICH with different geometrical parameters in the forward di- rection [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Following the detector committee advisory panel, the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='5T magnet configuration has been chosen as the base- line configuration, it has been mentioned that both ECCE and ATHENA proto collaborations are capable of delivering the entire EIC physics program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Hence, the new EPIC collabo- ration has been formed taking advantage of lessons learnt by the both ECCE and ATHENA collaborations, within EPIC a re- optimization of the forward RICH is ongoing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' The choice of the backward RICH technology is currently under discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Nevertheless, the software framework is almost identical to that used in the ATHENA framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Keeping the IRT software as a baseline, optimization of the dRICH, pfRICH is already ongo- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' For the tuning and characterization studies, single photon resolution, and the number of detected photons per particle will be used also for characterizing the RICH detectors of the EPIC collaboration [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Acknowledgement One of the authors (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='Chattejee) is supported by the Eu- ropean Union’s Horizon 2020 Research and Innovation Programme under Grant Agreement AIDAinnova - No 101004761 The work of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Dilks and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Vossen is supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Department of Energy, Office of Science, Office of Nuclear Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Khalek, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=', Science requirements and detector concepts for the electron-ion collider: Eic yellow report (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' 2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='05419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='org/abs/2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='05419 [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Adam, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=', Athena detector proposal – a totally hermetic electron nucleus apparatus proposed for ip6 at the electron-ion collider, JINST 17 (2022) 10 P10019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Adkins, et al, Design of the ecce detector for the electron ion collider (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='02580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='org/abs/2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='02580 [4] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Alarcon, et al, Core – a compact detector for the eic (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='00496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='org/abs/2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='00496 [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Contalbrigo, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=', Aerogel mass production for the clas12 rich: Novel characterization methods and optical performance, NIM A 876 (2017) 168–172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' [6] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Albrecht, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=', Vuv absorbing vapours in n-perfluorocarbons, NIMA A 510 (3) (2003) 262–272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' doi:https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='1016/ S0168-9002(03)01867-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='com/science/article/pii/ S0168900203018679 [7] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Photonics, S13361-3050ae-08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='hamamatsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='com/us/en/product/type/ S13361-3050AE-08/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='html [8] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Photonics, Mppc arrays, s13361-3050 series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='hamamatsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='com/resources/pdf/ssd/ s13361-3050_series_kapd1054e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='pdf [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Abjean, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=', Refractive index of hexafluoroethane (c2f6) in the 300- 150 nm wavelength range, NIMA 354 (2) (1995) 417–418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' doi:https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='1016/0168-9002(94)01006-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='com/science/article/pii/ 0168900294010064 [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Burner, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Goad, Measurement of the specific refractivities of cf4 and c2f6, The Journal of Chemical Physics 73 (2) (1980) 990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Frank, et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' al, Dd4hep: A detector description toolkit for high en- ergy physics experiments, Journal of Physics: Conference Series 513 (2) (2014) 022010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='1088/1742-6596/513/2/022010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' URL https://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='1088/1742-6596/513/2/022010 [12] Juggler webpage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' URL https://eicweb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='anl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='gov/EIC/juggler [13] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='Akopov, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=', The hermes dual-radiator ring imaging cherenkov de- tector, NIMA 479 (2002) 511–530.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Kiselev, Inverse ray tracing library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' URL https://eicweb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='anl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='gov/EIC/irt [15] Pid supplementary materials wiki page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' URL https://wiki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='bnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='gov/athena/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content='php/Particle_ Identification [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Arratia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE_T4oBgHgl3EQf1Ryp/content/2301.08334v1.pdf'} +page_content=' Sekula, A delphes card for the eic yellow-report detector (2021).' metadata={'source': 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ORGANISATION FOR NUCLEAR RESEARCH (CERN) +Submitted to: Phys. Rev. D. +CERN-EP-2022-235 +January 10, 2023 +Search for nonresonant pair production of Higgs +bosons in the 𝒃 ¯𝒃𝒃 ¯𝒃 final state in 𝒑 𝒑 collisions at +√𝒔 = 13 TeV with the ATLAS detector +The ATLAS Collaboration +A search for nonresonant Higgs boson pair production in the 𝑏 ¯𝑏𝑏 ¯𝑏 final state is presented. The +analysis uses 126 fb−1 of 𝑝𝑝 collision data at √𝑠 = 13 TeV collected with the ATLAS detector +at the Large Hadron Collider, and targets both the gluon–gluon fusion and vector-boson fusion +production modes. No evidence of the signal is found and the observed (expected) upper limit +on the cross-section for nonresonant Higgs boson pair production is determined to be 5.4 +(8.1) times the Standard Model predicted cross-section at 95% confidence level. Constraints +are placed on modifiers to the 𝐻𝐻𝐻 and 𝐻𝐻𝑉𝑉 couplings. The observed (expected) 2𝜎 +constraints on the 𝐻𝐻𝐻 coupling modifier, 𝜅𝜆, are determined to be [−3.5, 11.3] ([−5.4, 11.4]), +while the corresponding constraints for the 𝐻𝐻𝑉𝑉 coupling modifier, 𝜅2𝑉 , are [−0.0, 2.1] +([−0.1, 2.1]). In addition, constraints on relevant coefficients are derived in the context of the +Standard Model effective field theory and Higgs effective field theory, and upper limits on +the 𝐻𝐻 production cross-section are placed in seven Higgs effective field theory benchmark +scenarios. +© 2023 CERN for the benefit of the ATLAS Collaboration. +Reproduction of this article or parts of it is allowed as specified in the CC-BY-4.0 license. +arXiv:2301.03212v1 [hep-ex] 9 Jan 2023 + +CERN1 Introduction +The discovery of the 125 GeV Higgs boson (𝐻) [1–4] at the Large Hadron Collider (LHC) has prompted +a broad research program to investigate its properties and compare the measurements with the Standard +Model (SM) predictions. Of particular interest is the search for nonresonant Higgs boson pair, known +as di-Higgs (HH), production. This process has a strong dependence on the Higgs self-coupling, which +is a key ingredient of the electroweak symmetry breaking mechanism and a sensitive probe for physics +beyond the SM (BSM physics) in various scenarios, such as two-Higgs-doublet models [5], composite +Higgs models [6], twin Higgs models [7], and the minimal supersymmetric extension of the SM [8, 9]. +The Higgs self-coupling also plays a fundamental role in understanding the stability of the universe [10]. +The dominant SM HH production process is gluon–gluon fusion (ggF). Its cross-section, for a Higgs boson +mass 𝑚𝐻 = 125 GeV, calculated at next-to-next-to-leading order (NNLO) including finite top-quark-mass +effects [11], is 31.05 fb at a center-of-mass energy √𝑠 = 13 TeV. The two dominant leading-order Feynman +diagrams contributing to this process are shown in Figure 1, where Figure 1(a) is commonly referred to as +the box diagram and Figure 1(b) as the triangle diagram. The triangle diagram introduces the dependence +on the trilinear Higgs self-coupling, 𝜆, shown by the red vertex in Figure 1(b), which can be expressed +in terms of its modifier, 𝜅𝜆.1 In the SM, these two diagrams interfere destructively. As a result, the HH +production cross-section and kinematic properties depend critically on the value of 𝜅𝜆. +g +g +H +H +κt +κt +(a) +g +g +H +H +κt +κt +(b) +Figure 1: The two leading-order gluon–gluon fusion di-Higgs production Feynman diagrams: (a) the box diagram; +(b) the triangle diagram. +The HH production process with the second-highest cross-section in the SM is vector-boson fusion (VBF), +with a calculated value of 1.73 fb at next-to-next-to-next-to-leading order (N3LO) [12], for 𝑚𝐻 = 125 GeV +at √𝑠 = 13 TeV. Figure 2 illustrates the Feynman diagrams involved in di-Higgs production via vector-boson +fusion at leading order (LO). The coupling modifiers 𝜅𝜆, 𝜅𝑉 and 𝜅2𝑉 are respectively shown at the HHH, +HVV and HHVV interaction vertices, where 𝑉 stands for the gauge vector bosons W or Z. In the SM, the +divergences in the Figures 2(b) and 2(c) diagrams exactly cancel out due to perturbative unitarity. As +𝜅𝑉 and 𝜅2𝑉 depart from their SM value of one, this canceling out no longer occurs, introducing a linear +dependence of the cross-section on the effective center-of-mass energy of the incoming vector bosons [13]. +Therefore, the Higgs bosons produced in non-SM 𝜅𝑉 /𝜅2𝑉 scenarios are expected to be more energetic +and more central in the detector on average. This increase in the energy of Higgs bosons with increasing +deviation from the SM continues up to the scale of some new physics, which is required to unitarize the +total amplitude. +1 A coupling modifier, 𝜅, is defined as the ratio of the modified coupling to its SM value, 𝜅 = 𝑐/𝑐SM. By definition, 𝜅 = 1 denotes +the value of the coupling predicted by the SM. +2 + +H +H +q +q +q +q +H +V +V +κλ +κV +(a) +H +H +q +q +q +q +V +V +κ2V +(b) +H +H +q +q +q +q +V +V +κV +κV +(c) +Figure 2: The three tree-level vector-boson fusion di-Higgs production Feynman diagrams. +The analysis described in this paper targets the HH process in the bbbb final state, in both the ggF and +VBF production modes, using the data collected by ATLAS between 2016 and 2018, during Run 2 of the +LHC. Assuming the SM branching ratio of 58.2% for H → bb [14, 15], about one third of di-Higgs events +decay into bbbb, making it the most abundant di-Higgs final state. However, as this is a fully hadronic +final state, the analysis faces the challenge of large backgrounds, which originate mostly from nonresonant +QCD production of multiple heavy (b/t) quarks, as well as from light-quark-initiated jets misidentified as +originating from heavy quarks. +The results are interpreted in terms of constraints on the 𝜅𝜆 and 𝜅2𝑉 coupling modifiers, assuming 𝜅𝑉 = 1. +The analysis also provides one- and two-dimensional constraints on relevant couplings in the SM effective +field theory (SMEFT) [16–18] and Higgs effective field theory (HEFT) [19, 20] frameworks. In the SMEFT +framework, the effects of new physics may be described with an effective Lagrangian: +LSMEFT = LSM + 1 +Λ2 +∑︁ +𝑘 +𝑐(6) +𝑘 𝑂 (6) +𝑘 +, +where LSM represents the SM Lagrangian, 𝑂𝑘 are higher-dimensional local operators, 𝑐𝑘 are the Wilson +coefficients, and Λ is the mass scale of the new physics phenomena (set to 1 TeV for this result). The +analysis considers operators 𝑂𝑘 in the Warsaw basis, which provides a complete set of operators allowed by +SM gauge symmetries at dimension six [21] (dimension-five operators introduce lepton and baryon number +violation, and are therefore ignored in this result). The five operators relevant to the HH process and their +coefficients, 𝑐H, 𝑐H□, 𝑐𝑡H, 𝑐𝑡𝐺, and 𝑐H𝐺, are listed in Table 1 [22]. The computation of amplitudes from +the above Lagrangian includes three terms: a pure SM term, a “quadratic” term of order (1/Λ4) including +purely new physics, and a “linear” term of order (1/Λ2) accounting for the interference between the SM +and new physics. The SMEFT constraints calculated in this analysis include both the linear and quadratic +new physics terms. +In the HEFT framework, new physics in the electroweak sector is described through anomalous couplings +of the Higgs boson. The organization of the HEFT Lagrangian is guided by chiral perturbation theory [23], +with the low-energy dynamics of electroweak symmetry breaking described using a nonlinear realization of +the gauge symmetry group 𝑆𝑈(2)𝐿 ×𝑈(1)𝑌 . One advantage of the HEFT framework is that the anomalous +single-Higgs-boson and HH couplings are defined separately, allowing simplified HH interpretations. In +the HEFT Lagrangian, ggF HH production is described at LO by five relevant operators and their associated +Wilson coefficients: 𝑐HHH, 𝑐𝑡𝑡H, 𝑐𝑔𝑔H, 𝑐𝑔𝑔HH, and 𝑐𝑡𝑡HH. In this formalism, 𝑐HHH is equivalent to 𝜅𝜆 and +𝑐𝑡𝑡H is equivalent to the modifier for the coupling between the Higgs boson and top quark, 𝜅𝑡, shown by the +3 + +Table 1: The five relevant SMEFT coefficients and their corresponding dimension-6 operators, as defined in the +Warsaw basis [21, 22]. +Wilson Coefficient +Operator +𝑐H +(H†H)3 +𝑐H□ +(H†H)□(H†H) +𝑐tH +(H†H)( ¯𝑄 ˜H𝑡) +𝑐H𝐺 +H†H𝐺 𝐴 +𝜇𝜈𝐺 𝜇𝜈 +𝐴 +𝑐t𝐺 +( ¯𝑄𝜎𝜇𝜈𝑇 𝐴t) ˜H𝐺 𝐴 +𝜇𝜈 +light blue vertex in Figure 1. Fixing 𝑐𝑡𝑡H = 𝑐HHH = 1 and 𝑐𝑔𝑔H = 𝑐𝑔𝑔HH = 𝑐𝑡𝑡HH = 0 restores the SM. At +next-to-leading order (NLO), seven HEFT benchmark models (BM) [24] have been defined using cluster +analysis [25] to probe a wide variety of characteristic shapes of the 𝑚HH spectrum resulting from different +BSM scenarios. The values of the coefficients used to define these scenarios are given in Table 2. +Table 2: The values of the HEFT Wilson coefficients in the SM and in seven BSM benchmark models, as defined in +Ref. [24]. +Benchmark Model +𝑐𝐻 𝐻 𝐻 +𝑐𝑡𝑡𝐻 +𝑐𝑔𝑔𝐻 +𝑐𝑔𝑔𝐻 𝐻 +𝑐𝑡𝑡𝐻 𝐻 +SM +1 +1 +0 +0 +0 +BM1 +3.94 +0.94 +1/2 +1/3 +−1/3 +BM2 +6.84 +0.61 +0.0 +−1/3 +1/3 +BM3 +2.21 +1.05 +1/2 +1/2 +−1/3 +BM4 +2.79 +0.61 +−1/2 +1/6 +1/3 +BM5 +3.95 +1.17 +1/6 +−1/2 +−1/3 +BM6 +5.68 +0.83 +−1/2 +1/3 +1/3 +BM7 +−0.10 +0.94 +1/6 +−1/6 +1 +The ATLAS Collaboration has previously published search results for nonresonant HH → bbbb production +using 27 fb−1 of early Run 2 data [26], and a dedicated search for VBF HH production in 126 fb−1 of +data collected between 2016 and 2018 [27]. The present analysis benefits from the use of the 2016–2018 +data for both production channels and also takes advantage of improvements in jet reconstruction and +in the identification of jets arising from the decays of b-quarks (“𝑏-tagging”) achieved by the ATLAS +Collaboration since the publication of Ref. [26]. In addition, the analysis employs a fully data-driven +technique for the background estimation, which uses an artificial neural network to perform a kinematic +reweighting of data from an alternative phase space to model the background in the region of interest. The +CMS Collaboration has also published results of a search for nonresonant HH → bbbb with its full Run 2 +dataset [28], setting the observed (expected) upper limit on the HH cross-section at 3.9 (7.8) times the +SM predicted cross-section, and restricting the allowed interval for 𝜅𝜆 to [−2.3, 9.4] ([−5.0, 12.0]), both +at 95% confidence level (CL). A more recent CMS HH → bbbb publication [29], in which the analysis +exploits topologies arising from highly energetic Higgs boson decays into 𝑏 ¯𝑏, sets the observed (expected) +upper limit at 9.9 (5.1) times the SM cross-section expectation, and restricts the allowed interval for 𝜅2𝑉 to +[0.62, 1.41] ([0.66, 1.37]), at 95% CL. Other searches for nonresonant 𝐻𝐻 production were performed by +ATLAS and CMS in the 𝑏 ¯𝑏𝜏+𝜏− [30, 31], 𝑏 ¯𝑏𝛾𝛾 [32, 33], 𝑏 ¯𝑏ℓ+𝜈ℓ−𝜈 [34, 35] decay channels, as well as by +4 + +ATLAS in the 𝑏 ¯𝑏𝑞𝑞ℓ𝜈 [36], 𝑊𝑊∗𝛾𝛾 [37] and 𝑊𝑊∗𝑊𝑊∗ [38] decay channels. Amongst them, the most +sensitive results to date from ATLAS come from the 𝑏 ¯𝑏𝛾𝛾 analysis, which sets the observed (expected) +95% CL upper limit on the SM nonresonant 𝐻𝐻 cross-section at 4.2 (5.7) times the SM expectation and +restricts the corresponding 𝜅𝜆 interval to [−1.5, 6.7] ([−2.4, 7.7]). The most sensitive results to date from +CMS come from the combination of the 𝑏 ¯𝑏𝑍𝑍, multilepton, 𝑏 ¯𝑏𝛾𝛾, 𝑏 ¯𝑏𝜏𝜏, and bbbb analyses, which set +the observed (expected) 95% CL upper limit on the SM nonresonant 𝐻𝐻 cross-section at 3.4 (2.5) times +the SM expectation and restricts the corresponding observed 𝜅𝜆 interval to [−1.24, 6.49] [39]. +This document is structured as follows. The ATLAS detector and the data and simulated events used in +the analysis are described in Sections 2 and 3, respectively. Section 4 presents the reconstruction and +identification of physics objects in this analysis and Section 5 details the event selection and categorization. +The background modeling method is described in Section 6, the systematic uncertainties are detailed in +Section 7 and, finally, the results are reported in Section 8 and the conclusion is given in Section 9. +2 ATLAS detector +The ATLAS detector [40] at the LHC covers nearly the entire solid angle around the collision point.2 It +consists of an inner tracking detector surrounded by a thin superconducting solenoid, electromagnetic and +hadron calorimeters, and a muon spectrometer incorporating three large superconducting air-core toroidal +magnets. +The inner-detector (ID) system is immersed in a 2 T axial magnetic field and provides charged-particle +tracking in the range |𝜂| < 2.5. The high-granularity silicon pixel detector covers the vertex region and +typically provides four space-point measurements per track, the first hit normally being in the insertable +B-layer installed before Run 2 [41, 42]. Following the pixel detector is the silicon microstrip tracker, which +usually provides eight measurements per track. These silicon detectors are surrounded by the transition +radiation tracker, which enables radially extended track reconstruction up to |𝜂| = 2.0. +The calorimeter system covers the pseudorapidity range |𝜂| < 4.9. Within |𝜂| < 3.2, electromagnetic +calorimetry is provided by barrel and endcap high-granularity lead/liquid-argon (LAr) calorimeters, with +an additional thin LAr presampler covering |𝜂| < 1.8 to correct for energy loss in material upstream of +the calorimeters. Hadron calorimetry is provided by the steel/scintillator-tile calorimeter, segmented into +three barrel structures within |𝜂| < 1.7, and two copper/LAr hadron endcap calorimeters. The solid angle +coverage is completed with forward copper/LAr and tungsten/LAr calorimeter modules optimized for +electromagnetic and hadronic energy measurements respectively. +The muon spectrometer (MS) comprises separate trigger and high-precision tracking chambers measuring +the deflection of muons in a magnetic field generated by the superconducting air-core toroidal magnets. +The field integral of the toroids ranges between 2.0 and 6.0 T·m across most of the detector. A set of +precision chambers covers the region |𝜂| < 2.7 with three layers of monitored drift tubes, complemented by +cathode-strip chambers in the forward region, where the background is highest. The muon trigger system +2 ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the center of the detector +and the 𝑧-axis along the beam pipe. The 𝑥-axis points from the IP to the center of the LHC ring, and the 𝑦-axis points +upwards. Cylindrical coordinates (𝑟, 𝜙) are used in the transverse plane, 𝜙 being the azimuthal angle around the 𝑧-axis. +The pseudorapidity is defined in terms of the polar angle 𝜃 as 𝜂 = − ln tan(𝜃/2). Angular distance is measured in units of +Δ𝑅 ≡ +√︃ +(Δ𝜂)2 + (Δ𝜙)2. +5 + +covers the range |𝜂| < 2.4 with resistive-plate chambers in the barrel, and thin-gap chambers in the endcap +regions. +Interesting events are selected by the first-level trigger system implemented in custom hardware, followed +by selections made by algorithms implemented in software in the high-level trigger [43]. The first-level +trigger accepts events from the 40 MHz bunch crossings at a rate below 100 kHz, which the high-level +trigger reduces in order to record events to disk at about 1 kHz. +An extensive software suite [44] is used in data simulation, in the reconstruction and analysis of real and +simulated data, in detector operations, and in the trigger and data acquisition systems of the experiment. +3 Data and simulated samples +3.1 Data sample +This analysis is performed in LHC proton–proton (pp) collision data at √𝑠 = 13 TeV collected between +2016 and 2018. Only data collected during stable beam conditions are used, with all relevant detector +systems functional [45], corresponding to an integrated luminosity of 126 fb−1. During 2016 data taking, a +fraction of the data (8.3 fb−1) was affected by an inefficiency in the online primary vertex reconstruction, +which reduced the efficiency of the 𝑏-tagging algorithms in the trigger; those events were not retained for +further analysis, resulting in an integrated luminosity of 24.6 fb−1 for the 2016 dataset. The integrated +luminosities of the 2017 and 2018 datasets are 43.7 fb−1 and 57.7 fb−1, respectively. +The analysis uses events that satisfy either of two types of trigger signatures, each with different requirements +on the number of jets and their 𝑏-tagging status [46]. The jets used are reconstructed with the anti-𝑘𝑡 +algorithm [47, 48], with a radius parameter of 𝑅 = 0.4. The 𝑏-tagging is performed at the trigger level with +the MV2c20 algorithm in 2016 and the MV2c10 algorithm in 2017 and 2018 [46], with a range of b-jet +identification efficiency operating points from 40% to 70% (as calculated from simulated tt samples.) The +first of the two trigger signatures used for selecting bbbb events requires two 𝑏-jets plus one additional jet +(“2𝑏1j”), while the second requires two 𝑏-jets plus two additional jets (“2𝑏2j”). The minimum transverse +energy (𝐸T) requirement on the jets is 35 GeV for all jets used in the 2𝑏2j trigger. In the 2𝑏1j trigger, the +b-tagged jets must have 𝐸T > 55 GeV, while the requirement on the minimum 𝐸T of the additional jet is +between 100 and 150 GeV, depending on the year of data taking. +3.2 Simulated samples +Monte Carlo (MC) simulation is used for the modeling of signal events, as well as to produce event samples +of background processes for cross-checks and validation studies. The Higgs boson mass is set to 125 GeV +in the simulation. All samples were processed by the ATLAS simulation framework [49] and the detector +response was simulated with Geant4 [50]. +The ggF signal process was simulated using the Powheg Box v2 generator [51–53] at NLO, including finite +top-quark-mass effects, using the PDF4LHC15 [54] parton distribution function (PDF) set. Parton showers +and hadronization were simulated with Pythia 8.244 [55] with the A14 set of tuned parameters [56] and +the NNPDF2.3lo PDF set [57]. The SM ggF HH cross-section was taken as 𝜎ggF = 31.05 fb, calculated at +NNLO including finite top-quark-mass effects [11]. Signal samples for the ggF process were generated +6 + +explicitly for coupling modifier values of 𝜅𝜆 = 1 and 10. A reweighting method is used to obtain a ggF +signal sample at each 𝜅𝜆 value, as described in Ref. [58]: scale factors are derived as a function of 𝜅𝜆 +in bins of the generator-level invariant mass of the HH system by performing a linear combination of +generator-level samples at three different 𝜅𝜆 values (𝜅𝜆 = 0, 1, and 20). The 𝜅𝜆 = 10 ggF signal sample is +used to validate the derived scale factors; this generated sample and the signal sample obtained from the +reweighting method are found to agree within the statistical precision of the simulated sample. Additional +generator-level ggF HH signal samples without parton showering were produced with Powheg Box v2 for +the 𝜅𝜆 = 0 and 20 coupling modifier configurations to provide a basis for the 𝜅𝜆 reweighting, along with the +SM ggF sample. For the reweighted ggF signal, the NNLO cross-section as a function of 𝜅𝜆 is taken from +Ref. [11]. In order to assess parton showering uncertainties, alternative ggF samples were generated using +the Powheg Box v2 generator at NLO with the PDF4LHC15 PDF set, interfaced to Herwig 7.1.6 [59] +for parton showering and hadronization using the Herwig 7.1-default set of tuned parameters [60] and +MMHT2014lo PDF set [61]. +To extract SMEFT coefficient constraints, parton-level ggF HH samples were generated with Mad- +Graph5_aMC@NLO [62–64] with the SMEFT@NLO model [65] for a variety of SMEFT coefficients. +A finely spaced multidimensional grid of signal samples was obtained using a LO-derived reweighting +procedure in the generator-level invariant mass of the HH system; this procedure is similar to that used +to obtain 𝜅𝜆 variations for the ggF signal, as described above. To extract HEFT coefficient constraints, a +similar NLO-derived reweighting procedure was applied to the simulated reconstruction-level ggF signal +sample to produce a variety of HEFT signal scenarios, including the seven benchmark scenarios defined +in Section 1, following the prescription outlined in Refs. [66, 67]. Additional 𝐾-factors were applied +to the SMEFT samples; these 𝐾-factors were derived using the ratio of the NLO cross-section to the +LO cross-section at the equivalent HEFT point, as obtained using the HEFT to SMEFT translation from +Ref. [24].3 +The VBF signal process was simulated using MadGraph 2.7.3 [63] at LO with the NNPDF3.0nlo PDF +set [68], interfaced with Pythia 8.244 for parton showering and hadronization using the A14 set of tuned +parameters and NNPDF2.3lo PDF set. Signal samples for the VBF process were generated explicitly for +coupling modifier values of (𝜅𝜆, 𝜅2𝑉 , 𝜅𝑉 ) = (1, 1, 1), (1, 1.5, 1), (2, 1, 1), (10, 1, 1), (1, 1, 0.5), (−5, 1, 0.5), +(0, 1, 1), (1, 0, 1), and (1, 3, 1). A linear combination of the first six of the listed samples is used to derive +distributions for a finer granularity of 𝜅2𝑉 values, following a technique used previously to generate 𝜅𝜆 +distributions [69]. The specific basis of six samples utilized is chosen to avoid large statistical uncertainties +in the reweighted signal samples resulting from sparsely populated areas of kinematic phase space. The +generated VBF signal samples not included in the linear combination basis – (𝜅𝜆, 𝜅2𝑉 , 𝜅𝑉 ) = (0, 1, 1), +(1, 0, 1), and (1, 3, 1) – were used to validate the performance of the combination method. These generated +samples and the corresponding signal samples obtained from the combination method were found to agree +within the statistical precision of the simulated samples. The cross-section for the VBF 𝐻𝐻 process, +evaluated at N3LO in QCD, is 1.73 fb in the SM [12, 70–72]. For the reweighted VBF signal points, the +N3LO to LO cross-section ratio at the SM value is calculated, and this factor is applied to the cross-sections +at each 𝜅𝜆, 𝜅2𝑉 , and 𝜅𝑉 point. In order to assess parton showering uncertainties, alternative LO samples +were generated using MadGraph 2.7.3 with the NNPDF3.0nlo PDF set, interfaced to Herwig 7.0.4 with +the Herwig 7.1-default set of tuned parameters and MMHT2014lo PDF set for parton showering and +hadronization. +Top-quark pair production (tt) and multijet background processes were simulated in order to validate the +3 Variations in the 𝑐𝑡𝐺 Wilson coefficient were neglected when calculating 𝐾-factors because the corresponding chromomagnetic +operator does not appear at LO within HEFT. +7 + +background modeling procedure. The tt sample was simulated at NLO in 𝛼s using Powheg Box v2 [73]. +Parton showering, hadronization, and the underlying event were modeled using Pythia 8.230. The matrix +element calculation uses NNPDF3.0nlo as the PDF set, while the parton shower and underlying-event +modeling uses NNPDF2.3lo and the A14 set of tuned parameters. The damping parameter ℎdamp, which +effectively regulates radiation at high 𝑝T, was set to 1.5 times the top quark’s mass. The tt simulation is +normalized using the value of the inclusive cross-section calculated with Top++ 2.0 [74, 75]. This accounts +for NNLO corrections in 𝛼s, including next-to-next-to-leading logarithmic (NNLL) resummation of soft +gluon terms. The multijet background samples were modeled using Pythia 8.235. This simulates pure +QCD 2-to-2 interactions at LO in 𝛼s. Events were showered using the parton shower native to Pythia, +which includes radiation and splitting that can result in additional jets. The A14 set of tuned parameters +and the NNPDF2.3lo PDF set were used. +Other background processes, such as SM Higgs boson, HH (in other final states) and electroweak diboson +production, have been estimated to give negligible contributions to the selected event yields and are +therefore not included. +The effect of multiple interactions in the same and neighboring bunch crossings (pileup) was modeled +by overlaying each simulated hard-scattering event with inelastic pp events generated with Pythia 8.186 +using the NNPDF2.3lo PDF set and the A3 set of tuned parameters [76]. Additionally, for all HH signal +samples, heavy-flavor decays were modeled using EvtGen 1.7.0 [77]. +4 Object reconstruction +Primary vertices from pp interactions are reconstructed [78] using at least two charged-particle tracks +with transverse momentum (𝑝T) above 500 MeV measured with the ID. The vertex with the largest sum of +squared track momenta (� 𝑝2 +T) is taken as the hard-scatter primary vertex. +Hadronic jets are reconstructed using the anti-𝑘𝑡 algorithm with radius parameter 𝑅 = 0.4. The jet +clustering uses particle-flow objects as inputs [79]. Particle-flow objects are charged-particle tracks +matched to the hard-scatter vertex and calorimeter energy clusters after applying an energy subtraction +algorithm that removes the calorimeter deposits associated with good-quality tracks from any vertex. The +tracking information helps to improve the energy resolution of the calorimeter clusters and reduce the +impact from pileup. The momenta of reconstructed jets are calibrated in a multistep procedure [80]. Jets +with 𝑝T < 60 GeV and |𝜂| < 2.4 must also satisfy a requirement based on the output of the multivariate “jet +vertex tagger” (JVT) algorithm [81], which is used to identify and reject jets in which much of the energy +originates from pileup interactions. Correction factors are applied to the simulated events to compensate +for differences between the JVT efficiencies in data and simulation. In the HH → bbbb analysis, jets are +discarded if they fail the “Tight” JVT working point, corresponding to an average efficiency of 96% for jets +from the hard-scatter vertex. +Jets with radius parameter 𝑅 = 0.4 are also reconstructed from topological clusters of energy deposits in +the calorimeter [82] and calibrated in the same way as the jets reconstructed from particle-flow objects. +These jets are used exclusively for the purpose of applying quality criteria to identify events which are +consistent with noise in the calorimeter or noncollision background [83]. Events containing at least one +such jet with 𝑝T > 20 GeV, satisfying the JVT requirement, but not these quality criteria, are rejected. +The identification of jets originating from 𝑏-quarks is performed by the DL1r algorithm [84], which is +applied to all jets with |𝜂| < 2.5. DL1r is based on a multivariate classification technique combining +8 + +information from the impact parameters of ID tracks, the presence of displaced secondary vertices, and +the reconstructed flight paths of 𝑏- and 𝑐-hadrons inside the jet. The DL1r working point used in the +HH → bbbb analysis is the one that gives 77% efficiency for jets associated with true 𝑏-hadrons in +simulated tt events. At this working point, the light-jet (charm-jet) rejection measured in tt simulation is +about a factor of 130 (4.9). The calibration of the DL1r algorithm is performed separately for each jet +type [85, 86] and correction factors are derived and applied to the simulated samples to compensate for +differences between the 𝑏-tagging efficiencies in data and simulation. +Muons are reconstructed by matching ID tracks with either MS tracks or aligned individual hits in the MS +and performing a combined track fit [87]. They are required to have 𝑝T > 4 GeV and |𝜂| < 2.5, and to +satisfy “Medium” identification criteria based on track-quality variables. Muons are used only to apply +energy corrections to jets. +A momentum correction is applied to 𝑏-tagged jets to account for energy lost to soft out-of-cone radiation +and to muons and neutrinos in semileptonic 𝑏-hadron decays. This correction follows the procedure used +in Ref. [88] and consists of two steps. First, a search is performed for muons located near the jet which fall +within a cone of variable size Δ𝑅(𝜇, jet) < min �0.4, 0.04 + 10/𝑝𝜇 +T GeV� around the jet axis. If a muon is +found, its four-momentum is added to that of the jet, and the energy deposited in the calorimeter by the +muon is subtracted from the jet to avoid double counting; this is computed according to the description in +Ref. [89]. In the second step, a global scale factor is applied to each 𝑏-tagged jet according to its 𝑝T and +whether or not it has a muon associated with it. These scale factors are derived from simulation. +5 Analysis selection and categorization +The analysis utilizes a set of criteria to select HH → bbbb candidate events, including dedicated +requirements to separate events into orthogonal ggF and VBF signal regions. “Forward” and “central” jets +are used with the following selection criteria: +• central jets: |𝜂| < 2.5 and 𝑝T > 40 GeV; +• forward jets: 2.5 < |𝜂| < 4.5 and 𝑝T > 30 GeV. +An initial “preselection” is applied to all events, which requires at least four central jets with 𝑝T > 40 GeV, +at least two of which are 𝑏-tagged. As described in Section 3, the events considered in this analysis are +selected online through the 2𝑏2j or 2𝑏1j trigger signatures. In order to simplify the modeling of trigger +efficiencies, a further selection is applied using offline kinematic quantities. Events are selected if they +have a leading4 jet with 𝑝T > 170 GeV, a third leading jet with 𝑝T > 70 GeV, and pass the 2𝑏1j trigger, or +if they fail either of the two jet-𝑝T requirements and pass the 2𝑏2j trigger. This selection step leads to about +a 10% loss of signal efficiency, but enables the reliable calculation of simulation-to-data correction factors +for estimating the trigger efficiency in the remaining HH → bbbb signal events, depending on which of +the above two trigger classes they belong to. +Events passing the above preselection are required to contain at least four central jets passing the b-tagging +requirement outlined in Section 4. The four highest-𝑝T b-tagged jets are chosen to reconstruct the decays of +the two Higgs bosons. In about 75% of simulated signal events reaching this selection stage, these four jets +can be matched one-to-one (within Δ𝑅 < 0.3) to the four b-quarks from the decays of the Higgs bosons. +4 In this document, terms like “leading”, “subleading” etc for physics objects refer to the ordering of these objects in decreasing +𝑝T. +9 + +In signal events where this matching fails, one of the b-quarks from the Higgs boson decays typically +produces a jet that is outside the analysis acceptance. +From the four selected b-tagged jets, there are three possible combinatorial pairings to form the two Higgs +boson candidates. Of those three configurations, the analysis selects the one in which the higher-𝑝T jet +pair has the smallest Δ𝑅 separation. In the simulated samples with SM coupling values, for which the +analysis was mainly optimized, this method gives the correct pairing in around 90% of those signal events +in which the four b-tagged jets are correctly matched to the b-quarks from the decays of the Higgs bosons. +While the pairing accuracy drops for values of the coupling modifiers 𝜅𝜆 and 𝜅2𝑉 that result in softer 𝑝T +spectra for the produced Higgs bosons, this pairing method leads to a smoothly varying distribution of +the expected background in the plane of the invariant masses of the two Higgs boson candidates, which +facilitates the data-driven background estimation described in Section 6. +Events are then subjected to additional selections designed to separate out those consistent with the VBF +production mode. For this, events must contain at least two additional jets, central or forward; b-tagged jets +are excluded. The two jets forming the pair with the largest invariant mass (𝑚jj) are chosen as the “VBF +jets”. The VBF jet pair is required to satisfy 𝑚jj > 1 TeV, and the pseudorapidity separation between the +two jets, |Δ𝜂jj|, must satisfy |Δ𝜂jj| > 3. Lastly, the transverse component of the momentum vector sum of +the two VBF jets and the four jets forming the Higgs boson candidates is required to be less than 65 GeV. +Events satisfying the above criteria enter the VBF signal region, while those failing to satisfy any of these +criteria are considered further in the ggF signal region. +Events satisfying either the ggF or VBF selections are required to satisfy additional selection criteria +designed to reduce the background and improve the analysis sensitivity. In order to suppress the tt +background, a top-veto discriminant 𝑋Wt is defined as: +𝑋Wt = min +������� +� +� +�� +𝑚jj − 𝑚W +0.1𝑚jj +�2 ++ +� +𝑚jj𝑏 − 𝑚t +0.1𝑚jj𝑏 +�2 ������� +, +where 𝑚𝑊 = 80.4 GeV and 𝑚𝑡 = 172.5 GeV are the nominal W boson and top quark masses, and 𝑚jj and +𝑚jj𝑏 are the invariant masses of W boson and top quark candidates formed from jet combinations in each +event. The “minimum” refers to the minimum value from all possible jet combinations (of one b-tagged jet +and two additional untagged jets) that would give a W boson candidate and a corresponding top candidate. +The factor of 0.1 in the denominators is chosen to approximate the experimental dijet mass resolution. The +W boson candidates are formed from any pair of central jets in the event and the top quark candidates are +then reconstructed by pairing the W boson candidates with any remaining b-tagged Higgs boson candidate +jets. The 𝑋Wt discriminant is designed to quantify the likelihood that an event contains a hadronic top +quark decay. Events with 𝑋Wt < 1.5 are rejected. This reduces the tt background by a factor of about 2 in +simulated events, for a small loss of signal efficiency, of around 15%, and a similar reduction in the non-tt, +multijet background. +In order to further reduce the overall background contamination, events in the ggF signal region are also +required to have reconstructed Higgs bosons that satisfy a pseudorapidity separation |Δ𝜂HH| < 1.5 . No +such requirement is imposed in the VBF signal region, since SM VBF HH signal events tend to have a +larger |Δ𝜂HH|. +10 + +A final analysis selection criterion to test the compatibility of events with the HH decay is applied in both +the ggF and VBF selections. A discriminant 𝑋HH is defined as: +𝑋HH = +� +� +�� +𝑚H1 − 124 GeV +0.1 𝑚H1 +�2 ++ +� +𝑚H2 − 117 GeV +0.1 𝑚H2 +�2 +, +where 𝑚𝐻1 and 𝑚𝐻2 are the masses of the leading and subleading reconstructed Higgs boson candidates +respectively. The values of 124 GeV and 117 GeV in the 𝑋HH definition are chosen in accord with the +centers of the 𝑚𝐻1 and 𝑚𝐻2 distributions for correctly paired signal events from simulation. Events are +required to have 𝑋HH < 1.6 to be included in the Signal Region (SR) of the analysis. +Both the ggF and VBF signal regions are subdivided into a number of orthogonal categories in order to +better isolate the HH signal and improve the analysis sensitivity. The 𝑋HH and |Δ𝜂HH| quantities are used to +define six orthogonal ggF categories. The categories are defined by two intervals in 𝑋HH, with boundaries +at 0, 0.95, and 1.6, and three in |Δ𝜂HH|, with boundaries at 0, 0.5, 1, and 1.5 . In the VBF signal region, +two categories are defined using the |Δ𝜂HH| quantity, with the dividing boundary at |Δ𝜂HH| = 1.5 . The +|Δ𝜂HH| < 1.5 category is more sensitive to VBF signals with non-SM couplings, while the |Δ𝜂HH| > 1.5 +category is more sensitive to SM VBF production. +The reconstructed invariant mass of the Higgs boson candidate pair, 𝑚HH, is used as the discriminating +variable for all analysis regions and categories when extracting results, as detailed in Section 8. The +𝑚HH distribution is found to have significant separation power between background and signal, for all the +different values of coupling modifiers. The binning of the 𝑚HH distributions may vary between categories +and is chosen in order to both maintain discrimination power and limit the expected statistical uncertainty in +each bin to less than approximately 30%. This 30% limit ensures that the assumptions used in the statistical +procedure, outlined in Section 8, are satisfied. In the VBF signal region, only events with 𝑚HH > 400 GeV +are considered, as the background in the lower 𝑚HH region was found to be inadequately modeled by the +data-driven method described in Section 6 when applied to the 3b1f control data sample (also described in +Section 6). For the ggF signal region, no requirements on 𝑚HH are applied. +All the selection steps of the analysis are summarized in Figure 3. The yields in the data and the simulated +signal samples for some typical coupling values are shown in Table 3. This sample of data events is referred +to as 4𝑏 events hereafter. +6 Background modeling +After the selection described above, about 90% of the background events come from multijet processes +(excluding top quark production), with the approximately 10% remainder almost entirely composed of tt +events. This background composition was determined by applying the full event selection to simulated +samples of the various processes and comparing the yields with the total background estimate in the SR; it +is purely meant to be indicative and is not used for deriving any results. The background is modeled using +the fully data-driven technique described below. +The background estimation makes use of an alternative set of events, which pass the same b-jet triggers +and satisfy all the same selection criteria as the 4b events, with one difference: they are required to contain +exactly two b-tagged jets. This sample, referred to hereafter as “2b”, has about two orders of magnitude +more events than the 4b sample, hence the presence of any HH → bbbb signal in it is negligible, making it +11 + +(1) +Pass trigger class +(2) +≥ 4 central jets +(4) +≥ 6 central or +forward jets +(5) +VBF Jets +|Δηjj| > 3, +mjj > 1 TeV +(6) +(∑pj)T < 65 GeV +Yes +Yes +Yes +Yes +(7. ggF) +|ΔηHH| < 1.5 +No +No +No +(7. VBF) +XWt > 1.5 +Yes +(8. ggF) +XWt > 1.5 +(9. ggF) +XHH < 1.6 +(8. VBF) +XHH < 1.6 +(3) +≥ 4 b-tagged +central jets +Yes +Yes +Yes +Yes +ggF Selection +VBF Selection +VBF SR +ggF SR +Yes +(9. VBF) +mHH > 400 GeV +Yes +Yes +Figure 3: A flowchart summarizing the nine selection criteria used for the VBF and ggF analysis selections. Events +must satisfy selection criteria 1–3 in order to be considered for either analysis signal region. Events failing to satisfy +any of the selection criteria 4–6 are considered for inclusion in the ggF signal region, while those satisfying selection +criteria 4–6 are considered for the VBF signal region. +Table 3: The yields of data and various example ggF and VBF HH signal models at each step of the analysis selection. +The “Preselection” entry denotes an initial selection requiring at least four jets with 𝑝T > 40 GeV, at least two of +which are 𝑏-tagged. Events which satisfy the “VBF selection” requirements are considered as part of the VBF +signal region of the analysis, while the rest are considered for the ggF signal region. The signal yields are taken +from simulation and are normalized by their theoretical cross-sections and the integrated luminosity of 126 fb−1. +Corrections for differences in the 𝑏-tagging efficiency and trigger acceptance between data and simulation are applied +starting from the “Trigger class” requirement. +Data +ggF Signal +VBF Signal +SM +𝜿𝝀 = 10 +SM +𝜿2𝑽 = 0 +Common preselection +Preselection +5.70 × 108 +530 +7300 +22 +630 +Trigger class +2.49 × 108 +380 +5300 +16 +410 +ggF selection +Fail VBF selection +2.46 × 108 +380 +5200 +14 +330 +At least 4 b-tagged central jets +1.89 × 106 +86 +1000 +1.9 +65 +|Δ𝜂HH| < 1.5 +1.03 × 106 +72 +850 +0.94 +46 +𝑋Wt > 1.5 +7.51 × 105 +60 +570 +0.74 +43 +𝑋HH < 1.6 (ggF signal region) +1.62 × 104 +29 +180 +0.24 +23 +VBF selection +Pass VBF selection +3.30 × 106 +5.2 +81 +2.2 +71 +At least 4 b-tagged central jets +2.71 × 104 +1.1 +15 +0.74 +28 +𝑋Wt > 1.5 +2.18 × 104 +1.0 +11 +0.67 +26 +𝑋HH < 1.6 +5.02 × 102 +0.48 +3.1 +0.33 +17 +𝑚HH > 400 GeV (VBF signal region) +3.57 × 102 +0.43 +1.8 +0.30 +16 +suitable for the background estimation. The jets selected to form the two Higgs boson candidates in the 2b +events are the two b-tagged jets and the two untagged central jets with the highest 𝑝T (excluding the VBF +12 + +jets in the VBF categories). +The kinematic properties of the 2𝑏 and 4𝑏 events are not expected to be identical, partly due to different +processes contributing to the two samples, but also due to differences in the trigger acceptance and because +the probability of tagging a b-jet varies as a function of jet 𝑝T and 𝜂. Therefore, a reweighting function is +required, which, when applied to the 2𝑏 events, maps their kinematic distributions onto the corresponding +4𝑏 distributions. This function is derived using the 2𝑏 and 4𝑏 events in a Control Region (CR) surrounding +the SR in the reconstructed (𝑚𝐻1, 𝑚𝐻2) plane and then applied to the 2𝑏 events in the SR to produce the +background estimate. The “inner edge” of the CR is defined by 𝑋HH = 1.6 and the “outer edge” by the +circle: +𝑅CR = +√︂� +𝑚H1 − 1.05 · 124 GeV +�2 ++ +� +𝑚H2 − 1.05 · 117 GeV +�2 += 45 GeV . +The shift of the center of the above circle by a factor of 1.05, relative to 𝑋HH = 0, is found to be the optimal +trade-off between having a good number of events outside of the SR and avoiding the low 𝑚𝐻1/𝑚𝐻2 +regions, where the differences between 2𝑏 and 4𝑏 kinematic distributions are larger. The CR is split into +four roughly equal directional quadrants, defined by 45◦ and 135◦ lines passing through the SR center, (124, +117) GeV. The four quadrants are given labels based on compass directions: the upper quadrant QN, the +lower QS, the left QW, and the right QE. The above lines also define four quadrants, with the same names as +above, in the SR. Events in CR QN and QS, hereafter referred to as CR1, are used to derive the reweighting +function for the nominal background estimate, while an alternative reweighting function, derived from the +CR events in QE and QW (referred to hereafter as CR2) is used to define a systematic uncertainty related to +the reweighting function interpolation into the SR, as detailed in Section 7. The boundaries of the SR, CR1, +and CR2 in the reconstructed (𝑚𝐻1, 𝑚𝐻2) plane are shown in Figure 4. The horizontal and vertical bands +of lower event density around 80 GeV visible in these plots are caused by the 𝑋Wt selection criterion. +60 +80 +100 +120 +140 +160 +180 +200 +mH1 [GeV] +60 +80 +100 +120 +140 +160 +180 +200 +mH2 [GeV] +ATLAS +ps = 13 TeV, 126 fb 1 +ggF selection, XWt > 1.5 +4b data +SR +CR1 +CR2 +100 +200 +300 +400 +500 +Entries / (3 GeV)2 +(a) +60 +80 +100 +120 +140 +160 +180 +200 +mH1 [GeV] +60 +80 +100 +120 +140 +160 +180 +200 +mH2 [GeV] +ATLAS +ps = 13 TeV, 126 fb 1 +ggF selection, XWt > 1.5 +4b data +SR +CR1 +CR2 +100 +200 +300 +400 +500 +Entries / (3 GeV)2 +(b) +Figure 4: The mass planes of the reconstructed Higgs boson candidates for the (a) ggF and (b) VBF signal regions of +the analysis, shown for the 4b data events. In (a), the analysis selection up to step 8 (as outlined in Figure 3) of the +ggF selection has been applied, while in (b), the analysis selection up to step 7 of the VBF selection has been applied. +The continuous red line describes the Signal Region (SR), the dashed line describes Control Region 1 (CR1) and the +dotted line describes Control Region 2 (CR2). +The reweighting function has the form: +13 + +𝑤(�𝑥) = 𝑝4𝑏(�𝑥) +𝑝2𝑏(�𝑥) , +(1) +where 𝑝4𝑏(�𝑥) and 𝑝2𝑏(�𝑥) are the probability density functions for 4𝑏 and 2𝑏 data, respectively, over a set of +kinematic variables �𝑥. The computation of 𝑤(�𝑥) is a density ratio estimation problem, for which a variety +of approaches exist. The method employed in this analysis is modified from Refs. [90, 91] and makes use of +an artificial neural network (NN). This NN is trained on 2𝑏 and 4𝑏 CR1 data (or CR2 data, for determining +systematic uncertainties, as described Section 7). The training minimizes the following loss function: +L(𝑤(�𝑥)) = +∫ +𝑑�𝑥 +�√︁ +𝑤(�𝑥)𝑝2𝑏(�𝑥) + +1 +√︁ +𝑤(�𝑥) +𝑝4𝑏(�𝑥) +� +, +The function in Eq. (1) minimizes this loss by equalizing the contributions from the two terms. The +kinematic variables used to make up �𝑥 are listed in Table 4 for the ggF and VBF signal regions; they are +among those kinematic variables that exhibit larger differences between the 2𝑏 and 4𝑏 events. The NN +used in the ggF signal region has three densely connected hidden layers of 50 nodes, each with a rectified +linear unit activation function [92], and a single-node linear output. A similar architecture is chosen for the +NN used in the VBF signal region, except that only 20 nodes are used in each of the three hidden layers. +This reflects the fact that the 2𝑏 and 4𝑏 sample sizes in the VBF signal region are nearly two orders of +magnitude smaller than the corresponding ones in the ggF signal region. This is also the reason behind the +choice to perform the NN training in the VBF signal region for all data-taking years together, with the year +index as a one-hot encoded input feature.5 For the ggF signal region, a dedicated reweighting is derived for +each year separately, which, thanks to the adequate sample sizes, deals better with the different levels of +disparity between 2b and 4b distributions, due to the differences in the trigger conditions from year to year. +Finally, in order to ensure that there are adequate numbers of 4b events for both the ggF and VBF NN +trainings, these trainings are performed inclusively, before separating the events into the |Δ𝜂HH| categories +(the CR events have 𝑋HH > 1.6, hence it would not be possible to separate them into the 𝑋HH categories +defined for the SR events). Both |Δ𝜂HH| and 𝑋HH are found to be insensitive to the kinematic reweighting, +and so the inclusive training is not expected to introduce any additional bias when separating the events +into the various categories. +In order to estimate and mitigate the impact of the varying initial conditions and limited size of the +training samples on the NN training, the deep ensembles technique [93] is used together with a bootstrap +resampling [94] of the training data. This entails constructing a set of training datasets by sampling with +replacement from the original dataset. In this analysis, this is approximated by the usage of different +random training weights, following a Poisson distribution with 𝜇 = 1, for each event in each training. The +NN is trained independently on each element of this set, using different initial conditions each time. This +results in an ensemble of reweighting functions. Each reweighting function is further multiplied by a +normalization factor, such that the number of reweighted 2𝑏 events is equal to the number of 4𝑏 events in +the region where the NN is trained. In this analysis, the ensembles comprise 100 reweighting functions +each, hence 100 weights are calculated for each 2𝑏 event in the SR. The background estimate uses the +mean of these weights for each event, and the variation of the background predictions from the ensemble +5 One-hot encoding is a standard technique in machine learning. For example, for the data-taking years in the VBF reweighting, +instead of presenting the year numbers as input features to the NN, one-hot encoding uses three input features: (1, 0, 0) for +2016, (0, 1, 0) for 2017, and (0, 0, 1) for 2018. +14 + +Table 4: The set of input variables used for the 2b to 4b reweighting in the ggF and VBF channels respectively. +ggF +VBF +1. log(𝑝T) of the 2nd leading Higgs boson +candidate jet +2. log(𝑝T) of the 4th leading Higgs boson +candidate jet +3. log(Δ𝑅) between the closest two Higgs +boson candidate jets +4. log(Δ𝑅) between the other two Higgs +boson candidate jets +5. Average absolute 𝜂 value of the Higgs +boson candidate jets +6. log(𝑝T) of the di-Higgs system +7. Δ𝑅 between the two Higgs boson candi- +dates +8. Δ𝜙 between jets in the leading Higgs +boson candidate +9. Δ𝜙 between jets in the subleading Higgs +boson candidate +10. log(𝑋Wt) +11. Number of jets in the event +12. Trigger class index as one-hot encoder +1. Maximum dijet mass from the possible +pairings of the four Higgs boson candi- +date jets +2. Minimum dijet mass from the possible +pairings of the four Higgs boson candi- +date jets +3. Energy of the leading Higgs boson can- +didate +4. Energy of the subleading Higgs boson +candidate +5. Second-smallest Δ𝑅 between the jets +in the leading Higgs boson candidate +(from the three possible pairings for the +leading Higgs candidate) +6. Average absolute 𝜂 value of the four +Higgs boson candidate jets +7. log(𝑋Wt) +8. Trigger class index as one-hot encoder +9. Year index as one-hot encoder (for years +inclusive training) +of reweighting functions is used to estimate a systematic uncertainty for the stability of the NN training +procedure, as described in Section 7. +The effect of the above reweighting procedure in CR1, where the reweighting function is derived, is +illustrated in Figure 5 for the 𝑚HH distribution of the ggF-selected events and in Figure 6 for the 𝑋Wt +distribution of the VBF-selected events. The reweighted “2b” distributions agree with the corresponding +“4b” distributions to within about 10% for most of the phase space, with some larger deviations observed in +bins near the tails of the distributions where fewer data events are available. A large number of additional +kinematic variables were also studied before and after applying the reweighting in order to validate the +performance of the NN. For all variables, the level of agreement, as quantified by the 𝜒2 metric, either +improves after the reweighting or, for variables where the “2b” and “4b” distributions are already similar, +changes only slightly. +The background modeling procedure was tested and found to produce good results in a large simulated tt +sample and a much smaller sample of simulated (non-tt) multijet events in the SR. The procedure was also +tested in several control data samples orthogonal to the nominal event selection, where the presence of any +HH signal is negligible and the 4𝑏 events in the corresponding SR can be compared with the reweighted +SR 2𝑏 events without any bias. These samples, summarized in Table 5, include: (a) events satisfying +all the 2𝑏/4𝑏 ggF selection criteria, with the difference that the |Δ𝜂HH| < 1.5 cut is inverted; (b) events +satisfying all the 2𝑏/4𝑏 selection criteria, except that the center of the SR (and hence also of CR1 and CR2) +is shifted, to avoid any overlap with the nominal SR; and (c) events that satisfy all the same 4𝑏 selection +criteria, except that, in terms of b-tagging, they contain exactly three b-tagged jets, and the fourth jet is +taken as the highest-𝑝T jet that fails a looser working point of the b-tagging algorithm (one that gives 85% +15 + +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +Events / 25 GeV +ATLAS +ps = 13 TeV, 2018 57.7 fb 1 +ggF CR1 +Normalized 2b Data +Stat. Error +4b Data +300 +400 +500 +600 +700 +800 +900 +1000 +mHH [GeV] +0.5 +1.0 +1.5 +4b / 2b +(a) +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +Events / 25 GeV +ATLAS +ps = 13 TeV, 2018 57.7 fb 1 +ggF CR1 +Normalized 2b Data +Stat. Error +4b Data +300 +400 +500 +600 +700 +800 +900 +1000 +mHH [GeV] +0.5 +1.0 +1.5 +4b / 2b +(b) +Figure 5: Comparison of the 2b (yellow histogram with hatching) and 4b (black points with error bars) 𝑚HH +distributions, for events in Control Region 1 (CR1) of the ggF signal region from the 2018 data: (a) before the +kinematic reweighting of the 2b events, with only a normalization factor applied; and (b) after the kinematic +reweighting of the 2b events. The error bars indicate the statistical uncertainty of the 4b data, while the hatching +indicates the statistical uncertainty of the 2b data. The latter is only the Poisson uncertainty of the 2b data, in (a), +while in (b), it also includes the uncertainty from the bootstrap procedure described in Section 7. The hatching in (a) +is narrower than the line width of the plotted histogram. +efficiency for b-jets in simulated tt events). This sample, hereafter referred to as 3b1f, has about one order +of magnitude more events than the 4𝑏 sample and a negligible amount of HH signal; hence it is used to +derive a nonclosure systematic uncertainty for the reweighting procedure, as discussed in Section 7. No +significant background modeling nonclosure was observed in the other control data samples. +7 Systematic uncertainties +The uncertainties with the greatest impact on the analysis sensitivity are those arising from the data-driven +background estimate described in Section 6. These uncertainties have two main sources: the limited sample +sizes in the CR and SR, and physical differences between the CR, where the 2b reweighting function is +derived, and the SR, where it is applied. +As described in Section 6, the ensemble of 100 reweighting functions results in 100 separate background +predictions. An 𝑚HH histogram can be constructed from each of these predictions, and the standard +deviation of the predictions in each bin is taken as the bootstrap uncertainty. The uncertainty is treated as +uncorrelated across 𝑚HH bins. +An additional statistical uncertainty results from the limited sample size of the 2b SR dataset in which the +trained background reweighting network is applied to obtain the final background estimate. A Poisson +uncertainty is taken for each 𝑚HH bin, which is combined in quadrature with the bootstrap uncertainty +described above. +For the background estimate, the uncertainty component related to the kinematic differences between +the SR and CR1 is evaluated by using alternative predictions from the CR2 region. Four alternative +16 + +0 +25 +50 +75 +100 +125 +150 +175 +200 +Events / 0.5 +ATLAS +ps = 13 TeV, 126 fb 1 +VBF CR1 +Normalized 2b Data +Stat. Error +4b Data +2 +3 +4 +5 +6 +7 +8 +9 +10 +XWt +0.5 +1.0 +1.5 +4b / 2b +(a) +0 +25 +50 +75 +100 +125 +150 +175 +200 +Events / 0.5 +ATLAS +ps = 13 TeV, 126 fb 1 +VBF CR1 +Normalized 2b Data +Stat. Error +4b Data +2 +3 +4 +5 +6 +7 +8 +9 +10 +XWt +0.5 +1.0 +1.5 +4b / 2b +(b) +Figure 6: Comparison of the 2b (yellow histogram with hatching) and 4b (black points with error bars) 𝑋Wt +distributions, for events in Control Region 1 (CR1) of the VBF signal region: (a) before the kinematic reweighting of +the 2b events, with only a normalization factor applied; and (b) after the kinematic reweighting of the 2b events. The +error bars indicate the statistical uncertainty on the 4b data, while the hatching indicates the statistical uncertainty +on the 2b data. The latter is only the Poisson uncertainty on the 2b data, in (a), while in (b), it also includes the +uncertainty from the bootstrap procedure described in Section 7. +background estimates are produced by applying the CR1-derived weights to three of the SR quadrants, +and CR2-derived weights to the one remaining SR quadrant. For example, one alternative background +estimate is obtained by applying CR1-derived weights to QS, QE and QW, and CR2-derived weights to +QN. Each of these four background predictions is symmetrized around the nominal 𝑚HH distribution +to construct a two-sided uncertainty. Since the 𝑚HH distribution differs across the four SR quadrants, +substituting the CR2-based prediction for the CR1-based prediction in each of the four SR quadrants +separately and utilizing a four-component uncertainty gives the fit model greater flexibility to describe +these 𝑚HH variations with finer granularity. In the ggF signal region, these uncertainties are taken to be +uncorrelated across the datasets from the three different years. In both the ggF and VBF signal regions, the +uncertainty is treated as correlated across the analysis categories. +An additional closure uncertainty is estimated by applying the full background modeling procedure to +the 3b1f sample instead of the 4b sample. The predicted 3b1f 𝑚HH distribution in the various analysis +categories is then compared with the observed 3b1f data in the SR. For the VBF signal region, no statistically +significant difference between the prediction and observation is found, and hence no additional uncertainty +is applied. For the ggF signal region, an additional uncertainty is evaluated in each category from the +observed differences between the predicted and observed 3b1f 𝑚HH distributions. For 𝑚HH bins in which +the predicted and observed values differ by less than 1𝜎, where 𝜎 is obtained from all other background +modeling uncertainties combined, no additional uncertainty is applied. For 𝑚HH bins where the predicted +and observed values differ by more than 1𝜎, the amount beyond 1𝜎 is averaged with the corresponding +amounts in the two adjacent bins, to limit the impact of statistical fluctuations, and is symmetrized around +the nominal prediction to construct a two-sided uncertainty. This nonclosure uncertainty has a much +smaller impact on the analysis sensitivity than the other sources of background modeling uncertainty. +Several detector modeling uncertainties are evaluated and included. These affect only the signal description, +as the background is estimated entirely from data. Uncertainties in the jet energy scale and resolution, +17 + +Table 5: A summary of all the data samples used in the analysis. For each control sample, the variations from the +nominal analysis are noted explicitly. +Data Sample +Definition +Usage +Signal Region +(SR) +Events with 𝑋HH < 1.6 +Defines signal region in the 𝑚H1–𝑚H2 +plane +Control Region +(CR) +Events with 𝑋HH > 1.6 and 𝑅CR < 45 GeV +Defines control region in the 𝑚H1–𝑚H2 +plane for background estimation (ggF +and VBF) +Shifted valida- +tion regions +Shift the center of the SR in the 𝑚H1–𝑚H2 +plane to avoid overlap with the nominal SR +Background estimation validation (ggF +only) +4b +Events with at least 4 b-tagged central jets +Final analysis sample (ggF and VBF) +2b +Events with exactly 2 b-tagged central jets +plus at least two additional untagged central +jets +Background estimation (ggF and VBF) +3b1f +Events with exactly 3 b-tagged central jets +plus at least one central jet failing a looser +b-tagging requirement +Background estimation validation (ggF +and VBF), additional background mod- +eling uncertainty (ggF only) +Reverse |Δ𝜂HH| +2b and 4b events with |Δ𝜂HH| > 1.5 +Background estimation validation (ggF +only) +as well as the JVT, are treated according to the prescription in Refs. [80, 81]. Additional uncertainties +arising from the correction of the simulated pileup distribution are treated according to the prescription in +Ref. [95]. Uncertainties in the 𝑏-tagging efficiency are treated according to the prescription in Ref. [96]. +Uncertainties in the trigger efficiencies are evaluated from measurements of per-jet online efficiencies for +both jet reconstruction and 𝑏-tagging, which are used to compute event-level uncertainties. These are then +applied to the simulated events as overall weight variations. The uncertainty in the integrated luminosity +used in this analysis is in the range 2.0%–2.4% for the three years of data taking and 1.7% for the entire +dataset [97], obtained using the LUCID-2 detector for the primary luminosity measurements [98]. +Several sources of theoretical uncertainty affecting the signal models are considered as described below. +Uncertainties due to modeling of the parton shower and underlying event are evaluated by comparing +results between two generators for these parts of the calculation: the nominal Pythia 8 and the alternative +Herwig 7. This is found to have an effect of roughly 10% on the ggF and VBF signal acceptances, and +a negligible impact on the shape of the 𝑚HH distributions. The parton showering uncertainty is derived +within each analysis SR category; the uncertainty is observed to reach approximately 40% for a given +18 + +production mode in some categories in which the acceptance is small for that mode. Uncertainties in the +matrix element calculation are evaluated by raising and lowering the factorization and renormalization +scales used in the generator by a factor of two, both independently and simultaneously. This results in +an effect of typically 2% for both ggF and VBF, with a maximum effect of about 6% in certain analysis +categories. PDF uncertainties are evaluated using the PDF4LHC_NLO_MC set [54] by calculating the +signal acceptance for each replica and taking the standard deviation. The magnitude of this uncertainty +is typically found to be less than 1% in both the ggF and VBF signal acceptances, with a maximum +magnitude of approximately 2%. Theoretical uncertainties in the H → bb branching ratio [14] are included, +amounting to an approximately 3.5% overall uncertainty in the signal normalization. The dependence +of the branching ratio uncertainty on 𝜅𝜆 is neglected. Theoretical uncertainties in the ggF and VBF HH +cross-sections arising from uncertainties in the PDF and 𝛼s, as well as the choice of renormalization scheme +and the scale of the top quark’s mass, are taken from Refs. [11, 14, 99]. The cross-section uncertainties are +included in the derivation of the upper limits on the ggF, VBF, and combined HH signal strengths, as well +as the likelihood-based constraints on the values of the 𝜅𝜆 and 𝜅2𝑉 modifiers, as presented in Section 8. +An additional signal modeling systematic uncertainty is evaluated for the SMEFT and HEFT measurements. +The 𝑚HH spectra of reweighted SMEFT/HEFT signal samples are compared against explicitly generated +samples for a select number of coefficient variations. A two-component normalization uncertainty is derived +by taking the average of the relative deviations across the 𝑚HH bins in the ranges of 280 < 𝑚HH < 936 GeV +and 𝑚HH > 936 GeV. The use of separate components in the low- and high-𝑚HH regions prevents the level +of agreement in the more populated low-𝑚HH region from overconstraining the uncertainty in the more +sparsely populated high-𝑚HH region. +8 Results +The analysis results are obtained using a maximum-likelihood fit performed in bins of reconstructed 𝑚HH. +For the ggF signal region, the fit is performed simultaneously across the different data-taking years (2016, +2017, and 2018), while for the VBF signal region, the fit is performed inclusively on the data from all +years. +The likelihood function used to construct the test statistic has a standard form, consisting of a product of +Poisson distributions for the yields in each bin and constraint functions for nuisance parameters describing +systematic uncertainties. For uncertainties due to the limited sample size in data or simulation, the constraint +is a Poisson distribution. For all other systematic uncertainties, the constraint is a Gaussian distribution. +Where systematic uncertainties are deemed to be uncorrelated, independent nuisance parameters are +introduced. Uncertainties in the luminosity and signal modeling are treated as fully correlated between the +analysis categories and, for ggF, the data-taking years. Each component of the quadrant-derived uncertainty +covering the kinematic differences between the SR and CR1 regions is correlated across the data-taking +years for the ggF region. The components are correlated across analysis categories within the ggF and VBF +signal regions, but not between the ggF and VBF signal regions. All other uncertainties in the background +model are treated as uncorrelated across the different categories and data-taking years. The statistical +model is implemented using RooFit [100]. +The hypothesis of the presence of a signal is tested using the profile likelihood ratio [101]. The signal +strength of the combined ggF and VBF signal process, 𝜇ggF+VBF (= 𝜎ggF+VBF/𝜎SM +ggF+VBF), is chosen as the +parameter of interest (POI) and is a free parameter in the fit. The relative contributions of the ggF and VBF +19 + +signals to the total signal model are fixed to their predicted values. The profile likelihood ratio takes the +following form: +−2Δ ln 𝜆(𝜇) = −2 ln +� +𝐿(𝜇, ˆˆ𝜽) +𝐿( ˆ𝜇, ˆ𝜽) +� +, +where 𝜇 is the POI and 𝜽 represents the nuisance parameters. The numerator represents the conditional +maximum-likelihood fit, in which the nuisance parameters are set to their profiled values ˆˆ𝜽 for which the +likelihood is maximized for a fixed value of 𝜇. The denominator represents the unconditional likelihood fit, +where both 𝜇 and 𝜽 are set to the values which jointly maximize the likelihood, ˆ𝜇 and ˆ𝜽, respectively. +The observed distributions in 𝑚HH, as well as the predicted background and example signal shapes, +are presented in Figure 7 for each of the six ggF categories (with all data-taking years combined, for +presentation purposes). The distributions of the expected background are obtained using the best-fit values +of the nuisance parameters in the fit to the data with the background-only hypothesis. The corresponding +𝑚HH distributions in the two VBF categories are shown in Figure 8. The signal shape for 𝜅2𝑉 = 0 in +Figure 8(a) clearly shows the impact of the divergences in Figures 2(b) and 2(c) not canceling out as +discussed in Section 1. While the deviations from the SM studied in this analysis are below the level that +violates unitarity, this behavior makes the VBF topology in this analysis particularly sensitive to 𝜅2𝑉 . The +observed number of data events, predicted number of background events, and expected number of signal +events for the SM ggF and VBF signals are summarized for each of the analysis categories in Table 6. +Table 6: The yields in each analysis category of the data, expected background, and expected SM ggF and VBF +signals. The expected background yields are obtained using a fit to the data with the background-only hypothesis; the +quoted uncertainties are the sum in quadrature of all the per-bin systematic uncertainties. The expected signal yields +are obtained from simulation. +Category +Data +Expected +ggF Signal +VBF Signal +Background +SM +SM +ggF signal region +|Δ𝜂HH| < 0.5, 𝑋HH < 0.95 +1940 +1935 ± 25 +7.0 +0.038 +|Δ𝜂HH| < 0.5, 𝑋HH > 0.95 +3602 +3618 ± 37 +6.5 +0.036 +0.5 < |Δ𝜂HH| < 1.0, 𝑋HH < 0.95 +1924 +1874 ± 21 +5.1 +0.037 +0.5 < |Δ𝜂HH| < 1.0, 𝑋HH > 0.95 +3540 +3492 ± 35 +4.7 +0.040 +|Δ𝜂HH| > 1.0, 𝑋HH < 0.95 +1880 +1739 ± 22 +2.9 +0.043 +|Δ𝜂HH| > 1.0, 𝑋HH > 0.95 +3285 +3212 ± 37 +2.8 +0.041 +VBF signal region +|Δ𝜂HH| < 1.5 +116 +125.3 ± 4.4 +0.37 +0.090 +|Δ𝜂HH| > 1.5 +241 +230.6 ± 5.3 +0.06 +0.21 +An upper limit on the combined ggF and VBF HH signal strength 𝜇ggF+VBF is computed using the +asymptotic formula [101] and based on the CLs method [102]. The observed (expected) 95% CL upper +limit on 𝜇ggF+VBF is found to be 5.4 (8.1). The expected upper limits are obtained using a background-only +hypothesis, excluding a 𝐻𝐻 signal. The upper limit on the combined 𝜇ggF+VBF, as well as upper limits on +the individual 𝜇ggF (= 𝜎ggF/𝜎SM +ggF) and 𝜇VBF (= 𝜎VBF/𝜎SM +VBF), are summarized in Table 7. For the individual +𝜇ggF and 𝜇VBF limits, the results are derived by treating the other production mode (VBF when placing +limits on 𝜇ggF, and vice-versa) as a background process, with its normalization only loosely constrained +in the fit. Compared to the previous ATLAS measurement of ggF HH production in the bbbb decay +20 + +0 +100 +200 +300 +400 +500 +Events / 25 GeV +ATLAS +p +s = 13 TeV, 126 fb 1 +ggF Signal Region +| +HH| < 0.5, XHH < 0.95 +Post-Fit Background +Stat. + Syst. Error +4b Data +400 x SM HH +200 x += 6 HH +300 +400 +500 +600 +700 +800 +900 +1000 +mHH [GeV] +0.5 +1.0 +1.5 +Data/Pred +(a) +0 +100 +200 +300 +400 +Events / 25 GeV +ATLAS +p +s = 13 TeV, 126 fb 1 +ggF Signal Region +0.5 < | +HH| < 1.0, XHH < 0.95 +Post-Fit Background +Stat. + Syst. Error +4b Data +400 x SM HH +200 x += 6 HH +300 +400 +500 +600 +700 +800 +900 +1000 +mHH [GeV] +0.5 +1.0 +1.5 +Data/Pred +(b) +0 +50 +100 +150 +200 +250 +300 +350 +Events / 25 GeV +ATLAS +p +s = 13 TeV, 126 fb 1 +ggF Signal Region +| +HH| > 1.0, XHH < 0.95 +Post-Fit Background +Stat. + Syst. Error +4b Data +400 x SM HH +200 x += 6 HH +300 +400 +500 +600 +700 +800 +900 +1000 +mHH [GeV] +0.5 +1.0 +1.5 +Data/Pred +(c) +0 +200 +400 +600 +800 +Events / 25 GeV +ATLAS +p +s = 13 TeV, 126 fb 1 +ggF Signal Region +| +HH| < 0.5, XHH > 0.95 +Post-Fit Background +Stat. + Syst. Error +4b Data +400 x SM HH +200 x += 6 HH +300 +400 +500 +600 +700 +800 +900 +1000 +mHH [GeV] +0.5 +1.0 +1.5 +Data/Pred +(d) +0 +100 +200 +300 +400 +500 +600 +700 +800 +Events / 25 GeV +ATLAS +p +s = 13 TeV, 126 fb 1 +ggF Signal Region +0.5 < | +HH| < 1.0, XHH > 0.95 +Post-Fit Background +Stat. + Syst. Error +4b Data +400 x SM HH +200 x += 6 HH +300 +400 +500 +600 +700 +800 +900 +1000 +mHH [GeV] +0.5 +1.0 +1.5 +Data/Pred +(e) +0 +100 +200 +300 +400 +500 +600 +700 +Events / 25 GeV +ATLAS +p +s = 13 TeV, 126 fb 1 +ggF Signal Region +| +HH| > 1.0, XHH > 0.95 +Post-Fit Background +Stat. + Syst. Error +4b Data +400 x SM HH +200 x += 6 HH +300 +400 +500 +600 +700 +800 +900 +1000 +mHH [GeV] +0.5 +1.0 +1.5 +Data/Pred +(f) +Figure 7: Distributions of the reconstructed 𝑚HH in data (shown by the black points) and the estimated background +(shown by the yellow histograms), in each of the six |Δ𝜂HH|, 𝑋HH categories in the ggF signal region: (a) |Δ𝜂HH| < 0.5, +𝑋HH < 0.95; (b) 0.5 < |Δ𝜂HH| < 1.0, 𝑋HH < 0.95; (c) |Δ𝜂HH| > 1.0, 𝑋HH < 0.95; (d) |Δ𝜂HH| < 0.5, 𝑋HH > 0.95; +(e) 0.5 < |Δ𝜂HH| < 1.0, 𝑋HH > 0.95; and (f) |Δ𝜂HH| > 1.0, 𝑋HH > 0.95. The contributions from the different +data-taking years are combined in each category for presentation purposes. The hatching shows the total uncertainty +of the background estimate. The distribution of the expected background is obtained using the best-fit values of the +nuisance parameters in the fit to the data with the background-only hypothesis. Distributions of the SM and 𝜅𝜆 = 6 +signal models are overlaid, scaled so as to be visible on the plot, and the scaling for each signal model is the same +across the six categories. The lower panels show the ratio of the observed data yield to the predicted background in +each bin. Events in the underflow and overflow bins are counted in the yields of the initial and final bins respectively. +channel (using 27 fb−1 of early Run 2 data) [26], the upper limit on the ggF cross-section is over 50% lower, +with approximately 20% of this improvement arising from advances in analysis techniques and object +reconstruction. Similarly, compared to the previous ATLAS measurement of VBF HH production in the +bbbb decay channel, which used 126 fb−1 of data collected between 2016 and 2018 [27], the upper limit on +the VBF HH cross-section is over 75% lower, with this improvement arising entirely from advances in +analysis technique and object reconstruction. +The total uncertainty in the upper limit of the cross-section is dominated by the uncertainty sources related +to the background modeling procedure and theoretical predictions. With only the statistical uncertainties +of the reweighted 2b data, observed 4b data, and simulated signal samples included in the fit, the expected +upper limit on 𝜇ggF+VBF is found to be 6.0 times the SM prediction. Including the uncertainty sources +resulting from the background estimation (the bootstrap uncertainty, the uncertainty from the kinematic +differences between the SR and CR1, and, in the ggF signal region, the 3b1f nonclosure uncertainty), +the expected upper limit on 𝜇ggF+VBF is relaxed to 7.1 times the SM prediction. The further reduction +of sensitivity to the value of 8.1, as quoted in Table 7, is driven primarily by the uncertainties arising +from theoretical predictions. The relative impact of the various sources of systematic uncertainty on the +21 + +0 +10 +20 +30 +40 +50 +60 +70 +80 +Events / 50 GeV +ATLAS +p +s = 13 TeV, 126 fb 1 +VBF Signal Region +| +HH| < 1.5 +Post-Fit Background +Stat. + Syst. Error +4b Data +1500 x SM HH +50 x += 6 HH +10 x 2V = 0 HH +400 +500 +600 +700 +800 +900 +mHH [GeV] +0.5 +1.0 +1.5 +Data/Pred +(a) +0 +10 +20 +30 +40 +50 +60 +Events / 50 GeV +ATLAS +p +s = 13 TeV, 126 fb 1 +VBF Signal Region +| +HH| > 1.5 +Post-Fit Background +Stat. + Syst. Error +4b Data +1000 x SM HH +200 x += 6 HH +40 x 2V = 0 HH +400 +600 +800 +1000 +1200 +1400 +mHH [GeV] +0.5 +1.0 +1.5 +Data/Pred +(b) +Figure 8: Distributions of the reconstructed 𝑚HH in data (shown by the black points), the estimated background +(shown by the yellow histograms), in each of the two |Δ𝜂HH| categories in the VBF signal region: (a) |Δ𝜂HH| < 1.5 +and (b) |Δ𝜂HH| > 1.5. The hatching shows the total uncertainty of the background estimate. The distribution of the +expected background is obtained using the best-fit values of the nuisance parameters in the fit to the data with the +background-only hypothesis. Distributions for three choices of couplings are shown: the SM, 𝜅𝜆 = 6, and 𝜅2𝑉 = 0 +(with all other couplings set to their SM values in the last two models), scaled so as to be visible on the plot. The +lower panels show the ratio of the observed data yield to the predicted background in each bin. Events in the overflow +bins are counted in the yields of the final bins. +Table 7: The observed and expected upper limits on the SM ggF HH production cross-section 𝜎ggF, SM VBF +HH production cross-section 𝜎VBF, and combined SM ggF and VBF HH production cross-section 𝜎ggF+VBF at the +95% CL, expressed as multiples of the corresponding SM cross-sections. The expected values are shown with +corresponding one- and two-standard-deviation error bounds, and they are obtained using a background-only fit to +the data. When extracting the limits on 𝜎ggF+VBF, the relative contributions of ggF and VBF production to the total +cross-section are fixed to the SM prediction. +Observed Limit +−2𝝈 +−1𝝈 +Expected Limit ++1𝝈 ++2𝝈 +𝜇ggF +5.5 +4.4 +5.9 +8.2 +12.4 +19.6 +𝜇VBF +130 +70 +100 +130 +190 +280 +𝜇ggF+VBF +5.4 +4.3 +5.8 +8.1 +12.2 +19.1 +expected upper limit on 𝜇ggF+VBF is summarized in Table 8. +Constraints are placed on the 𝜅𝜆 and 𝜅2𝑉 modifiers using two different interpretations, the first named +the “95% CL” method and the second named the “profile likelihood ratio” method. The former uses the +signal strength 𝜇 as the POI, while the latter uses the vector of coupling modifiers 𝜿 = (𝜅𝜆, 𝜅2𝑉 ). The +95% CL method allows for interpretation as a traditional search for an arbitrarily normalized set of signals +with different shapes against an estimated background, while the profile likelihood ratio method allows for +interpretation as to whether the data are compatible with the specific cross-section and shape predictions +of the 𝜅 framework. The 95% CL results presented here offer a consistent comparison with previous +ATLAS HH measurements. The constraints obtained from the two interpretations are not expected to be +identical, as the two strategies employ slightly different physical assumptions. In the profile likelihood +22 + +Table 8: Breakdown of the dominant systematic uncertainties. The impact of the uncertainties on the expected upper +limit on 𝜇ggF+VBF when re-evaluating the profile likelihood ratio after fixing the nuisance parameter(s) in question +to its (their) best-fit value(s), while all remaining nuisance parameters remain free to float. The impact is shown +in %. Only (groups of) systematic uncertainties that have an impact of at least 1% are shown. The impact of each +experimental source of systematic uncertainty described in the text, as well as of all of them together, is less than 1%. +Source of Uncertainty +Δ𝜇/𝜇 +Theory uncertainties +Theory uncertainty in signal cross-section +−9.0% +All other theory uncertainties +−1.4% +Background modeling uncertainties +Bootstrap uncertainty +−7.1% +CR to SR extrapolation uncertainty +−7.5% +3b1f nonclosure uncertainty +−2.0% +ratio interpretation, the signal strength is fixed to the prediction obtained for a specific coupling modifier +configuration, while for the 95% CL interpretation, the signal strength is allowed to float. The profile +likelihood ratio method utilizes a hypothesis consisting of the predicted background plus the SM HH +signal, while the 95% CL results utilize a hypothesis containing only the predicted background and no HH +signal. Given the relatively small size of the SM HH signal compared to the predicted background, the use +of different hypotheses is not expected to have a significant effect. Additionally, 2𝜎-level constraints are +quoted from the profile likelihood ratio interpretation, as opposed to 95% CL constraints. +The 95% CL constraints on 𝜅𝜆 and 𝜅2𝑉 are obtained by determining the 95% CL upper limits on the +cross-section as a function of these coupling modifiers, 𝜇ggF+VBF(𝜅𝜆, 𝜅2𝑉 ). Values of the coupling modifiers +(𝜅𝜆, 𝜅2𝑉 ) are excluded if the predicted cross-section of the signal model obtained with that configuration is +excluded at the 95% CL. The H → bb branching ratio is fixed to the SM prediction in the likelihood fit +and any dependence on 𝜅𝜆 is ignored. Upper limits on the HH signal strength as a function of 𝜅𝜆 and 𝜅2𝑉 +are shown in Figure 9, and the exclusion boundaries are summarized in Table 9. With the values of the +other modifiers (𝜅𝑉 and either 𝜅2𝑉 or 𝜅𝜆, respectively) fixed to their SM value of 1, values of 𝜅𝜆 beyond +[−3.9, 11.1] and values of 𝜅2𝑉 beyond [−0.03, 2.11] are excluded. +Table 9: The observed and expected constraints on the 𝜅𝜆 and 𝜅2𝑉 coupling modifiers at 95% CL. For each modifier, +the constraints were extracted with all other modifiers fixed to the SM prediction. +Parameter +Expected Constraint +Observed Constraint +Lower +Upper +Lower +Upper +𝜅𝜆 +−4.6 +10.8 +−3.9 +11.1 +𝜅2𝑉 +−0.05 +2.12 +−0.03 +2.11 +Figure 10 shows the 95% CL exclusion limits in the two-dimensional plane of the 𝜅𝜆–𝜅2𝑉 modifier space. +The alternative coupling modifier constraints are obtained using the profile likelihood ratio interpretation, +with the coupling modifiers 𝜿 = (𝜅𝜆, 𝜅2𝑉 ) as the POIs, rather than the signal strength 𝜇: +−2Δ ln 𝐿(𝜿) = −2 ln +� +𝐿(𝜿, ˆˆ𝜽) +𝐿( ˆ𝜿, ˆ𝜽) +� +. +23 + +20 +15 +10 +5 +0 +5 +10 +15 +20 + ( 2V=1.0, V=1.0) +101 +102 +103 +104 +105 +106 +95% CL Limit on +ggF + VBF HH [fb] +ATLAS +p +s = 13 TeV, 126 fb 1 +Combined ggF and VBF Regions +Observed: + +[-3.9, 11.1] +Expected: + +[-4.6, 10.8] +Observed Limit +Expected Limit +Expected Limit ±1 +Expected Limit ±2 +Theory Prediction +SM Prediction +(a) +2 +1 +0 +1 +2 +3 +4 +2V ( +=1.0, V=1.0) +100 +101 +102 +103 +104 +105 +95% CL Limit on +VBF HH [fb] +ATLAS +p +s = 13 TeV, 126 fb 1 +Combined ggF and VBF Regions +Observed: 2V +[-0.03, 2.11] +Expected: 2V +[-0.05, 2.12] +Observed Limit +Expected Limit +Expected Limit ±1 +Expected Limit ±2 +Theory Prediction +SM Prediction +(b) +Figure 9: The observed 95% CL exclusion limits as a function of (a) 𝜅𝜆 (obtained using the signal strength 𝜇ggF+VBF +as the POI) and (b) 𝜅2𝑉 (obtained using the signal strength 𝜇VBF as the POI) from the combined ggF and VBF signal +regions, as shown by the solid black line. In each case, the value of the other modifier is fixed to 1. The blue and +yellow bands show respectively the 1𝜎 and 2𝜎 bands around the expected exclusion limits, which are shown by +the dashed black line. The expected exclusion limits are obtained using a fit to the data with the background-only +hypothesis. The dark red line shows in (a) the predicted combined ggF and VBF HH cross-section as a function +of 𝜅𝜆 and in (b) the predicted VBF HH cross-section as a function of 𝜅2𝑉 . The dark pink bands surrounding the +predicted cross-section lines indicate the theoretical uncertainty of the cross-section, as taken from Ref. [99]. The +band in (b) is smaller than the width of the plotted line. +2 +0 +2 +4 +2V +20 +10 +0 +10 +20 +ATLAS +p +s = 13 TeV, 126 fb 1 +V=1.0 +Observed limit (95% CL) +Expected +±1 +±2 +Standard Model +(a) +Figure 10: The observed 95% CL exclusion limit obtained using the signal strength 𝜇ggF+VBF as the POI in the +two-dimensional 𝜅𝜆 vs. 𝜅2𝑉 space, obtained from the combined ggF and VBF signal model, as shown by the solid +black line. The blue and yellow bands show respectively the 1𝜎 and 2𝜎 bands around the expected exclusion limits, +which are shown by the dashed black line. The star denotes the SM prediction (𝜅𝜆 = 𝜅2𝑉 = 1). +A scan of the profile likelihood ratio is performed as a function of the coupling modifiers at discrete points +to produce the curves shown in Figure 11. The best-fit value of 𝜅𝜆 is found to be 6.2 from the profile +likelihood scan. The observed pull of the best-fit 𝜅𝜆 value away from the SM value is due to a slight excess +in the observed data in the ggF signal region, specifically in the low-𝑚HH range. The particular signal +model in which 𝜅𝜆 is close to 6 is favored due to a balance between two competing effects: the 𝑚HH +24 + +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 + ( 2V=1.0, V=1.0) +0 +2 +4 +6 +8 +10 +2 ln(L) +ATLAS +ps = 13 TeV, 126 fb 1 +Combined ggF and VBF Regions +Expected 2 constraints: + + +[-5.4, 11.4] +Observed 2 constraints: + + +[-3.5, 11.3] +Best fit + = 6.2 +Observed +Expected +±2 +(a) +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +2V ( +=1.0, V=1.0) +0 +5 +10 +15 +20 +25 +30 +35 +2 ln(L) +ATLAS +ps = 13 TeV, 126 fb 1 +Combined ggF and VBF Regions +Expected 2 constraints: + 2V +[-0.1, 2.1] +Observed 2 constraints: + 2V +[-0.0, 2.1] +Best fit 2V = 1.0 +Observed +Expected +±2 +(b) +Figure 11: The observed profile likelihood ratio scans for the (a) 𝜅𝜆 and (b) 𝜅2𝑉 coupling modifiers, shown by the +solid black line, using the coupling modifiers 𝜿 as the POIs. In each case, the value of the other parameter is fixed +to 1. The dashed blue line shows the expected profile likelihood ratio, as obtained using a fit to the data with the +background-only hypothesis. The pink line indicates the 2𝜎 exclusion boundary. +2 +1 +0 +1 +2 +3 +4 +2V +20 +15 +10 +5 +0 +5 +10 +15 +20 +25 +ATLAS +p +s = 13 TeV, 126 fb 1 +Combined ggF and VBF Regions +Observed ±2 +Observed ± 1 +SM Prediction +Best Fit +(a) +2 +1 +0 +1 +2 +3 +4 +2V +20 +15 +10 +5 +0 +5 +10 +15 +20 +25 +ATLAS +p +s = 13 TeV, 126 fb 1 +Combined ggF and VBF Regions +Expected ±2 +Expected ± 1 +SM Prediction +(b) +Figure 12: (a) The observed profile likelihood ratio exclusion limits for the two-dimensional 𝜅𝜆 vs. 𝜅2𝑉 modifier +space, shown by the solid dark purple line at the 1𝜎 level and the dashed turquoise line at the 2𝜎 level. The black +cross denotes the best-fit values of (𝜅𝜆, 𝜅2𝑉 ). The expected exclusion limits are presented in (b), where the solid pink +line denotes the 1𝜎-level exclusion and the dashed orange line denotes the 2𝜎-level exclusion. For both the expected +and observed limit plots, the black star indicates the SM prediction (𝜅𝜆 = 𝜅2𝑉 = 1). +spectrum becomes softer as 𝜅𝜆 increases away from the SM, but the cross-section also grows beyond the +magnitude of the excess as 𝜅𝜆 increases much further. This slight excess also results in the deviation of the +observed limits in Figure 11 from the expected limits by about 1𝜎. No such excess is observed in the VBF +signal region, and the best-fit value of 𝜅2𝑉 from the likelihood scan is found to be 1.0. With the values +of the other modifiers (𝜅𝑉 and either 𝜅2𝑉 or 𝜅𝜆, respectively) fixed to their SM value of 1, the observed +(expected) 2𝜎 allowed range for 𝜅𝜆 is found to be [−3.5, 11.3] ([−5.4, 11.4]) and the corresponding range +for 𝜅2𝑉 is [−0.0, 2.1] ([−0.1, 2.1]). +25 + +The exclusion constraints obtained using the profile likelihood ratio method are also presented in the +two-dimensional 𝜅𝜆–𝜅2𝑉 coupling modifier space, similarly to the 95% CL constraints described above. +The excluded regions are presented in Figure 12. With both modifiers able to float in the two-dimensional +fit that combines both the ggF and VBF signal regions, the fit converges to 𝜅𝜆 and 𝜅2𝑉 values slightly +different from the ones where the minimum is found in the fits with a single parameter free. +In addition to constraints on the ggF and VBF HH cross-sections and the 𝜅𝜆 and 𝜅2𝑉 coupling modifiers, +constraints for relevant coefficients can be derived from the ggF selection of the analysis in the SMEFT and +HEFT frameworks, as outlined in Section 1. The VBF HH process was ignored for both the SMEFT and +HEFT results; including the VBF HH process as a background was found to have a negligible effect on +the extracted parameter limits. The slight dependence of the H → bb branching fraction on the SMEFT +and HEFT coefficients is also ignored, as the impact on the analysis sensitivity is small. Constraints +on the SMEFT coefficients are extracted by considering the 95% CL exclusion of the cross-section as +a function of SMEFT parameter, as was done for the 𝜅𝜆 and 𝜅2𝑉 constraints discussed previously. The +extracted constraints on individual parameters in the scenario where the other parameters are fixed to 0 +are summarized in Table 10. Limits approaching or exceeding ±4𝜋 should be interpreted with caution +because of the potential impact from effects such as missing higher-order model contributions. The +exclusion limits are also presented in two-dimensional SMEFT coefficient subspaces. The exclusion limits +for each coefficient versus the 𝑐H coefficient (with the remaining three coefficients fixed to 0) are shown +in Figure 13. The upper limits on the HEFT ggF HH production cross-section in the seven benchmark +models are presented in Figure 14. The spread of sensitivity between the seven benchmark models reflects +the different signal kinematics and, hence, shapes of the signal 𝑚HH distributions. The different variation +between observed and expected limits is linked to the slight excess observed in the low 𝑚HH region, as +discussed earlier. The red crosses in Figure 14 indicate the predicted HH cross-sections from the respective +benchmark models. As can be seen, BM3, BM5 and BM7 are observed to be excluded with more than 95% +confidence. Constraints are placed on the values of 𝑐𝑔𝑔HH and 𝑐𝑡𝑡HH, with all other HEFT coefficients fixed +to SM values. The observed (expected) constraints on 𝑐𝑔𝑔HH are found to be [−0.36, 0.78] ([−0.42, 0.75]), +while the observed (expected) constraints on 𝑐𝑡𝑡HH are found to be [−0.55, 0.51] ([−0.46, 0.40]). +Table 10: The extracted upper and lower limits on the SMEFT parameters to which the analysis is sensitive. For +each parameter, the constraints are provided assuming the other parameters are fixed to 0. The VBF HH process is +ignored for this result. +Parameter +Expected Constraint +Observed Constraint +Lower +Upper +Lower +Upper +𝑐H +−20 +11 +−22 +11 +𝑐H𝐺 +−0.056 +0.049 +−0.067 +0.060 +𝑐H□ +−9.3 +13.9 +−8.9 +14.5 +𝑐𝑡H +−10.0 +6.4 +−10.7 +6.2 +𝑐𝑡𝐺 +−0.97 +0.94 +−1.12 +1.15 +26 + +40 +30 +20 +10 +0 +10 +20 +30 +cH +0.10 +0.05 +0.00 +0.05 +0.10 +0.15 +cHG +ATLAS +p +s = 13 TeV, 126 fb 1 +cH =0.0, ctG=0.0, ctH=0.0 +Observed Limit (95% CL) +Expected Limit (95% CL) +Expected Limit ±1 +Expected Limit ±2 +SM Prediction +(a) +40 +30 +20 +10 +0 +10 +20 +30 +cH +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +ctG +ATLAS +p +s = 13 TeV, 126 fb 1 +cH =0.0, ctH=0.0, cHG=0.0 +Observed Limit (95% CL) +Expected Limit (95% CL) +Expected Limit ±1 +Expected Limit ±2 +SM Prediction +(b) +40 +30 +20 +10 +0 +10 +20 +30 +cH +20 +10 +0 +10 +20 +30 +40 +ctH +ATLAS +p +s = 13 TeV, 126 fb 1 +cH =0.0, ctG=0.0, cHG=0.0 +Observed Limit (95% CL) +Expected Limit (95% CL) +Expected Limit ±1 +Expected Limit ±2 +SM Prediction +(c) +40 +30 +20 +10 +0 +10 +20 +30 +cH +30 +20 +10 +0 +10 +20 +30 +40 +50 +cH +ATLAS +p +s = 13 TeV, 126 fb 1 +ctH=0.0, ctG=0.0, cHG=0.0 +Observed Limit (95% CL) +Expected Limit (95% CL) +Expected Limit ±1 +Expected Limit ±2 +SM Prediction +(d) +Figure 13: The observed 95% CL exclusion limits on the SMEFT coefficients in the two-dimensional spaces (a) +𝑐H𝐺 vs. 𝑐H, (b) 𝑐𝑡𝐺 vs. 𝑐H, (c) 𝑐𝑡H vs. 𝑐H, and (d) 𝑐H□ vs. 𝑐H, shown by the solid black lines. The dashed black +line indicates the expected 95% CL exclusion limits. The shaded blue band indicates the ±1𝜎 uncertainty of the +exclusion limits, while the yellow band indicates the ±2𝜎 uncertainty. The values of the other three coefficients for +each plot are fixed to 0. The VBF HH process is ignored for this result. +27 + +102 +103 +ggF, HH [fb] +SM +BM1 +BM2 +BM3 +BM4 +BM5 +BM6 +BM7 +170 +(230) +290 +(200) +360 +(250) +110 +(74) +130 +(130) +110 +(95) +190 +(180) +88 +(71) +Obs. +(Exp.) +ATLAS +ps = 13 TeV, 126 fb 1 +Observed +Expected +Theory Prediction +±1 +±2 +Figure 14: The observed 95% CL exclusion limits on the ggF HH production cross-section in the SM and each of the +seven HEFT benchmark models, given by the solid black points. The blue and yellow bands show respectively the +1𝜎 and 2𝜎 bands around the expected upper limits, which are shown by the open circles. The predicted ggF HH +production cross-section from each benchmark is indicated by a pink cross. The VBF HH process is ignored for this +result. +28 + +9 Conclusion +A search for nonresonant pair production of Higgs bosons in the bbbb final state was carried out, with +dedicated analyses for the ggF and VBF production modes, using 126 fb−1 of √𝑠 = 13 TeV 𝑝𝑝 collision +data collected by the ATLAS detector at the LHC. The sensitivity of the analyses is improved relative to +previous iterations by using more sophisticated background modeling techniques, event categorization and +improved jet reconstruction and flavor identification algorithms, in addition to the increased integrated +luminosity of the analyzed data. +No evidence of signal is found and the observed (expected) upper limit on the cross-section for nonresonant +Higgs boson pair production is determined to be 5.4 (8.1) times the Standard Model predicted cross-section +at 95% confidence level. Constraints are placed upon modifiers to the HHH and HHVV couplings. The +observed (expected) 2𝜎 constraints on the HHH coupling modifier, 𝜅𝜆, are determined to be [−3.5, 11.3] +([−5.4, 11.4]), while the corresponding constraints for the HHVV coupling modifier, 𝜅2𝑉 , are [−0.0, 2.1] +([−0.1, 2.1]). The results are also used to derive constraints on relevant coefficients in the SM effective +field theory and the Higgs effective field theory frameworks. +References +[1] +ATLAS Collaboration, Observation of a new particle in the search for the Standard Model Higgs +boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1, +arXiv: 1207.7214 [hep-ex]. +[2] +CMS Collaboration, +Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, +Phys. Lett. B 716 (2012) 30, arXiv: 1207.7235 [hep-ex]. +[3] +ATLAS Collaboration, +Observation of 𝐻 → 𝑏 ¯𝑏 decays and 𝑉𝐻 production with the ATLAS detector, +Phys. Lett. B 786 (2018) 59, arXiv: 1808.08238 [hep-ex]. +[4] +CMS Collaboration, Observation of Higgs Boson Decay to Bottom Quarks, +Phys. Rev. Lett. 121 (2018) 121801, arXiv: 1808.08242 [hep-ex]. +[5] +G. C. Branco et al., Theory and phenomenology of two-Higgs-doublet models, +Phys. Rept. 516 (2012) 1, arXiv: 1106.0034 [hep-ph]. +[6] +R. Gröber and M. Mühlleitner, Composite Higgs boson pair production at the LHC, +JHEP 06 (2011) 020, arXiv: 1012.1562 [hep-ph]. +[7] +Z. Chacko, Y. Nomura, M. Papucci, and G. Perez, +Natural little hierarchy from a partially goldstone twin Higgs, JHEP 01 (2006) 126, +arXiv: hep-ph/0510273. +[8] +P. Fayet, Supersymmetry and weak, electromagnetic and strong interactions, +Phys. Lett. B 64 (1976) 159. +[9] +P. Fayet, +Spontaneously broken supersymmetric theories of weak, electromagnetic and strong interactions, +Phys. Lett. B 69 (1977) 489. +29 + +[10] +G. Degrassi et al., Higgs mass and vacuum stability in the Standard Model at NNLO, +JHEP 08 (2012) 098, arXiv: 1205.6497. +[11] +M. Grazzini et al., Higgs boson pair production at NNLO with top quark mass effects, +JHEP 05 (2018) 059, arXiv: 1803.02463 [hep-ph]. +[12] +F. A. Dreyer and A. Karlberg, Vector-boson fusion Higgs pair production at N3LO, +Phys. Rev. D 98 (2018) 114016, arXiv: 1811.07906 [hep-ph]. +[13] +F. Bishara, R. Contino, and J. Rojo, +Higgs pair production in vector-boson fusion at the LHC and beyond, +Eur. Phys. J. C 77 (2017) 481, arXiv: 1611.03860 [hep-ph]. +[14] +D. de Florian et al., +Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector, (2016), +arXiv: 1610.07922 [hep-ph]. +[15] +A. Djouadi, J. Kalinowski, and M. Spira, HDECAY: A program for Higgs boson decays in the +Standard Model and its supersymmetric extension, Comput. Phys. Commun. 108 (1998) 56, +arXiv: hep-ph/9704448. +[16] +W. Buchmuller and D. Wyler, +Effective lagrangian analysis of new interactions and flavour conservation, +Nucl. Phys. B 268 (1986) 621. +[17] +C. N. Leung, S. T. Love, and S. Rao, +Low-energy manifestations of a new interactions scale: Operator analysis, +Z. Phys. C 31 (1986) 433. +[18] +I. Brivio and M. Trott, The standard model as an effective field theory, Phys. Rept. 793 (2019) 1, +arXiv: 1706.08945 [hep-ph]. +[19] +R. Alonso, M. Gavela, L. Merlo, S. Rigolin, and J. Yepes, +The effective chiral lagrangian for a light dynamical "Higgs particle", +Phys. Lett. B 722 (2012) 330, arXiv: 1212.3305 [hep-ph]. +[20] +G. Buchalla, O. Catà, and C. Krause, +Complete electroweak chiral Lagrangian with a light Higgs at NLO, Nucl. Phys. B 880 (2014) 552, +arXiv: 1307.5017 [hep-ph], Erratum: Nucl. Phys. B 913 (2016) 475. +[21] +B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek, +Dimension-six terms in the Standard Model Lagrangian, JHEP 10 (2010) 085, +arXiv: 1008.4884 [hep-ph]. +[22] +ATLAS Collaboration, Combined effective field theory interpretation of Higgs boson and weak +boson production and decay with ATLAS data and electroweak precision observables, +ATL-PHYS-PUB-2022-037, 2022, url: https://cds.cern.ch/record/2816369. +[23] +G. Buchalla, O. Catá, and C. Krause, On the power counting in effective field theories, +Phys. Lett. B 731 (2014) 80, arXiv: 1312.5624 [hep-ph]. +[24] +M. Capozi and G. Heinrich, +Exploring anomalous couplings in Higgs boson pair production through shape analysis, +JHEP 03 (2020) 091, arXiv: 1908.08923 [hep-ph]. +[25] +A. Carvalho et al., Higgs pair production: choosing benchmarks with cluster analysis, +JHEP 04 (2016) 126, arXiv: 1507.02245 [hep-ph]. +30 + +[26] +ATLAS Collaboration, Search for pair production of Higgs bosons in the 𝑏 ¯𝑏𝑏 ¯𝑏 final state using +proton–proton collisions at √𝑠 = 13 TeV with the ATLAS detector, JHEP 01 (2019) 030, +arXiv: 1804.06174 [hep-ex]. +[27] +ATLAS Collaboration, Search for the 𝐻𝐻 → 𝑏 ¯𝑏𝑏 ¯𝑏 process via vector-boson fusion production +using proton–proton collisions at √𝑠 = 13 TeV with the ATLAS detector, JHEP 07 (2020) 108, +arXiv: 2001.05178 [hep-ex], [Erratum: JHEP 01 (2021) 145]. +[28] +CMS Collaboration, Search for Higgs Boson Pair Production in the Four b Quark Final State in +Proton-Proton Collisions at s=13 TeV, Phys. Rev. Lett. 129 (2022) 081802, +arXiv: 2202.09617 [hep-ex]. +[29] +CMS Collaboration, Search for nonresonant pair production of highly energetic Higgs bosons +decaying to bottom quarks, (2022), arXiv: 2205.06667 [hep-ex]. +[30] +ATLAS Collaboration, Search for resonant and non-resonant Higgs boson pair production in the +𝑏 ¯𝑏𝜏+𝜏− decay channel using 13 TeV 𝑝𝑝 collision data from the ATLAS detector, (2022), +arXiv: 2209.10910 [hep-ex]. +[31] +CMS Collaboration, Search for Higgs boson pair production in events with two bottom quarks and +two tau leptons in proton–proton collisions at √𝑠 = 13 TeV, Phys. Lett. B 778 (2018) 101, +arXiv: 1707.02909 [hep-ex]. +[32] +ATLAS Collaboration, Search for Higgs boson pair production in the two bottom quarks plus two +photons final state in 𝑝𝑝 collisions at √𝑠 = 13 TeV with the ATLAS detector, +Phys. Rev. D 106 (2021) 052001, arXiv: 2112.11876 [hep-ex]. +[33] +CMS Collaboration, Search for nonresonant Higgs boson pair production in final states with two +bottom quarks and two photons in proton–proton collisions at √𝑠 = 13 TeV, JHEP 03 (2021) 257, +arXiv: 2011.12373 [hep-ex]. +[34] +ATLAS Collaboration, Search for non-resonant Higgs boson pair production in the 𝑏𝑏ℓ𝜈ℓ𝜈 final +state with the ATLAS detector in 𝑝𝑝 collisions at √𝑠 = 13 TeV, Phys. Lett. B 801 (2020) 135145, +arXiv: 1908.06765 [hep-ex]. +[35] +CMS Collaboration, Search for resonant and nonresonant Higgs boson pair production in the +𝑏 ¯𝑏ℓ𝜈ℓ𝜈 final state in proton–proton collisions at √𝑠 = 13 TeV, JHEP 01 (2018) 054, +arXiv: 1708.04188 [hep-ex]. +[36] +ATLAS Collaboration, Search for Higgs boson pair production in the 𝑏 ¯𝑏𝑊𝑊∗ decay mode at +√𝑠 = 13 TeV with the ATLAS detector, JHEP 04 (2019) 092, arXiv: 1811.04671 [hep-ex]. +[37] +ATLAS Collaboration, Search for Higgs boson pair production in the 𝛾𝛾𝑊𝑊∗ channel using 𝑝𝑝 +collision data recorded at √𝑠 = 13 TeV with the ATLAS detector, Eur. Phys. J. C 78 (2018) 1007, +arXiv: 1807.08567 [hep-ex]. +[38] +ATLAS Collaboration, Search for Higgs boson pair production in the 𝑊𝑊 (∗)𝑊𝑊 (∗) decay +channel using ATLAS data recorded at √𝑠 = 13 TeV, JHEP 05 (2019) 124, +arXiv: 1811.11028 [hep-ex]. +[39] +CMS Collaboration, +A portrait of the Higgs boson by the CMS experiment ten years after the discovery, +Nature 607 (2022) 60, arXiv: 2207.00043 [hep-ex]. +[40] +ATLAS Collaboration, The ATLAS Experiment at the CERN Large Hadron Collider, +JINST 3 (2008) S08003. +31 + +[41] +ATLAS Collaboration, ATLAS Insertable B-Layer: Technical Design Report, +ATLAS-TDR-19; CERN-LHCC-2010-013, 2010, +url: https://cds.cern.ch/record/1291633, Addendum: ATLAS-TDR-19-ADD-1; +CERN-LHCC-2012-009, 2012, url: https://cds.cern.ch/record/1451888. +[42] +B. Abbott et al., Production and integration of the ATLAS Insertable B-Layer, +JINST 13 (2018) T05008, arXiv: 1803.00844 [physics.ins-det]. +[43] +ATLAS Collaboration, Performance of the ATLAS trigger system in 2015, +Eur. Phys. J. C 77 (2017) 317, arXiv: 1611.09661 [hep-ex]. +[44] +ATLAS Collaboration, The ATLAS Collaboration Software and Firmware, +ATL-SOFT-PUB-2021-001, 2021, url: https://cds.cern.ch/record/2767187. +[45] +ATLAS Collaboration, +ATLAS data quality operations and performance for 2015–2018 data-taking, +JINST 15 (2020) P04003, arXiv: 1911.04632 [physics.ins-det]. +[46] +ATLAS Collaboration, Configuration and performance of the ATLAS 𝑏-jet triggers in Run 2, +Eur. Phys. J. C 81 (2021) 1087, arXiv: 2106.03584 [hep-ex]. +[47] +M. Cacciari, G. P. Salam, and G. Soyez, The anti-𝑘𝑡 jet clustering algorithm, JHEP 04 (2008) 063, +arXiv: 0802.1189 [hep-ph]. +[48] +M. Cacciari, G. P. Salam, and G. Soyez, FastJet user manual, Eur. Phys. J. C 72 (2012) 1896, +arXiv: 1111.6097 [hep-ph]. +[49] +ATLAS Collaboration, The ATLAS Simulation Infrastructure, Eur. Phys. J. C 70 (2010) 823, +arXiv: 1005.4568 [physics.ins-det]. +[50] +GEANT4 Collaboration, S. Agostinelli, et al., Geant4 – a simulation toolkit, +Nucl. Instrum. Meth. A 506 (2003) 250. +[51] +P. Nason, A New method for combining NLO QCD with shower Monte Carlo algorithms, +JHEP 11 (2004) 040, arXiv: hep-ph/0409146 [hep-ph]. +[52] +S. Frixione, P. Nason and C. Oleari, +Matching NLO QCD computations with parton shower simulations: the POWHEG method, +JHEP 11 (2007) 070, arXiv: 0709.2092 [hep-ph]. +[53] +S. Alioli, P. Nason, C. Oleari, and E. Re, A general framework for implementing NLO calculations +in shower Monte Carlo programs: the POWHEG BOX, JHEP 06 (2010) 043, +arXiv: 1002.2581 [hep-ph]. +[54] +J. Butterworth et al., PDF4LHC recommendations for LHC Run II, J. Phys. G 43 (2016) 023001, +arXiv: 1510.03865 [hep-ph]. +[55] +T. Sjöstrand et al., An introduction to PYTHIA 8.2, Comput. Phys. Commun. 191 (2015) 159, +arXiv: 1410.3012 [hep-ph]. +[56] +ATLAS Collaboration, ATLAS Pythia 8 tunes to 7 TeV data, ATL-PHYS-PUB-2014-021, 2014, +url: https://cds.cern.ch/record/1966419. +[57] +R. D. Ball et al., Parton distributions with LHC data, Nucl. Phys. B 867 (2013) 244, +arXiv: 1207.1303 [hep-ph]. +[58] +ATLAS Collaboration, Validation of signal Monte Carlo event generation in searches for Higgs +boson pairs with the ATLAS detector, ATL-PHYS-PUB-2019-007, 2019, +url: https://cds.cern.ch/record/2665057. +32 + +[59] +M. Bähr et al., Herwig++ Physics and Manual, Eur. Phys. J. C58 (2008) 639, +arXiv: 0803.0883 [hep-ph]. +[60] +S. Gieseke, C. Röhr, and A. Siodmok, Colour reconnections in Herwig++, +Eur. Phys. J. C 72 (2012) 2225, arXiv: 1206.0041 [hep-ph]. +[61] +L. A. Harland-Lang, A. D. Martin, P. Motylinski, and R. S. Thorne, +Parton distributions in the LHC era: MMHT 2014 PDFs, Eur. Phys. J. C 75 (2015) 204, +arXiv: 1412.3989 [hep-ph]. +[62] +ATLAS Collaboration, +A study of optimal parameter setting for MadGraph5_aMC@NLO + Pythia 8 matched setup, +ATL-PHYS-PUB-2015-048, 2015, url: https://cds.cern.ch/record/2103221. +[63] +J. Alwall et al., The automated computation of tree-level and next-to-leading order differential +cross sections, and their matching to parton shower simulations, JHEP 07 (2014) 079, +arXiv: 1405.0301 [hep-ph]. +[64] +J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer, and T. Stelzer, MadGraph 5 : Going Beyond, +JHEP 06 (2011) 128, arXiv: 1106.0522 [hep-ph]. +[65] +C. Degrande et al., Automated one-loop computations in the standard model effective field theory, +Phys. Rev. D 103 (2021) 096024, arXiv: 2008.11743 [hep-ph]. +[66] +G. Buchalla, M. Capozi, A. Celis, G. Heinrich, and L. Scyboz, Higgs boson pair production in +non-linear Effective Field Theory with full 𝑚𝑡-dependence at NLO QCD, JHEP 09 (2018) 057, +arXiv: 1806.05162 [hep-ph]. +[67] +ATLAS Collaboration, HEFT interpretations of Higgs boson pair searches in 𝑏 ¯𝑏𝛾𝛾 and 𝑏 ¯𝑏𝜏+𝜏− +final states and of their combination in ATLAS, ATL-PHYS-PUB-2022-019, 2022, +url: https://cds.cern.ch/record/2806411. +[68] +R. D. Ball et al., Parton distributions for the LHC run II, JHEP 04 (2015) 040, +arXiv: 1410.8849 [hep-ph]. +[69] +ATLAS Collaboration, Combination of searches for Higgs boson pairs in 𝑝𝑝 collisions at +√𝑠 = 13 TeV with the ATLAS detector, Phys. Lett. B 800 (2020) 135103, +arXiv: 1906.02025 [hep-ex]. +[70] +J. Baglio et al., The measurement of the Higgs self-coupling at the LHC: theoretical status, +JHEP 04 (2013) 151, arXiv: 1212.5581 [hep-ph]. +[71] +L.-S. Ling et al., +NNLO QCD corrections to Higgs pair production via vector boson fusion at hadron colliders, +Phys. Rev. D 89 (2014) 073001, arXiv: 1401.7754 [hep-ph]. +[72] +F. A. Dreyer and A. Karlberg, +Fully differential Vector-Boson Fusion Higgs Pair Production at Next-to-Next-to-Leading Order, +Phys. Rev. D 99 (2019) 074028, arXiv: 1811.07918 [hep-ph]. +[73] +J. M. Campbell, R. K. Ellis, P. Nason, and E. Re, +Top-pair production and decay at NLO matched with parton showers, JHEP 04 (2015) 114, +arXiv: 1412.1828 [hep-ph]. +[74] +M. Czakon and A. Mitov, +Top++: A program for the calculation of the top-pair cross-section at hadron colliders, +Comput. Phys. Commun. 185 (2014) 2930, arXiv: 1112.5675 [hep-ph]. +33 + +[75] +ATLAS Collaboration, +Improvements in 𝑡¯𝑡 modelling using NLO+PS Monte Carlo generators for Run 2, +ATL-PHYS-PUB-2018-009, 2018, url: https://cds.cern.ch/record/2630327. +[76] +ATLAS Collaboration, The Pythia 8 A3 tune description of ATLAS minimum bias and inelastic +measurements incorporating the Donnachie–Landshoff diffractive model, +ATL-PHYS-PUB-2016-017, 2016, url: https://cds.cern.ch/record/2206965. +[77] +D. J. Lange, The EvtGen particle decay simulation package, +Nucl. Instrum. Meth. A 462 (2001) 152. +[78] +ATLAS Collaboration, Vertex Reconstruction Performance of the ATLAS Detector at √𝑠 = 13 TeV, +ATL-PHYS-PUB-2015-026, 2015, url: https://cds.cern.ch/record/2037717. +[79] +ATLAS Collaboration, +Jet reconstruction and performance using particle flow with the ATLAS Detector, +Eur. Phys. J. C 77 (2017) 466, arXiv: 1703.10485 [hep-ex]. +[80] +ATLAS Collaboration, Jet energy scale and resolution measured in proton–proton collisions at +√𝑠 = 13 TeV with the ATLAS detector, Eur. Phys. J. C 81 (2020) 689, +arXiv: 2007.02645 [hep-ex]. +[81] +ATLAS Collaboration, Performance of pile-up mitigation techniques for jets in 𝑝𝑝 collisions at +√𝑠 = 8 TeV using the ATLAS detector, Eur. Phys. J. C 76 (2016) 581, +arXiv: 1510.03823 [hep-ex]. +[82] +ATLAS Collaboration, +Topological cell clustering in the ATLAS calorimeters and its performance in LHC Run 1, +Eur. Phys. J. C 77 (2017) 490, arXiv: 1603.02934 [hep-ex]. +[83] +ATLAS Collaboration, +Selection of jets produced in 13 TeV proton–proton collisions with the ATLAS detector, +ATLAS-CONF-2015-029, 2015, url: https://cds.cern.ch/record/2037702. +[84] +ATLAS Collaboration, ATLAS flavour-tagging algorithms for the LHC Run 2 𝑝𝑝 collision dataset, +(2022), arXiv: 2211.16345 [physics.data-an]. +[85] +ATLAS Collaboration, Calibration of light-flavour 𝑏-jet mistagging rates using ATLAS +proton–proton collision data at √𝑠 = 13 TeV, ATLAS-CONF-2018-006, 2018, +url: https://cds.cern.ch/record/2314418. +[86] +ATLAS Collaboration, Measurement of the c-jet mistagging efficiency in 𝑡¯𝑡 events using pp +collision data at √𝑠 = 13 TeV collected with the ATLAS detector, Eur. Phys. J. C 82 (2022) 95, +arXiv: 2109.10627 [hep-ex]. +[87] +ATLAS Collaboration, Muon reconstruction and identification efficiency in ATLAS using the full +Run 2 𝑝𝑝 collision data set at √𝑠 = 13 TeV, Eur. Phys. J. C 81 (2021) 578, +arXiv: 2012.00578 [hep-ex]. +[88] +ATLAS Collaboration, Evidence for the 𝐻 → 𝑏 ¯𝑏 decay with the ATLAS detector, +JHEP 12 (2017) 024, arXiv: 1708.03299 [hep-ex]. +[89] +ATLAS Collaboration, Muon reconstruction performance of the ATLAS detector in proton–proton +collision data at √𝑠 = 13 TeV, Eur. Phys. J. C 76 (2016) 292, arXiv: 1603.05598 [hep-ex]. +[90] +G. V. Moustakides and K. Basioti, +Training Neural Networks for Likelihood/Density Ratio Estimation, 2019, +arXiv: 1911.00405 [eess.SP]. +34 + +[91] +T. Kanamori, S. Hido, and M. Sugiyama, +A Least-squares Approach to Direct Importance Estimation, +Journal of Machine Learning Research 10 (2009) 1391, +url: http://jmlr.org/papers/v10/kanamori09a.html. +[92] +V. Nair and G. E. Hinton, “Rectified Linear Units Improve Restricted Boltzmann Machines,” +Proceedings of the 27th International Conference on International Conference on Machine +Learning, ICML’10, Haifa, Israel: Omnipress, 2010 807, isbn: 9781605589077, +url: https://dl.acm.org/doi/10.5555/3104322.3104425. +[93] +B. Lakshminarayanan, A. Pritzel, and C. Blundell, +Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles, 2017, +arXiv: 1612.01474 [stat.ML]. +[94] +B. Efron, Bootstrap Methods: Another Look at the Jackknife, Ann. Statist. 7 (1979) 1. +[95] +ATLAS Collaboration, Measurement of the Inelastic Proton–Proton Cross Section at √𝑠 = 13 TeV +with the ATLAS Detector at the LHC, Phys. Rev. Lett. 117 (2016) 182002, +arXiv: 1606.02625 [hep-ex]. +[96] +ATLAS Collaboration, ATLAS 𝑏-jet identification performance and efficiency measurement with 𝑡¯𝑡 +events in 𝑝𝑝 collisions at √𝑠 = 13 TeV, Eur. Phys. J. C 79 (2019) 970, +arXiv: 1907.05120 [hep-ex]. +[97] +ATLAS Collaboration, +Luminosity determination in 𝑝𝑝 collisions at √𝑠 = 13 TeV using the ATLAS detector at the LHC, +ATLAS-CONF-2019-021, 2019, url: https://cds.cern.ch/record/2677054. +[98] +G. Avoni et al., The new LUCID-2 detector for luminosity measurement and monitoring in ATLAS, +JINST 13 (2018) P07017. +[99] +J. Baglio et al., 𝑔𝑔 → 𝐻𝐻: Combined uncertainties, Phys. Rev. D 103 (2021) 056002, +arXiv: 2008.11626 [hep-ph]. +[100] +W. Verkerke and D. Kirkby, The RooFit toolkit for data modeling, 2003, +arXiv: physics/0306116 [physics.data-an]. +[101] +G. Cowan, K. Cranmer, E. Gross, and O. Vitells, +Asymptotic formulae for likelihood-based tests of new physics, Eur. Phys. J. C 71 (2011) 1554, +arXiv: 1007.1727 [physics.data-an], Erratum: Eur. Phys. J. C 73 (2013) 2501. +[102] +A. L. Read, Presentation of search results: the 𝐶𝐿𝑆 technique, J. Phys. G 28 (2002) 2693. +35 + diff --git a/gNE1T4oBgHgl3EQffAQh/content/tmp_files/load_file.txt b/gNE1T4oBgHgl3EQffAQh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..acfda6d0596d3f004262059ef1294987a69d327a --- /dev/null +++ b/gNE1T4oBgHgl3EQffAQh/content/tmp_files/load_file.txt @@ -0,0 +1,1532 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf,len=1531 +page_content='EUROPEAN ORGANISATION FOR NUCLEAR RESEARCH (CERN) Submitted to: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' CERN-EP-2022-235 January 10, 2023 Search for nonresonant pair production of Higgs bosons in the 𝒃 ¯𝒃𝒃 ¯𝒃 final state in 𝒑 𝒑 collisions at √𝒔 = 13 TeV with the ATLAS detector The ATLAS Collaboration A search for nonresonant Higgs boson pair production in the 𝑏 ¯𝑏𝑏 ¯𝑏 final state is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The analysis uses 126 fb−1 of 𝑝𝑝 collision data at √𝑠 = 13 TeV collected with the ATLAS detector at the Large Hadron Collider, and targets both the gluon–gluon fusion and vector-boson fusion production modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' No evidence of the signal is found and the observed (expected) upper limit on the cross-section for nonresonant Higgs boson pair production is determined to be 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1) times the Standard Model predicted cross-section at 95% confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Constraints are placed on modifiers to the 𝐻𝐻𝐻 and 𝐻𝐻𝑉𝑉 couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The observed (expected) 2𝜎 constraints on the 𝐻𝐻𝐻 coupling modifier, 𝜅𝜆, are determined to be [−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3] ([−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4]), while the corresponding constraints for the 𝐻𝐻𝑉𝑉 coupling modifier, 𝜅2𝑉 , are [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1] ([−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In addition, constraints on relevant coefficients are derived in the context of the Standard Model effective field theory and Higgs effective field theory, and upper limits on the 𝐻𝐻 production cross-section are placed in seven Higgs effective field theory benchmark scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' © 2023 CERN for the benefit of the ATLAS Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Reproduction of this article or parts of it is allowed as specified in the CC-BY-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='03212v1 [hep-ex] 9 Jan 2023 CERN1 Introduction The discovery of the 125 GeV Higgs boson (𝐻) [1–4] at the Large Hadron Collider (LHC) has prompted a broad research program to investigate its properties and compare the measurements with the Standard Model (SM) predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Of particular interest is the search for nonresonant Higgs boson pair, known as di-Higgs (HH), production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This process has a strong dependence on the Higgs self-coupling, which is a key ingredient of the electroweak symmetry breaking mechanism and a sensitive probe for physics beyond the SM (BSM physics) in various scenarios, such as two-Higgs-doublet models [5], composite Higgs models [6], twin Higgs models [7], and the minimal supersymmetric extension of the SM [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The Higgs self-coupling also plays a fundamental role in understanding the stability of the universe [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The dominant SM HH production process is gluon–gluon fusion (ggF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Its cross-section, for a Higgs boson mass 𝑚𝐻 = 125 GeV, calculated at next-to-next-to-leading order (NNLO) including finite top-quark-mass effects [11], is 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='05 fb at a center-of-mass energy √𝑠 = 13 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The two dominant leading-order Feynman diagrams contributing to this process are shown in Figure 1, where Figure 1(a) is commonly referred to as the box diagram and Figure 1(b) as the triangle diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The triangle diagram introduces the dependence on the trilinear Higgs self-coupling, 𝜆, shown by the red vertex in Figure 1(b), which can be expressed in terms of its modifier, 𝜅𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1 In the SM, these two diagrams interfere destructively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' As a result, the HH production cross-section and kinematic properties depend critically on the value of 𝜅𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' g g H H κt κt (a) g g H H κt κt (b) Figure 1: The two leading-order gluon–gluon fusion di-Higgs production Feynman diagrams: (a) the box diagram;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' (b) the triangle diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The HH production process with the second-highest cross-section in the SM is vector-boson fusion (VBF), with a calculated value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='73 fb at next-to-next-to-next-to-leading order (N3LO) [12], for 𝑚𝐻 = 125 GeV at √𝑠 = 13 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Figure 2 illustrates the Feynman diagrams involved in di-Higgs production via vector-boson fusion at leading order (LO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The coupling modifiers 𝜅𝜆, 𝜅𝑉 and 𝜅2𝑉 are respectively shown at the HHH, HVV and HHVV interaction vertices, where 𝑉 stands for the gauge vector bosons W or Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In the SM, the divergences in the Figures 2(b) and 2(c) diagrams exactly cancel out due to perturbative unitarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' As 𝜅𝑉 and 𝜅2𝑉 depart from their SM value of one, this canceling out no longer occurs, introducing a linear dependence of the cross-section on the effective center-of-mass energy of the incoming vector bosons [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Therefore, the Higgs bosons produced in non-SM 𝜅𝑉 /𝜅2𝑉 scenarios are expected to be more energetic and more central in the detector on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This increase in the energy of Higgs bosons with increasing deviation from the SM continues up to the scale of some new physics, which is required to unitarize the total amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 1 A coupling modifier, 𝜅, is defined as the ratio of the modified coupling to its SM value, 𝜅 = 𝑐/𝑐SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' By definition, 𝜅 = 1 denotes the value of the coupling predicted by the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 2 H H q q q q H V V κλ κV (a) H H q q q q V V κ2V (b) H H q q q q V V κV κV (c) Figure 2: The three tree-level vector-boson fusion di-Higgs production Feynman diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The analysis described in this paper targets the HH process in the bbbb final state, in both the ggF and VBF production modes, using the data collected by ATLAS between 2016 and 2018, during Run 2 of the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Assuming the SM branching ratio of 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='2% for H → bb [14, 15], about one third of di-Higgs events decay into bbbb, making it the most abundant di-Higgs final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' However, as this is a fully hadronic final state, the analysis faces the challenge of large backgrounds, which originate mostly from nonresonant QCD production of multiple heavy (b/t) quarks, as well as from light-quark-initiated jets misidentified as originating from heavy quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The results are interpreted in terms of constraints on the 𝜅𝜆 and 𝜅2𝑉 coupling modifiers, assuming 𝜅𝑉 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The analysis also provides one- and two-dimensional constraints on relevant couplings in the SM effective field theory (SMEFT) [16–18] and Higgs effective field theory (HEFT) [19, 20] frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In the SMEFT framework, the effects of new physics may be described with an effective Lagrangian: LSMEFT = LSM + 1 Λ2 ∑︁ 𝑘 𝑐(6) 𝑘 𝑂 (6) 𝑘 , where LSM represents the SM Lagrangian, 𝑂𝑘 are higher-dimensional local operators, 𝑐𝑘 are the Wilson coefficients, and Λ is the mass scale of the new physics phenomena (set to 1 TeV for this result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The analysis considers operators 𝑂𝑘 in the Warsaw basis, which provides a complete set of operators allowed by SM gauge symmetries at dimension six [21] (dimension-five operators introduce lepton and baryon number violation, and are therefore ignored in this result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The five operators relevant to the HH process and their coefficients, 𝑐H, 𝑐H□, 𝑐𝑡H, 𝑐𝑡𝐺, and 𝑐H𝐺, are listed in Table 1 [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The computation of amplitudes from the above Lagrangian includes three terms: a pure SM term, a “quadratic” term of order (1/Λ4) including purely new physics, and a “linear” term of order (1/Λ2) accounting for the interference between the SM and new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The SMEFT constraints calculated in this analysis include both the linear and quadratic new physics terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In the HEFT framework, new physics in the electroweak sector is described through anomalous couplings of the Higgs boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The organization of the HEFT Lagrangian is guided by chiral perturbation theory [23], with the low-energy dynamics of electroweak symmetry breaking described using a nonlinear realization of the gauge symmetry group 𝑆𝑈(2)𝐿 ×𝑈(1)𝑌 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' One advantage of the HEFT framework is that the anomalous single-Higgs-boson and HH couplings are defined separately, allowing simplified HH interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In the HEFT Lagrangian, ggF HH production is described at LO by five relevant operators and their associated Wilson coefficients: 𝑐HHH, 𝑐𝑡𝑡H, 𝑐𝑔𝑔H, 𝑐𝑔𝑔HH, and 𝑐𝑡𝑡HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In this formalism, 𝑐HHH is equivalent to 𝜅𝜆 and 𝑐𝑡𝑡H is equivalent to the modifier for the coupling between the Higgs boson and top quark, 𝜅𝑡, shown by the 3 Table 1: The five relevant SMEFT coefficients and their corresponding dimension-6 operators, as defined in the Warsaw basis [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Wilson Coefficient Operator 𝑐H (H†H)3 𝑐H□ (H†H)□(H†H) 𝑐tH (H†H)( ¯𝑄 ˜H𝑡) 𝑐H𝐺 H†H𝐺 𝐴 𝜇𝜈𝐺 𝜇𝜈 𝐴 𝑐t𝐺 ( ¯𝑄𝜎𝜇𝜈𝑇 𝐴t) ˜H𝐺 𝐴 𝜇𝜈 light blue vertex in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Fixing 𝑐𝑡𝑡H = 𝑐HHH = 1 and 𝑐𝑔𝑔H = 𝑐𝑔𝑔HH = 𝑐𝑡𝑡HH = 0 restores the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' At next-to-leading order (NLO), seven HEFT benchmark models (BM) [24] have been defined using cluster analysis [25] to probe a wide variety of characteristic shapes of the 𝑚HH spectrum resulting from different BSM scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The values of the coefficients used to define these scenarios are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Table 2: The values of the HEFT Wilson coefficients in the SM and in seven BSM benchmark models, as defined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Benchmark Model 𝑐𝐻 𝐻 𝐻 𝑐𝑡𝑡𝐻 𝑐𝑔𝑔𝐻 𝑐𝑔𝑔𝐻 𝐻 𝑐𝑡𝑡𝐻 𝐻 SM 1 1 0 0 0 BM1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='94 1/2 1/3 −1/3 BM2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 −1/3 1/3 BM3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='05 1/2 1/2 −1/3 BM4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='61 −1/2 1/6 1/3 BM5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='17 1/6 −1/2 −1/3 BM6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='83 −1/2 1/3 1/3 BM7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='94 1/6 −1/6 1 The ATLAS Collaboration has previously published search results for nonresonant HH → bbbb production using 27 fb−1 of early Run 2 data [26], and a dedicated search for VBF HH production in 126 fb−1 of data collected between 2016 and 2018 [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The present analysis benefits from the use of the 2016–2018 data for both production channels and also takes advantage of improvements in jet reconstruction and in the identification of jets arising from the decays of b-quarks (“𝑏-tagging”) achieved by the ATLAS Collaboration since the publication of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In addition, the analysis employs a fully data-driven technique for the background estimation, which uses an artificial neural network to perform a kinematic reweighting of data from an alternative phase space to model the background in the region of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The CMS Collaboration has also published results of a search for nonresonant HH → bbbb with its full Run 2 dataset [28], setting the observed (expected) upper limit on the HH cross-section at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='9 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='8) times the SM predicted cross-section, and restricting the allowed interval for 𝜅𝜆 to [−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4] ([−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0]), both at 95% confidence level (CL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' A more recent CMS HH → bbbb publication [29], in which the analysis exploits topologies arising from highly energetic Higgs boson decays into 𝑏 ¯𝑏, sets the observed (expected) upper limit at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='9 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1) times the SM cross-section expectation, and restricts the allowed interval for 𝜅2𝑉 to [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='62, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='41] ([0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='66, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='37]), at 95% CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Other searches for nonresonant 𝐻𝐻 production were performed by ATLAS and CMS in the 𝑏 ¯𝑏𝜏+𝜏− [30, 31], 𝑏 ¯𝑏𝛾𝛾 [32, 33], 𝑏 ¯𝑏ℓ+𝜈ℓ−𝜈 [34, 35] decay channels, as well as by 4 ATLAS in the 𝑏 ¯𝑏𝑞𝑞ℓ𝜈 [36], 𝑊𝑊∗𝛾𝛾 [37] and 𝑊𝑊∗𝑊𝑊∗ [38] decay channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Amongst them, the most sensitive results to date from ATLAS come from the 𝑏 ¯𝑏𝛾𝛾 analysis, which sets the observed (expected) 95% CL upper limit on the SM nonresonant 𝐻𝐻 cross-section at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='7) times the SM expectation and restricts the corresponding 𝜅𝜆 interval to [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='7] ([−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The most sensitive results to date from CMS come from the combination of the 𝑏 ¯𝑏𝑍𝑍, multilepton, 𝑏 ¯𝑏𝛾𝛾, 𝑏 ¯𝑏𝜏𝜏, and bbbb analyses, which set the observed (expected) 95% CL upper limit on the SM nonresonant 𝐻𝐻 cross-section at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5) times the SM expectation and restricts the corresponding observed 𝜅𝜆 interval to [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='24, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='49] [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This document is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The ATLAS detector and the data and simulated events used in the analysis are described in Sections 2 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Section 4 presents the reconstruction and identification of physics objects in this analysis and Section 5 details the event selection and categorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The background modeling method is described in Section 6, the systematic uncertainties are detailed in Section 7 and, finally, the results are reported in Section 8 and the conclusion is given in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 2 ATLAS detector The ATLAS detector [40] at the LHC covers nearly the entire solid angle around the collision point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='2 It consists of an inner tracking detector surrounded by a thin superconducting solenoid, electromagnetic and hadron calorimeters, and a muon spectrometer incorporating three large superconducting air-core toroidal magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The inner-detector (ID) system is immersed in a 2 T axial magnetic field and provides charged-particle tracking in the range |𝜂| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The high-granularity silicon pixel detector covers the vertex region and typically provides four space-point measurements per track, the first hit normally being in the insertable B-layer installed before Run 2 [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Following the pixel detector is the silicon microstrip tracker, which usually provides eight measurements per track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' These silicon detectors are surrounded by the transition radiation tracker, which enables radially extended track reconstruction up to |𝜂| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The calorimeter system covers the pseudorapidity range |𝜂| < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Within |𝜂| < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='2, electromagnetic calorimetry is provided by barrel and endcap high-granularity lead/liquid-argon (LAr) calorimeters, with an additional thin LAr presampler covering |𝜂| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='8 to correct for energy loss in material upstream of the calorimeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Hadron calorimetry is provided by the steel/scintillator-tile calorimeter, segmented into three barrel structures within |𝜂| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='7, and two copper/LAr hadron endcap calorimeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The solid angle coverage is completed with forward copper/LAr and tungsten/LAr calorimeter modules optimized for electromagnetic and hadronic energy measurements respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The muon spectrometer (MS) comprises separate trigger and high-precision tracking chambers measuring the deflection of muons in a magnetic field generated by the superconducting air-core toroidal magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The field integral of the toroids ranges between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 T·m across most of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' A set of precision chambers covers the region |𝜂| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='7 with three layers of monitored drift tubes, complemented by cathode-strip chambers in the forward region, where the background is highest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The muon trigger system 2 ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the center of the detector and the 𝑧-axis along the beam pipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The 𝑥-axis points from the IP to the center of the LHC ring, and the 𝑦-axis points upwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Cylindrical coordinates (𝑟, 𝜙) are used in the transverse plane, 𝜙 being the azimuthal angle around the 𝑧-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The pseudorapidity is defined in terms of the polar angle 𝜃 as 𝜂 = − ln tan(𝜃/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Angular distance is measured in units of Δ𝑅 ≡ √︃ (Δ𝜂)2 + (Δ𝜙)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 5 covers the range |𝜂| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4 with resistive-plate chambers in the barrel, and thin-gap chambers in the endcap regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Interesting events are selected by the first-level trigger system implemented in custom hardware, followed by selections made by algorithms implemented in software in the high-level trigger [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The first-level trigger accepts events from the 40 MHz bunch crossings at a rate below 100 kHz, which the high-level trigger reduces in order to record events to disk at about 1 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' An extensive software suite [44] is used in data simulation, in the reconstruction and analysis of real and simulated data, in detector operations, and in the trigger and data acquisition systems of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 3 Data and simulated samples 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1 Data sample This analysis is performed in LHC proton–proton (pp) collision data at √𝑠 = 13 TeV collected between 2016 and 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Only data collected during stable beam conditions are used, with all relevant detector systems functional [45], corresponding to an integrated luminosity of 126 fb−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' During 2016 data taking, a fraction of the data (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3 fb−1) was affected by an inefficiency in the online primary vertex reconstruction, which reduced the efficiency of the 𝑏-tagging algorithms in the trigger;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' those events were not retained for further analysis, resulting in an integrated luminosity of 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='6 fb−1 for the 2016 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The integrated luminosities of the 2017 and 2018 datasets are 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='7 fb−1 and 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='7 fb−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The analysis uses events that satisfy either of two types of trigger signatures, each with different requirements on the number of jets and their 𝑏-tagging status [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The jets used are reconstructed with the anti-𝑘𝑡 algorithm [47, 48], with a radius parameter of 𝑅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The 𝑏-tagging is performed at the trigger level with the MV2c20 algorithm in 2016 and the MV2c10 algorithm in 2017 and 2018 [46], with a range of b-jet identification efficiency operating points from 40% to 70% (as calculated from simulated tt samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=') The first of the two trigger signatures used for selecting bbbb events requires two 𝑏-jets plus one additional jet (“2𝑏1j”), while the second requires two 𝑏-jets plus two additional jets (“2𝑏2j”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The minimum transverse energy (𝐸T) requirement on the jets is 35 GeV for all jets used in the 2𝑏2j trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In the 2𝑏1j trigger, the b-tagged jets must have 𝐸T > 55 GeV, while the requirement on the minimum 𝐸T of the additional jet is between 100 and 150 GeV, depending on the year of data taking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='2 Simulated samples Monte Carlo (MC) simulation is used for the modeling of signal events, as well as to produce event samples of background processes for cross-checks and validation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The Higgs boson mass is set to 125 GeV in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' All samples were processed by the ATLAS simulation framework [49] and the detector response was simulated with Geant4 [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The ggF signal process was simulated using the Powheg Box v2 generator [51–53] at NLO, including finite top-quark-mass effects, using the PDF4LHC15 [54] parton distribution function (PDF) set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Parton showers and hadronization were simulated with Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='244 [55] with the A14 set of tuned parameters [56] and the NNPDF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3lo PDF set [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The SM ggF HH cross-section was taken as 𝜎ggF = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='05 fb, calculated at NNLO including finite top-quark-mass effects [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Signal samples for the ggF process were generated 6 explicitly for coupling modifier values of 𝜅𝜆 = 1 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' A reweighting method is used to obtain a ggF signal sample at each 𝜅𝜆 value, as described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [58]: scale factors are derived as a function of 𝜅𝜆 in bins of the generator-level invariant mass of the HH system by performing a linear combination of generator-level samples at three different 𝜅𝜆 values (𝜅𝜆 = 0, 1, and 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The 𝜅𝜆 = 10 ggF signal sample is used to validate the derived scale factors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' this generated sample and the signal sample obtained from the reweighting method are found to agree within the statistical precision of the simulated sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Additional generator-level ggF HH signal samples without parton showering were produced with Powheg Box v2 for the 𝜅𝜆 = 0 and 20 coupling modifier configurations to provide a basis for the 𝜅𝜆 reweighting, along with the SM ggF sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' For the reweighted ggF signal, the NNLO cross-section as a function of 𝜅𝜆 is taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In order to assess parton showering uncertainties, alternative ggF samples were generated using the Powheg Box v2 generator at NLO with the PDF4LHC15 PDF set, interfaced to Herwig 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='6 [59] for parton showering and hadronization using the Herwig 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1-default set of tuned parameters [60] and MMHT2014lo PDF set [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' To extract SMEFT coefficient constraints, parton-level ggF HH samples were generated with Mad- Graph5_aMC@NLO [62–64] with the SMEFT@NLO model [65] for a variety of SMEFT coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' A finely spaced multidimensional grid of signal samples was obtained using a LO-derived reweighting procedure in the generator-level invariant mass of the HH system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' this procedure is similar to that used to obtain 𝜅𝜆 variations for the ggF signal, as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' To extract HEFT coefficient constraints, a similar NLO-derived reweighting procedure was applied to the simulated reconstruction-level ggF signal sample to produce a variety of HEFT signal scenarios, including the seven benchmark scenarios defined in Section 1, following the prescription outlined in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [66, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Additional 𝐾-factors were applied to the SMEFT samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' these 𝐾-factors were derived using the ratio of the NLO cross-section to the LO cross-section at the equivalent HEFT point, as obtained using the HEFT to SMEFT translation from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3 The VBF signal process was simulated using MadGraph 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3 [63] at LO with the NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0nlo PDF set [68], interfaced with Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='244 for parton showering and hadronization using the A14 set of tuned parameters and NNPDF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3lo PDF set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Signal samples for the VBF process were generated explicitly for coupling modifier values of (𝜅𝜆, 𝜅2𝑉 , 𝜅𝑉 ) = (1, 1, 1), (1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5, 1), (2, 1, 1), (10, 1, 1), (1, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5), (−5, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5), (0, 1, 1), (1, 0, 1), and (1, 3, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' A linear combination of the first six of the listed samples is used to derive distributions for a finer granularity of 𝜅2𝑉 values, following a technique used previously to generate 𝜅𝜆 distributions [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The specific basis of six samples utilized is chosen to avoid large statistical uncertainties in the reweighted signal samples resulting from sparsely populated areas of kinematic phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The generated VBF signal samples not included in the linear combination basis – (𝜅𝜆, 𝜅2𝑉 , 𝜅𝑉 ) = (0, 1, 1), (1, 0, 1), and (1, 3, 1) – were used to validate the performance of the combination method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' These generated samples and the corresponding signal samples obtained from the combination method were found to agree within the statistical precision of the simulated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The cross-section for the VBF 𝐻𝐻 process, evaluated at N3LO in QCD, is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='73 fb in the SM [12, 70–72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' For the reweighted VBF signal points, the N3LO to LO cross-section ratio at the SM value is calculated, and this factor is applied to the cross-sections at each 𝜅𝜆, 𝜅2𝑉 , and 𝜅𝑉 point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In order to assess parton showering uncertainties, alternative LO samples were generated using MadGraph 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3 with the NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0nlo PDF set, interfaced to Herwig 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4 with the Herwig 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1-default set of tuned parameters and MMHT2014lo PDF set for parton showering and hadronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Top-quark pair production (tt) and multijet background processes were simulated in order to validate the 3 Variations in the 𝑐𝑡𝐺 Wilson coefficient were neglected when calculating 𝐾-factors because the corresponding chromomagnetic operator does not appear at LO within HEFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 7 background modeling procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The tt sample was simulated at NLO in 𝛼s using Powheg Box v2 [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Parton showering, hadronization, and the underlying event were modeled using Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The matrix element calculation uses NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0nlo as the PDF set, while the parton shower and underlying-event modeling uses NNPDF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3lo and the A14 set of tuned parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The damping parameter ℎdamp, which effectively regulates radiation at high 𝑝T, was set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 times the top quark’s mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The tt simulation is normalized using the value of the inclusive cross-section calculated with Top++ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 [74, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This accounts for NNLO corrections in 𝛼s, including next-to-next-to-leading logarithmic (NNLL) resummation of soft gluon terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The multijet background samples were modeled using Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This simulates pure QCD 2-to-2 interactions at LO in 𝛼s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Events were showered using the parton shower native to Pythia, which includes radiation and splitting that can result in additional jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The A14 set of tuned parameters and the NNPDF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3lo PDF set were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Other background processes, such as SM Higgs boson, HH (in other final states) and electroweak diboson production, have been estimated to give negligible contributions to the selected event yields and are therefore not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The effect of multiple interactions in the same and neighboring bunch crossings (pileup) was modeled by overlaying each simulated hard-scattering event with inelastic pp events generated with Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='186 using the NNPDF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3lo PDF set and the A3 set of tuned parameters [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Additionally, for all HH signal samples, heavy-flavor decays were modeled using EvtGen 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 4 Object reconstruction Primary vertices from pp interactions are reconstructed [78] using at least two charged-particle tracks with transverse momentum (𝑝T) above 500 MeV measured with the ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The vertex with the largest sum of squared track momenta (� 𝑝2 T) is taken as the hard-scatter primary vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Hadronic jets are reconstructed using the anti-𝑘𝑡 algorithm with radius parameter 𝑅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The jet clustering uses particle-flow objects as inputs [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Particle-flow objects are charged-particle tracks matched to the hard-scatter vertex and calorimeter energy clusters after applying an energy subtraction algorithm that removes the calorimeter deposits associated with good-quality tracks from any vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The tracking information helps to improve the energy resolution of the calorimeter clusters and reduce the impact from pileup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The momenta of reconstructed jets are calibrated in a multistep procedure [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Jets with 𝑝T < 60 GeV and |𝜂| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4 must also satisfy a requirement based on the output of the multivariate “jet vertex tagger” (JVT) algorithm [81], which is used to identify and reject jets in which much of the energy originates from pileup interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Correction factors are applied to the simulated events to compensate for differences between the JVT efficiencies in data and simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In the HH → bbbb analysis, jets are discarded if they fail the “Tight” JVT working point, corresponding to an average efficiency of 96% for jets from the hard-scatter vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Jets with radius parameter 𝑅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4 are also reconstructed from topological clusters of energy deposits in the calorimeter [82] and calibrated in the same way as the jets reconstructed from particle-flow objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' These jets are used exclusively for the purpose of applying quality criteria to identify events which are consistent with noise in the calorimeter or noncollision background [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Events containing at least one such jet with 𝑝T > 20 GeV, satisfying the JVT requirement, but not these quality criteria, are rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The identification of jets originating from 𝑏-quarks is performed by the DL1r algorithm [84], which is applied to all jets with |𝜂| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' DL1r is based on a multivariate classification technique combining 8 information from the impact parameters of ID tracks, the presence of displaced secondary vertices, and the reconstructed flight paths of 𝑏- and 𝑐-hadrons inside the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The DL1r working point used in the HH → bbbb analysis is the one that gives 77% efficiency for jets associated with true 𝑏-hadrons in simulated tt events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' At this working point, the light-jet (charm-jet) rejection measured in tt simulation is about a factor of 130 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The calibration of the DL1r algorithm is performed separately for each jet type [85, 86] and correction factors are derived and applied to the simulated samples to compensate for differences between the 𝑏-tagging efficiencies in data and simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Muons are reconstructed by matching ID tracks with either MS tracks or aligned individual hits in the MS and performing a combined track fit [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' They are required to have 𝑝T > 4 GeV and |𝜂| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5, and to satisfy “Medium” identification criteria based on track-quality variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Muons are used only to apply energy corrections to jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' A momentum correction is applied to 𝑏-tagged jets to account for energy lost to soft out-of-cone radiation and to muons and neutrinos in semileptonic 𝑏-hadron decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This correction follows the procedure used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [88] and consists of two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' First, a search is performed for muons located near the jet which fall within a cone of variable size Δ𝑅(𝜇, jet) < min �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='04 + 10/𝑝𝜇 T GeV� around the jet axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' If a muon is found, its four-momentum is added to that of the jet, and the energy deposited in the calorimeter by the muon is subtracted from the jet to avoid double counting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' this is computed according to the description in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In the second step, a global scale factor is applied to each 𝑏-tagged jet according to its 𝑝T and whether or not it has a muon associated with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' These scale factors are derived from simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 5 Analysis selection and categorization The analysis utilizes a set of criteria to select HH → bbbb candidate events, including dedicated requirements to separate events into orthogonal ggF and VBF signal regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' “Forward” and “central” jets are used with the following selection criteria: central jets: |𝜂| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 and 𝑝T > 40 GeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' forward jets: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 < |𝜂| < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 and 𝑝T > 30 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' An initial “preselection” is applied to all events, which requires at least four central jets with 𝑝T > 40 GeV, at least two of which are 𝑏-tagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' As described in Section 3, the events considered in this analysis are selected online through the 2𝑏2j or 2𝑏1j trigger signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In order to simplify the modeling of trigger efficiencies, a further selection is applied using offline kinematic quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Events are selected if they have a leading4 jet with 𝑝T > 170 GeV, a third leading jet with 𝑝T > 70 GeV, and pass the 2𝑏1j trigger, or if they fail either of the two jet-𝑝T requirements and pass the 2𝑏2j trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This selection step leads to about a 10% loss of signal efficiency, but enables the reliable calculation of simulation-to-data correction factors for estimating the trigger efficiency in the remaining HH → bbbb signal events, depending on which of the above two trigger classes they belong to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Events passing the above preselection are required to contain at least four central jets passing the b-tagging requirement outlined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The four highest-𝑝T b-tagged jets are chosen to reconstruct the decays of the two Higgs bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In about 75% of simulated signal events reaching this selection stage, these four jets can be matched one-to-one (within Δ𝑅 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3) to the four b-quarks from the decays of the Higgs bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 4 In this document, terms like “leading”, “subleading” etc for physics objects refer to the ordering of these objects in decreasing 𝑝T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 9 In signal events where this matching fails, one of the b-quarks from the Higgs boson decays typically produces a jet that is outside the analysis acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' From the four selected b-tagged jets, there are three possible combinatorial pairings to form the two Higgs boson candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Of those three configurations, the analysis selects the one in which the higher-𝑝T jet pair has the smallest Δ𝑅 separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In the simulated samples with SM coupling values, for which the analysis was mainly optimized, this method gives the correct pairing in around 90% of those signal events in which the four b-tagged jets are correctly matched to the b-quarks from the decays of the Higgs bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' While the pairing accuracy drops for values of the coupling modifiers 𝜅𝜆 and 𝜅2𝑉 that result in softer 𝑝T spectra for the produced Higgs bosons, this pairing method leads to a smoothly varying distribution of the expected background in the plane of the invariant masses of the two Higgs boson candidates, which facilitates the data-driven background estimation described in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Events are then subjected to additional selections designed to separate out those consistent with the VBF production mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' For this, events must contain at least two additional jets, central or forward;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' b-tagged jets are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The two jets forming the pair with the largest invariant mass (𝑚jj) are chosen as the “VBF jets”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The VBF jet pair is required to satisfy 𝑚jj > 1 TeV, and the pseudorapidity separation between the two jets, |Δ𝜂jj|, must satisfy |Δ𝜂jj| > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Lastly, the transverse component of the momentum vector sum of the two VBF jets and the four jets forming the Higgs boson candidates is required to be less than 65 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Events satisfying the above criteria enter the VBF signal region, while those failing to satisfy any of these criteria are considered further in the ggF signal region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Events satisfying either the ggF or VBF selections are required to satisfy additional selection criteria designed to reduce the background and improve the analysis sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In order to suppress the tt background, a top-veto discriminant 𝑋Wt is defined as: 𝑋Wt = min ������� � � �� 𝑚jj − 𝑚W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1𝑚jj �2 + � 𝑚jj𝑏 − 𝑚t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1𝑚jj𝑏 �2 ������� , where 𝑚𝑊 = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4 GeV and 𝑚𝑡 = 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 GeV are the nominal W boson and top quark masses, and 𝑚jj and 𝑚jj𝑏 are the invariant masses of W boson and top quark candidates formed from jet combinations in each event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The “minimum” refers to the minimum value from all possible jet combinations (of one b-tagged jet and two additional untagged jets) that would give a W boson candidate and a corresponding top candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1 in the denominators is chosen to approximate the experimental dijet mass resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The W boson candidates are formed from any pair of central jets in the event and the top quark candidates are then reconstructed by pairing the W boson candidates with any remaining b-tagged Higgs boson candidate jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The 𝑋Wt discriminant is designed to quantify the likelihood that an event contains a hadronic top quark decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Events with 𝑋Wt < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 are rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This reduces the tt background by a factor of about 2 in simulated events, for a small loss of signal efficiency, of around 15%, and a similar reduction in the non-tt, multijet background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In order to further reduce the overall background contamination, events in the ggF signal region are also required to have reconstructed Higgs bosons that satisfy a pseudorapidity separation |Δ𝜂HH| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' No such requirement is imposed in the VBF signal region, since SM VBF HH signal events tend to have a larger |Δ𝜂HH|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 10 A final analysis selection criterion to test the compatibility of events with the HH decay is applied in both the ggF and VBF selections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' A discriminant 𝑋HH is defined as: 𝑋HH = � � �� 𝑚H1 − 124 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1 𝑚H1 �2 + � 𝑚H2 − 117 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1 𝑚H2 �2 , where 𝑚𝐻1 and 𝑚𝐻2 are the masses of the leading and subleading reconstructed Higgs boson candidates respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The values of 124 GeV and 117 GeV in the 𝑋HH definition are chosen in accord with the centers of the 𝑚𝐻1 and 𝑚𝐻2 distributions for correctly paired signal events from simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Events are required to have 𝑋HH < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='6 to be included in the Signal Region (SR) of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Both the ggF and VBF signal regions are subdivided into a number of orthogonal categories in order to better isolate the HH signal and improve the analysis sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The 𝑋HH and |Δ𝜂HH| quantities are used to define six orthogonal ggF categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The categories are defined by two intervals in 𝑋HH, with boundaries at 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='95, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='6, and three in |Δ𝜂HH|, with boundaries at 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5, 1, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In the VBF signal region, two categories are defined using the |Δ𝜂HH| quantity, with the dividing boundary at |Δ𝜂HH| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The |Δ𝜂HH| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 category is more sensitive to VBF signals with non-SM couplings, while the |Δ𝜂HH| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 category is more sensitive to SM VBF production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The reconstructed invariant mass of the Higgs boson candidate pair, 𝑚HH, is used as the discriminating variable for all analysis regions and categories when extracting results, as detailed in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The 𝑚HH distribution is found to have significant separation power between background and signal, for all the different values of coupling modifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The binning of the 𝑚HH distributions may vary between categories and is chosen in order to both maintain discrimination power and limit the expected statistical uncertainty in each bin to less than approximately 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This 30% limit ensures that the assumptions used in the statistical procedure, outlined in Section 8, are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In the VBF signal region, only events with 𝑚HH > 400 GeV are considered, as the background in the lower 𝑚HH region was found to be inadequately modeled by the data-driven method described in Section 6 when applied to the 3b1f control data sample (also described in Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' For the ggF signal region, no requirements on 𝑚HH are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' All the selection steps of the analysis are summarized in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The yields in the data and the simulated signal samples for some typical coupling values are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This sample of data events is referred to as 4𝑏 events hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 6 Background modeling After the selection described above, about 90% of the background events come from multijet processes (excluding top quark production), with the approximately 10% remainder almost entirely composed of tt events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This background composition was determined by applying the full event selection to simulated samples of the various processes and comparing the yields with the total background estimate in the SR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' it is purely meant to be indicative and is not used for deriving any results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The background is modeled using the fully data-driven technique described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The background estimation makes use of an alternative set of events, which pass the same b-jet triggers and satisfy all the same selection criteria as the 4b events, with one difference: they are required to contain exactly two b-tagged jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This sample, referred to hereafter as “2b”, has about two orders of magnitude more events than the 4b sample, hence the presence of any HH → bbbb signal in it is negligible, making it 11 (1) Pass trigger class (2) ≥ 4 central jets (4) ≥ 6 central or forward jets (5) VBF Jets |Δηjj| > 3, mjj > 1 TeV (6) (∑pj)T < 65 GeV Yes Yes Yes Yes (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' ggF) |ΔηHH| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 No No No (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' VBF) XWt > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 Yes (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' ggF) XWt > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' ggF) XHH < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='6 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' VBF) XHH < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='6 (3) ≥ 4 b-tagged central jets Yes Yes Yes Yes ggF Selection VBF Selection VBF SR ggF SR Yes (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' VBF) mHH > 400 GeV Yes Yes Figure 3: A flowchart summarizing the nine selection criteria used for the VBF and ggF analysis selections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Events must satisfy selection criteria 1–3 in order to be considered for either analysis signal region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Events failing to satisfy any of the selection criteria 4–6 are considered for inclusion in the ggF signal region, while those satisfying selection criteria 4–6 are considered for the VBF signal region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Table 3: The yields of data and various example ggF and VBF HH signal models at each step of the analysis selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The “Preselection” entry denotes an initial selection requiring at least four jets with 𝑝T > 40 GeV, at least two of which are 𝑏-tagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Events which satisfy the “VBF selection” requirements are considered as part of the VBF signal region of the analysis, while the rest are considered for the ggF signal region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The signal yields are taken from simulation and are normalized by their theoretical cross-sections and the integrated luminosity of 126 fb−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Corrections for differences in the 𝑏-tagging efficiency and trigger acceptance between data and simulation are applied starting from the “Trigger class” requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Data ggF Signal VBF Signal SM 𝜿𝝀 = 10 SM 𝜿2𝑽 = 0 Common preselection Preselection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='70 × 108 530 7300 22 630 Trigger class 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='49 × 108 380 5300 16 410 ggF selection Fail VBF selection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='46 × 108 380 5200 14 330 At least 4 b-tagged central jets 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='89 × 106 86 1000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='9 65 |Δ𝜂HH| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='03 × 106 72 850 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='94 46 𝑋Wt > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='51 × 105 60 570 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='74 43 𝑋HH < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='6 (ggF signal region) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='62 × 104 29 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='24 23 VBF selection Pass VBF selection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='30 × 106 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='2 81 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='2 71 At least 4 b-tagged central jets 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='71 × 104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='74 28 𝑋Wt > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='18 × 104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='67 26 𝑋HH < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='02 × 102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='48 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='33 17 𝑚HH > 400 GeV (VBF signal region) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='57 × 102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='30 16 suitable for the background estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The jets selected to form the two Higgs boson candidates in the 2b events are the two b-tagged jets and the two untagged central jets with the highest 𝑝T (excluding the VBF 12 jets in the VBF categories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The kinematic properties of the 2𝑏 and 4𝑏 events are not expected to be identical, partly due to different processes contributing to the two samples, but also due to differences in the trigger acceptance and because the probability of tagging a b-jet varies as a function of jet 𝑝T and 𝜂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Therefore, a reweighting function is required, which, when applied to the 2𝑏 events, maps their kinematic distributions onto the corresponding 4𝑏 distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This function is derived using the 2𝑏 and 4𝑏 events in a Control Region (CR) surrounding the SR in the reconstructed (𝑚𝐻1, 𝑚𝐻2) plane and then applied to the 2𝑏 events in the SR to produce the background estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The “inner edge” of the CR is defined by 𝑋HH = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='6 and the “outer edge” by the circle: 𝑅CR = √︂� 𝑚H1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='05 · 124 GeV �2 + � 𝑚H2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='05 · 117 GeV �2 = 45 GeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The shift of the center of the above circle by a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='05, relative to 𝑋HH = 0, is found to be the optimal trade-off between having a good number of events outside of the SR and avoiding the low 𝑚𝐻1/𝑚𝐻2 regions, where the differences between 2𝑏 and 4𝑏 kinematic distributions are larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The CR is split into four roughly equal directional quadrants, defined by 45◦ and 135◦ lines passing through the SR center, (124, 117) GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The four quadrants are given labels based on compass directions: the upper quadrant QN, the lower QS, the left QW, and the right QE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The above lines also define four quadrants, with the same names as above, in the SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Events in CR QN and QS, hereafter referred to as CR1, are used to derive the reweighting function for the nominal background estimate, while an alternative reweighting function, derived from the CR events in QE and QW (referred to hereafter as CR2) is used to define a systematic uncertainty related to the reweighting function interpolation into the SR, as detailed in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The boundaries of the SR, CR1, and CR2 in the reconstructed (𝑚𝐻1, 𝑚𝐻2) plane are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The horizontal and vertical bands of lower event density around 80 GeV visible in these plots are caused by the 𝑋Wt selection criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 60 80 100 120 140 160 180 200 mH1 [GeV] 60 80 100 120 140 160 180 200 mH2 [GeV] ATLAS ps = 13 TeV, 126 fb 1 ggF selection, XWt > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 4b data SR CR1 CR2 100 200 300 400 500 Entries / (3 GeV)2 (a) 60 80 100 120 140 160 180 200 mH1 [GeV] 60 80 100 120 140 160 180 200 mH2 [GeV] ATLAS ps = 13 TeV, 126 fb 1 ggF selection, XWt > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 4b data SR CR1 CR2 100 200 300 400 500 Entries / (3 GeV)2 (b) Figure 4: The mass planes of the reconstructed Higgs boson candidates for the (a) ggF and (b) VBF signal regions of the analysis, shown for the 4b data events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In (a), the analysis selection up to step 8 (as outlined in Figure 3) of the ggF selection has been applied, while in (b), the analysis selection up to step 7 of the VBF selection has been applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The continuous red line describes the Signal Region (SR), the dashed line describes Control Region 1 (CR1) and the dotted line describes Control Region 2 (CR2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The reweighting function has the form: 13 𝑤(�𝑥) = 𝑝4𝑏(�𝑥) 𝑝2𝑏(�𝑥) , (1) where 𝑝4𝑏(�𝑥) and 𝑝2𝑏(�𝑥) are the probability density functions for 4𝑏 and 2𝑏 data, respectively, over a set of kinematic variables �𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The computation of 𝑤(�𝑥) is a density ratio estimation problem, for which a variety of approaches exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The method employed in this analysis is modified from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [90, 91] and makes use of an artificial neural network (NN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This NN is trained on 2𝑏 and 4𝑏 CR1 data (or CR2 data, for determining systematic uncertainties, as described Section 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The training minimizes the following loss function: L(𝑤(�𝑥)) = ∫ 𝑑�𝑥 �√︁ 𝑤(�𝑥)𝑝2𝑏(�𝑥) + 1 √︁ 𝑤(�𝑥) 𝑝4𝑏(�𝑥) � , The function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' (1) minimizes this loss by equalizing the contributions from the two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The kinematic variables used to make up �𝑥 are listed in Table 4 for the ggF and VBF signal regions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' they are among those kinematic variables that exhibit larger differences between the 2𝑏 and 4𝑏 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The NN used in the ggF signal region has three densely connected hidden layers of 50 nodes, each with a rectified linear unit activation function [92], and a single-node linear output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' A similar architecture is chosen for the NN used in the VBF signal region, except that only 20 nodes are used in each of the three hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This reflects the fact that the 2𝑏 and 4𝑏 sample sizes in the VBF signal region are nearly two orders of magnitude smaller than the corresponding ones in the ggF signal region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This is also the reason behind the choice to perform the NN training in the VBF signal region for all data-taking years together, with the year index as a one-hot encoded input feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 For the ggF signal region, a dedicated reweighting is derived for each year separately, which, thanks to the adequate sample sizes, deals better with the different levels of disparity between 2b and 4b distributions, due to the differences in the trigger conditions from year to year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Finally, in order to ensure that there are adequate numbers of 4b events for both the ggF and VBF NN trainings, these trainings are performed inclusively, before separating the events into the |Δ𝜂HH| categories (the CR events have 𝑋HH > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='6, hence it would not be possible to separate them into the 𝑋HH categories defined for the SR events).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Both |Δ𝜂HH| and 𝑋HH are found to be insensitive to the kinematic reweighting, and so the inclusive training is not expected to introduce any additional bias when separating the events into the various categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In order to estimate and mitigate the impact of the varying initial conditions and limited size of the training samples on the NN training, the deep ensembles technique [93] is used together with a bootstrap resampling [94] of the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This entails constructing a set of training datasets by sampling with replacement from the original dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In this analysis, this is approximated by the usage of different random training weights, following a Poisson distribution with 𝜇 = 1, for each event in each training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The NN is trained independently on each element of this set, using different initial conditions each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This results in an ensemble of reweighting functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Each reweighting function is further multiplied by a normalization factor, such that the number of reweighted 2𝑏 events is equal to the number of 4𝑏 events in the region where the NN is trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In this analysis, the ensembles comprise 100 reweighting functions each, hence 100 weights are calculated for each 2𝑏 event in the SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The background estimate uses the mean of these weights for each event, and the variation of the background predictions from the ensemble 5 One-hot encoding is a standard technique in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' For example, for the data-taking years in the VBF reweighting, instead of presenting the year numbers as input features to the NN, one-hot encoding uses three input features: (1, 0, 0) for 2016, (0, 1, 0) for 2017, and (0, 0, 1) for 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 14 Table 4: The set of input variables used for the 2b to 4b reweighting in the ggF and VBF channels respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' ggF VBF 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' log(𝑝T) of the 2nd leading Higgs boson candidate jet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' log(𝑝T) of the 4th leading Higgs boson candidate jet 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' log(Δ𝑅) between the closest two Higgs boson candidate jets 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' log(Δ𝑅) between the other two Higgs boson candidate jets 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Average absolute 𝜂 value of the Higgs boson candidate jets 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' log(𝑝T) of the di-Higgs system 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Δ𝑅 between the two Higgs boson candi- dates 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Δ𝜙 between jets in the leading Higgs boson candidate 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Δ𝜙 between jets in the subleading Higgs boson candidate 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' log(𝑋Wt) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Number of jets in the event 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Trigger class index as one-hot encoder 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Maximum dijet mass from the possible pairings of the four Higgs boson candi- date jets 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Minimum dijet mass from the possible pairings of the four Higgs boson candi- date jets 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Energy of the leading Higgs boson can- didate 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Energy of the subleading Higgs boson candidate 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Second-smallest Δ𝑅 between the jets in the leading Higgs boson candidate (from the three possible pairings for the leading Higgs candidate) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Average absolute 𝜂 value of the four Higgs boson candidate jets 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' log(𝑋Wt) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Trigger class index as one-hot encoder 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Year index as one-hot encoder (for years inclusive training) of reweighting functions is used to estimate a systematic uncertainty for the stability of the NN training procedure, as described in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The effect of the above reweighting procedure in CR1, where the reweighting function is derived, is illustrated in Figure 5 for the 𝑚HH distribution of the ggF-selected events and in Figure 6 for the 𝑋Wt distribution of the VBF-selected events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The reweighted “2b” distributions agree with the corresponding “4b” distributions to within about 10% for most of the phase space, with some larger deviations observed in bins near the tails of the distributions where fewer data events are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' A large number of additional kinematic variables were also studied before and after applying the reweighting in order to validate the performance of the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' For all variables, the level of agreement, as quantified by the 𝜒2 metric, either improves after the reweighting or, for variables where the “2b” and “4b” distributions are already similar, changes only slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The background modeling procedure was tested and found to produce good results in a large simulated tt sample and a much smaller sample of simulated (non-tt) multijet events in the SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The procedure was also tested in several control data samples orthogonal to the nominal event selection, where the presence of any HH signal is negligible and the 4𝑏 events in the corresponding SR can be compared with the reweighted SR 2𝑏 events without any bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' These samples, summarized in Table 5, include: (a) events satisfying all the 2𝑏/4𝑏 ggF selection criteria, with the difference that the |Δ𝜂HH| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 cut is inverted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' (b) events satisfying all the 2𝑏/4𝑏 selection criteria, except that the center of the SR (and hence also of CR1 and CR2) is shifted, to avoid any overlap with the nominal SR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' and (c) events that satisfy all the same 4𝑏 selection criteria, except that, in terms of b-tagging, they contain exactly three b-tagged jets, and the fourth jet is taken as the highest-𝑝T jet that fails a looser working point of the b-tagging algorithm (one that gives 85% 15 0 500 1000 1500 2000 2500 3000 3500 Events / 25 GeV ATLAS ps = 13 TeV, 2018 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='7 fb 1 ggF CR1 Normalized 2b Data Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Error 4b Data 300 400 500 600 700 800 900 1000 mHH [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 4b / 2b (a) 0 500 1000 1500 2000 2500 3000 3500 Events / 25 GeV ATLAS ps = 13 TeV, 2018 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='7 fb 1 ggF CR1 Normalized 2b Data Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Error 4b Data 300 400 500 600 700 800 900 1000 mHH [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 4b / 2b (b) Figure 5: Comparison of the 2b (yellow histogram with hatching) and 4b (black points with error bars) 𝑚HH distributions, for events in Control Region 1 (CR1) of the ggF signal region from the 2018 data: (a) before the kinematic reweighting of the 2b events, with only a normalization factor applied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' and (b) after the kinematic reweighting of the 2b events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The error bars indicate the statistical uncertainty of the 4b data, while the hatching indicates the statistical uncertainty of the 2b data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The latter is only the Poisson uncertainty of the 2b data, in (a), while in (b), it also includes the uncertainty from the bootstrap procedure described in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The hatching in (a) is narrower than the line width of the plotted histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' efficiency for b-jets in simulated tt events).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This sample, hereafter referred to as 3b1f, has about one order of magnitude more events than the 4𝑏 sample and a negligible amount of HH signal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' hence it is used to derive a nonclosure systematic uncertainty for the reweighting procedure, as discussed in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' No significant background modeling nonclosure was observed in the other control data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 7 Systematic uncertainties The uncertainties with the greatest impact on the analysis sensitivity are those arising from the data-driven background estimate described in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' These uncertainties have two main sources: the limited sample sizes in the CR and SR, and physical differences between the CR, where the 2b reweighting function is derived, and the SR, where it is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' As described in Section 6, the ensemble of 100 reweighting functions results in 100 separate background predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' An 𝑚HH histogram can be constructed from each of these predictions, and the standard deviation of the predictions in each bin is taken as the bootstrap uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The uncertainty is treated as uncorrelated across 𝑚HH bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' An additional statistical uncertainty results from the limited sample size of the 2b SR dataset in which the trained background reweighting network is applied to obtain the final background estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' A Poisson uncertainty is taken for each 𝑚HH bin, which is combined in quadrature with the bootstrap uncertainty described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' For the background estimate, the uncertainty component related to the kinematic differences between the SR and CR1 is evaluated by using alternative predictions from the CR2 region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Four alternative 16 0 25 50 75 100 125 150 175 200 Events / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 ATLAS ps = 13 TeV, 126 fb 1 VBF CR1 Normalized 2b Data Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Error 4b Data 2 3 4 5 6 7 8 9 10 XWt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 4b / 2b (a) 0 25 50 75 100 125 150 175 200 Events / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 ATLAS ps = 13 TeV, 126 fb 1 VBF CR1 Normalized 2b Data Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Error 4b Data 2 3 4 5 6 7 8 9 10 XWt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 4b / 2b (b) Figure 6: Comparison of the 2b (yellow histogram with hatching) and 4b (black points with error bars) 𝑋Wt distributions, for events in Control Region 1 (CR1) of the VBF signal region: (a) before the kinematic reweighting of the 2b events, with only a normalization factor applied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' and (b) after the kinematic reweighting of the 2b events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The error bars indicate the statistical uncertainty on the 4b data, while the hatching indicates the statistical uncertainty on the 2b data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The latter is only the Poisson uncertainty on the 2b data, in (a), while in (b), it also includes the uncertainty from the bootstrap procedure described in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' background estimates are produced by applying the CR1-derived weights to three of the SR quadrants, and CR2-derived weights to the one remaining SR quadrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' For example, one alternative background estimate is obtained by applying CR1-derived weights to QS, QE and QW, and CR2-derived weights to QN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Each of these four background predictions is symmetrized around the nominal 𝑚HH distribution to construct a two-sided uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Since the 𝑚HH distribution differs across the four SR quadrants, substituting the CR2-based prediction for the CR1-based prediction in each of the four SR quadrants separately and utilizing a four-component uncertainty gives the fit model greater flexibility to describe these 𝑚HH variations with finer granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In the ggF signal region, these uncertainties are taken to be uncorrelated across the datasets from the three different years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In both the ggF and VBF signal regions, the uncertainty is treated as correlated across the analysis categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' An additional closure uncertainty is estimated by applying the full background modeling procedure to the 3b1f sample instead of the 4b sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The predicted 3b1f 𝑚HH distribution in the various analysis categories is then compared with the observed 3b1f data in the SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' For the VBF signal region, no statistically significant difference between the prediction and observation is found, and hence no additional uncertainty is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' For the ggF signal region, an additional uncertainty is evaluated in each category from the observed differences between the predicted and observed 3b1f 𝑚HH distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' For 𝑚HH bins in which the predicted and observed values differ by less than 1𝜎, where 𝜎 is obtained from all other background modeling uncertainties combined, no additional uncertainty is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' For 𝑚HH bins where the predicted and observed values differ by more than 1𝜎, the amount beyond 1𝜎 is averaged with the corresponding amounts in the two adjacent bins, to limit the impact of statistical fluctuations, and is symmetrized around the nominal prediction to construct a two-sided uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This nonclosure uncertainty has a much smaller impact on the analysis sensitivity than the other sources of background modeling uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Several detector modeling uncertainties are evaluated and included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' These affect only the signal description, as the background is estimated entirely from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Uncertainties in the jet energy scale and resolution, 17 Table 5: A summary of all the data samples used in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' For each control sample, the variations from the nominal analysis are noted explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Data Sample Definition Usage Signal Region (SR) Events with 𝑋HH < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='6 Defines signal region in the 𝑚H1–𝑚H2 plane Control Region (CR) Events with 𝑋HH > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='6 and 𝑅CR < 45 GeV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='Defines control region in the 𝑚H1–𝑚H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='plane for background estimation (ggF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='and VBF) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='Shifted valida- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='tion regions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='Shift the center of the SR in the 𝑚H1–𝑚H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='plane to avoid overlap with the nominal SR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='Background estimation validation (ggF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='only) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='Events with at least 4 b-tagged central jets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='Final analysis sample (ggF and VBF) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='2b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='Events with exactly 2 b-tagged central jets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='plus at least two additional untagged central ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='jets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='Background estimation (ggF and VBF) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3b1f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='Events with exactly 3 b-tagged central jets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='plus at least one central jet failing a looser ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='b-tagging requirement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='Background estimation validation (ggF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='and VBF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' additional background mod- eling uncertainty (ggF only) Reverse |Δ𝜂HH| 2b and 4b events with |Δ𝜂HH| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 Background estimation validation (ggF only) as well as the JVT, are treated according to the prescription in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [80, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Additional uncertainties arising from the correction of the simulated pileup distribution are treated according to the prescription in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Uncertainties in the 𝑏-tagging efficiency are treated according to the prescription in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Uncertainties in the trigger efficiencies are evaluated from measurements of per-jet online efficiencies for both jet reconstruction and 𝑏-tagging, which are used to compute event-level uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' These are then applied to the simulated events as overall weight variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The uncertainty in the integrated luminosity used in this analysis is in the range 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0%–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4% for the three years of data taking and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='7% for the entire dataset [97], obtained using the LUCID-2 detector for the primary luminosity measurements [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Several sources of theoretical uncertainty affecting the signal models are considered as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Uncertainties due to modeling of the parton shower and underlying event are evaluated by comparing results between two generators for these parts of the calculation: the nominal Pythia 8 and the alternative Herwig 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This is found to have an effect of roughly 10% on the ggF and VBF signal acceptances, and a negligible impact on the shape of the 𝑚HH distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The parton showering uncertainty is derived within each analysis SR category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' the uncertainty is observed to reach approximately 40% for a given 18 production mode in some categories in which the acceptance is small for that mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Uncertainties in the matrix element calculation are evaluated by raising and lowering the factorization and renormalization scales used in the generator by a factor of two, both independently and simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This results in an effect of typically 2% for both ggF and VBF, with a maximum effect of about 6% in certain analysis categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' PDF uncertainties are evaluated using the PDF4LHC_NLO_MC set [54] by calculating the signal acceptance for each replica and taking the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The magnitude of this uncertainty is typically found to be less than 1% in both the ggF and VBF signal acceptances, with a maximum magnitude of approximately 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Theoretical uncertainties in the H → bb branching ratio [14] are included, amounting to an approximately 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5% overall uncertainty in the signal normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The dependence of the branching ratio uncertainty on 𝜅𝜆 is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Theoretical uncertainties in the ggF and VBF HH cross-sections arising from uncertainties in the PDF and 𝛼s, as well as the choice of renormalization scheme and the scale of the top quark’s mass, are taken from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [11, 14, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The cross-section uncertainties are included in the derivation of the upper limits on the ggF, VBF, and combined HH signal strengths, as well as the likelihood-based constraints on the values of the 𝜅𝜆 and 𝜅2𝑉 modifiers, as presented in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' An additional signal modeling systematic uncertainty is evaluated for the SMEFT and HEFT measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The 𝑚HH spectra of reweighted SMEFT/HEFT signal samples are compared against explicitly generated samples for a select number of coefficient variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' A two-component normalization uncertainty is derived by taking the average of the relative deviations across the 𝑚HH bins in the ranges of 280 < 𝑚HH < 936 GeV and 𝑚HH > 936 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The use of separate components in the low- and high-𝑚HH regions prevents the level of agreement in the more populated low-𝑚HH region from overconstraining the uncertainty in the more sparsely populated high-𝑚HH region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 8 Results The analysis results are obtained using a maximum-likelihood fit performed in bins of reconstructed 𝑚HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' For the ggF signal region, the fit is performed simultaneously across the different data-taking years (2016, 2017, and 2018), while for the VBF signal region, the fit is performed inclusively on the data from all years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The likelihood function used to construct the test statistic has a standard form, consisting of a product of Poisson distributions for the yields in each bin and constraint functions for nuisance parameters describing systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' For uncertainties due to the limited sample size in data or simulation, the constraint is a Poisson distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' For all other systematic uncertainties, the constraint is a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Where systematic uncertainties are deemed to be uncorrelated, independent nuisance parameters are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Uncertainties in the luminosity and signal modeling are treated as fully correlated between the analysis categories and, for ggF, the data-taking years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Each component of the quadrant-derived uncertainty covering the kinematic differences between the SR and CR1 regions is correlated across the data-taking years for the ggF region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The components are correlated across analysis categories within the ggF and VBF signal regions, but not between the ggF and VBF signal regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' All other uncertainties in the background model are treated as uncorrelated across the different categories and data-taking years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The statistical model is implemented using RooFit [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The hypothesis of the presence of a signal is tested using the profile likelihood ratio [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The signal strength of the combined ggF and VBF signal process, 𝜇ggF+VBF (= 𝜎ggF+VBF/𝜎SM ggF+VBF), is chosen as the parameter of interest (POI) and is a free parameter in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The relative contributions of the ggF and VBF 19 signals to the total signal model are fixed to their predicted values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The profile likelihood ratio takes the following form: −2Δ ln 𝜆(𝜇) = −2 ln � 𝐿(𝜇, ˆˆ𝜽) 𝐿( ˆ𝜇, ˆ𝜽) � , where 𝜇 is the POI and 𝜽 represents the nuisance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The numerator represents the conditional maximum-likelihood fit, in which the nuisance parameters are set to their profiled values ˆˆ𝜽 for which the likelihood is maximized for a fixed value of 𝜇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The denominator represents the unconditional likelihood fit, where both 𝜇 and 𝜽 are set to the values which jointly maximize the likelihood, ˆ𝜇 and ˆ𝜽, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The observed distributions in 𝑚HH, as well as the predicted background and example signal shapes, are presented in Figure 7 for each of the six ggF categories (with all data-taking years combined, for presentation purposes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The distributions of the expected background are obtained using the best-fit values of the nuisance parameters in the fit to the data with the background-only hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The corresponding 𝑚HH distributions in the two VBF categories are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The signal shape for 𝜅2𝑉 = 0 in Figure 8(a) clearly shows the impact of the divergences in Figures 2(b) and 2(c) not canceling out as discussed in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' While the deviations from the SM studied in this analysis are below the level that violates unitarity, this behavior makes the VBF topology in this analysis particularly sensitive to 𝜅2𝑉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The observed number of data events, predicted number of background events, and expected number of signal events for the SM ggF and VBF signals are summarized for each of the analysis categories in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Table 6: The yields in each analysis category of the data, expected background, and expected SM ggF and VBF signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The expected background yields are obtained using a fit to the data with the background-only hypothesis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' the quoted uncertainties are the sum in quadrature of all the per-bin systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The expected signal yields are obtained from simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Category Data Expected ggF Signal VBF Signal Background SM SM ggF signal region |Δ𝜂HH| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5, 𝑋HH < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='95 1940 1935 ± 25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='038 |Δ𝜂HH| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5, 𝑋HH > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='95 3602 3618 ± 37 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 < |Δ𝜂HH| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, 𝑋HH < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='95 1924 1874 ± 21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 < |Δ𝜂HH| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, 𝑋HH > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='95 3540 3492 ± 35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='040 |Δ𝜂HH| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, 𝑋HH < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='95 1880 1739 ± 22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='043 |Δ𝜂HH| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, 𝑋HH > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='95 3285 3212 ± 37 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='041 VBF signal region |Δ𝜂HH| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 116 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='090 |Δ𝜂HH| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 241 230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='6 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='21 An upper limit on the combined ggF and VBF HH signal strength 𝜇ggF+VBF is computed using the asymptotic formula [101] and based on the CLs method [102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The observed (expected) 95% CL upper limit on 𝜇ggF+VBF is found to be 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The expected upper limits are obtained using a background-only hypothesis, excluding a 𝐻𝐻 signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The upper limit on the combined 𝜇ggF+VBF, as well as upper limits on the individual 𝜇ggF (= 𝜎ggF/𝜎SM ggF) and 𝜇VBF (= 𝜎VBF/𝜎SM VBF), are summarized in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' For the individual 𝜇ggF and 𝜇VBF limits, the results are derived by treating the other production mode (VBF when placing limits on 𝜇ggF, and vice-versa) as a background process, with its normalization only loosely constrained in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Compared to the previous ATLAS measurement of ggF HH production in the bbbb decay 20 0 100 200 300 400 500 Events / 25 GeV ATLAS p s = 13 TeV, 126 fb 1 ggF Signal Region | HH| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5, XHH < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='95 Post-Fit Background Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' + Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Error 4b Data 400 x SM HH 200 x = 6 HH 300 400 500 600 700 800 900 1000 mHH [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 Data/Pred (a) 0 100 200 300 400 Events / 25 GeV ATLAS p s = 13 TeV, 126 fb 1 ggF Signal Region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 < | HH| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, XHH < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='95 Post-Fit Background Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' + Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Error 4b Data 400 x SM HH 200 x = 6 HH 300 400 500 600 700 800 900 1000 mHH [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 Data/Pred (b) 0 50 100 150 200 250 300 350 Events / 25 GeV ATLAS p s = 13 TeV, 126 fb 1 ggF Signal Region | HH| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, XHH < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='95 Post-Fit Background Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' + Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Error 4b Data 400 x SM HH 200 x = 6 HH 300 400 500 600 700 800 900 1000 mHH [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 Data/Pred (c) 0 200 400 600 800 Events / 25 GeV ATLAS p s = 13 TeV, 126 fb 1 ggF Signal Region | HH| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5, XHH > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='95 Post-Fit Background Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' + Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Error 4b Data 400 x SM HH 200 x = 6 HH 300 400 500 600 700 800 900 1000 mHH [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 Data/Pred (d) 0 100 200 300 400 500 600 700 800 Events / 25 GeV ATLAS p s = 13 TeV, 126 fb 1 ggF Signal Region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 < | HH| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, XHH > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='95 Post-Fit Background Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' + Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Error 4b Data 400 x SM HH 200 x = 6 HH 300 400 500 600 700 800 900 1000 mHH [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 Data/Pred (e) 0 100 200 300 400 500 600 700 Events / 25 GeV ATLAS p s = 13 TeV, 126 fb 1 ggF Signal Region | HH| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, XHH > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='95 Post-Fit Background Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' + Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Error 4b Data 400 x SM HH 200 x = 6 HH 300 400 500 600 700 800 900 1000 mHH [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 Data/Pred (f) Figure 7: Distributions of the reconstructed 𝑚HH in data (shown by the black points) and the estimated background (shown by the yellow histograms), in each of the six |Δ𝜂HH|, 𝑋HH categories in the ggF signal region: (a) |Δ𝜂HH| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5, 𝑋HH < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='95;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 < |Δ𝜂HH| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, 𝑋HH < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='95;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' (c) |Δ𝜂HH| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, 𝑋HH < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='95;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' (d) |Δ𝜂HH| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5, 𝑋HH > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='95;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 < |Δ𝜂HH| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, 𝑋HH > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='95;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' and (f) |Δ𝜂HH| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, 𝑋HH > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The contributions from the different data-taking years are combined in each category for presentation purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The hatching shows the total uncertainty of the background estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The distribution of the expected background is obtained using the best-fit values of the nuisance parameters in the fit to the data with the background-only hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Distributions of the SM and 𝜅𝜆 = 6 signal models are overlaid, scaled so as to be visible on the plot, and the scaling for each signal model is the same across the six categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The lower panels show the ratio of the observed data yield to the predicted background in each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Events in the underflow and overflow bins are counted in the yields of the initial and final bins respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' channel (using 27 fb−1 of early Run 2 data) [26], the upper limit on the ggF cross-section is over 50% lower, with approximately 20% of this improvement arising from advances in analysis techniques and object reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Similarly, compared to the previous ATLAS measurement of VBF HH production in the bbbb decay channel, which used 126 fb−1 of data collected between 2016 and 2018 [27], the upper limit on the VBF HH cross-section is over 75% lower, with this improvement arising entirely from advances in analysis technique and object reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The total uncertainty in the upper limit of the cross-section is dominated by the uncertainty sources related to the background modeling procedure and theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' With only the statistical uncertainties of the reweighted 2b data, observed 4b data, and simulated signal samples included in the fit, the expected upper limit on 𝜇ggF+VBF is found to be 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 times the SM prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Including the uncertainty sources resulting from the background estimation (the bootstrap uncertainty, the uncertainty from the kinematic differences between the SR and CR1, and, in the ggF signal region, the 3b1f nonclosure uncertainty), the expected upper limit on 𝜇ggF+VBF is relaxed to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1 times the SM prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The further reduction of sensitivity to the value of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1, as quoted in Table 7, is driven primarily by the uncertainties arising from theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The relative impact of the various sources of systematic uncertainty on the 21 0 10 20 30 40 50 60 70 80 Events / 50 GeV ATLAS p s = 13 TeV, 126 fb 1 VBF Signal Region | HH| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 Post-Fit Background Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' + Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Error 4b Data 1500 x SM HH 50 x = 6 HH 10 x 2V = 0 HH 400 500 600 700 800 900 mHH [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 Data/Pred (a) 0 10 20 30 40 50 60 Events / 50 GeV ATLAS p s = 13 TeV, 126 fb 1 VBF Signal Region | HH| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 Post-Fit Background Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' + Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Error 4b Data 1000 x SM HH 200 x = 6 HH 40 x 2V = 0 HH 400 600 800 1000 1200 1400 mHH [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 Data/Pred (b) Figure 8: Distributions of the reconstructed 𝑚HH in data (shown by the black points), the estimated background (shown by the yellow histograms), in each of the two |Δ𝜂HH| categories in the VBF signal region: (a) |Δ𝜂HH| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 and (b) |Δ𝜂HH| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The hatching shows the total uncertainty of the background estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The distribution of the expected background is obtained using the best-fit values of the nuisance parameters in the fit to the data with the background-only hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Distributions for three choices of couplings are shown: the SM, 𝜅𝜆 = 6, and 𝜅2𝑉 = 0 (with all other couplings set to their SM values in the last two models), scaled so as to be visible on the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The lower panels show the ratio of the observed data yield to the predicted background in each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Events in the overflow bins are counted in the yields of the final bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Table 7: The observed and expected upper limits on the SM ggF HH production cross-section 𝜎ggF, SM VBF HH production cross-section 𝜎VBF, and combined SM ggF and VBF HH production cross-section 𝜎ggF+VBF at the 95% CL, expressed as multiples of the corresponding SM cross-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The expected values are shown with corresponding one- and two-standard-deviation error bounds, and they are obtained using a background-only fit to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' When extracting the limits on 𝜎ggF+VBF, the relative contributions of ggF and VBF production to the total cross-section are fixed to the SM prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Observed Limit −2𝝈 −1𝝈 Expected Limit +1𝝈 +2𝝈 𝜇ggF 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='6 𝜇VBF 130 70 100 130 190 280 𝜇ggF+VBF 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='2 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1 expected upper limit on 𝜇ggF+VBF is summarized in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Constraints are placed on the 𝜅𝜆 and 𝜅2𝑉 modifiers using two different interpretations, the first named the “95% CL” method and the second named the “profile likelihood ratio” method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The former uses the signal strength 𝜇 as the POI, while the latter uses the vector of coupling modifiers 𝜿 = (𝜅𝜆, 𝜅2𝑉 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The 95% CL method allows for interpretation as a traditional search for an arbitrarily normalized set of signals with different shapes against an estimated background, while the profile likelihood ratio method allows for interpretation as to whether the data are compatible with the specific cross-section and shape predictions of the 𝜅 framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The 95% CL results presented here offer a consistent comparison with previous ATLAS HH measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The constraints obtained from the two interpretations are not expected to be identical, as the two strategies employ slightly different physical assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In the profile likelihood 22 Table 8: Breakdown of the dominant systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The impact of the uncertainties on the expected upper limit on 𝜇ggF+VBF when re-evaluating the profile likelihood ratio after fixing the nuisance parameter(s) in question to its (their) best-fit value(s), while all remaining nuisance parameters remain free to float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The impact is shown in %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Only (groups of) systematic uncertainties that have an impact of at least 1% are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The impact of each experimental source of systematic uncertainty described in the text, as well as of all of them together, is less than 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Source of Uncertainty Δ𝜇/𝜇 Theory uncertainties Theory uncertainty in signal cross-section −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0% All other theory uncertainties −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4% Background modeling uncertainties Bootstrap uncertainty −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1% CR to SR extrapolation uncertainty −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5% 3b1f nonclosure uncertainty −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0% ratio interpretation, the signal strength is fixed to the prediction obtained for a specific coupling modifier configuration, while for the 95% CL interpretation, the signal strength is allowed to float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The profile likelihood ratio method utilizes a hypothesis consisting of the predicted background plus the SM HH signal, while the 95% CL results utilize a hypothesis containing only the predicted background and no HH signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Given the relatively small size of the SM HH signal compared to the predicted background, the use of different hypotheses is not expected to have a significant effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Additionally, 2𝜎-level constraints are quoted from the profile likelihood ratio interpretation, as opposed to 95% CL constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The 95% CL constraints on 𝜅𝜆 and 𝜅2𝑉 are obtained by determining the 95% CL upper limits on the cross-section as a function of these coupling modifiers, 𝜇ggF+VBF(𝜅𝜆, 𝜅2𝑉 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Values of the coupling modifiers (𝜅𝜆, 𝜅2𝑉 ) are excluded if the predicted cross-section of the signal model obtained with that configuration is excluded at the 95% CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The H → bb branching ratio is fixed to the SM prediction in the likelihood fit and any dependence on 𝜅𝜆 is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Upper limits on the HH signal strength as a function of 𝜅𝜆 and 𝜅2𝑉 are shown in Figure 9, and the exclusion boundaries are summarized in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' With the values of the other modifiers (𝜅𝑉 and either 𝜅2𝑉 or 𝜅𝜆, respectively) fixed to their SM value of 1, values of 𝜅𝜆 beyond [−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='9, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1] and values of 𝜅2𝑉 beyond [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='03, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='11] are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Table 9: The observed and expected constraints on the 𝜅𝜆 and 𝜅2𝑉 coupling modifiers at 95% CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' For each modifier, the constraints were extracted with all other modifiers fixed to the SM prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Parameter Expected Constraint Observed Constraint Lower Upper Lower Upper 𝜅𝜆 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='8 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='9 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1 𝜅2𝑉 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='11 Figure 10 shows the 95% CL exclusion limits in the two-dimensional plane of the 𝜅𝜆–𝜅2𝑉 modifier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The alternative coupling modifier constraints are obtained using the profile likelihood ratio interpretation, with the coupling modifiers 𝜿 = (𝜅𝜆, 𝜅2𝑉 ) as the POIs, rather than the signal strength 𝜇: −2Δ ln 𝐿(𝜿) = −2 ln � 𝐿(𝜿, ˆˆ𝜽) 𝐿( ˆ𝜿, ˆ𝜽) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 23 20 15 10 5 0 5 10 15 20 ( 2V=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, V=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0) 101 102 103 104 105 106 95% CL Limit on ggF + VBF HH [fb] ATLAS p s = 13 TeV, 126 fb 1 Combined ggF and VBF Regions Observed: [-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='9, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1] Expected: [-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='6, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='8] Observed Limit Expected Limit Expected Limit ±1 Expected Limit ±2 Theory Prediction SM Prediction (a) 2 1 0 1 2 3 4 2V ( =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, V=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0) 100 101 102 103 104 105 95% CL Limit on VBF HH [fb] ATLAS p s = 13 TeV, 126 fb 1 Combined ggF and VBF Regions Observed: 2V [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='03, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='11] Expected: 2V [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='05, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='12] Observed Limit Expected Limit Expected Limit ±1 Expected Limit ±2 Theory Prediction SM Prediction (b) Figure 9: The observed 95% CL exclusion limits as a function of (a) 𝜅𝜆 (obtained using the signal strength 𝜇ggF+VBF as the POI) and (b) 𝜅2𝑉 (obtained using the signal strength 𝜇VBF as the POI) from the combined ggF and VBF signal regions, as shown by the solid black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In each case, the value of the other modifier is fixed to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The blue and yellow bands show respectively the 1𝜎 and 2𝜎 bands around the expected exclusion limits, which are shown by the dashed black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The expected exclusion limits are obtained using a fit to the data with the background-only hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The dark red line shows in (a) the predicted combined ggF and VBF HH cross-section as a function of 𝜅𝜆 and in (b) the predicted VBF HH cross-section as a function of 𝜅2𝑉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The dark pink bands surrounding the predicted cross-section lines indicate the theoretical uncertainty of the cross-section, as taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The band in (b) is smaller than the width of the plotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 2 0 2 4 2V 20 10 0 10 20 ATLAS p s = 13 TeV, 126 fb 1 V=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 Observed limit (95% CL) Expected ±1 ±2 Standard Model (a) Figure 10: The observed 95% CL exclusion limit obtained using the signal strength 𝜇ggF+VBF as the POI in the two-dimensional 𝜅𝜆 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 𝜅2𝑉 space, obtained from the combined ggF and VBF signal model, as shown by the solid black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The blue and yellow bands show respectively the 1𝜎 and 2𝜎 bands around the expected exclusion limits, which are shown by the dashed black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The star denotes the SM prediction (𝜅𝜆 = 𝜅2𝑉 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' A scan of the profile likelihood ratio is performed as a function of the coupling modifiers at discrete points to produce the curves shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The best-fit value of 𝜅𝜆 is found to be 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='2 from the profile likelihood scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The observed pull of the best-fit 𝜅𝜆 value away from the SM value is due to a slight excess in the observed data in the ggF signal region, specifically in the low-𝑚HH range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The particular signal model in which 𝜅𝜆 is close to 6 is favored due to a balance between two competing effects: the 𝑚HH 24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 ( 2V=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, V=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0) 0 2 4 6 8 10 2 ln(L) ATLAS ps = 13 TeV, 126 fb 1 Combined ggF and VBF Regions Expected 2 constraints: [-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4] Observed 2 constraints: [-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3] Best fit = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='2 Observed Expected ±2 (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 2V ( =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, V=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0) 0 5 10 15 20 25 30 35 2 ln(L) ATLAS ps = 13 TeV, 126 fb 1 Combined ggF and VBF Regions Expected 2 constraints: 2V [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1] Observed 2 constraints: 2V [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1] Best fit 2V = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 Observed Expected ±2 (b) Figure 11: The observed profile likelihood ratio scans for the (a) 𝜅𝜆 and (b) 𝜅2𝑉 coupling modifiers, shown by the solid black line, using the coupling modifiers 𝜿 as the POIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In each case, the value of the other parameter is fixed to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The dashed blue line shows the expected profile likelihood ratio, as obtained using a fit to the data with the background-only hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The pink line indicates the 2𝜎 exclusion boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 2 1 0 1 2 3 4 2V 20 15 10 5 0 5 10 15 20 25 ATLAS p s = 13 TeV, 126 fb 1 Combined ggF and VBF Regions Observed ±2 Observed ± 1 SM Prediction Best Fit (a) 2 1 0 1 2 3 4 2V 20 15 10 5 0 5 10 15 20 25 ATLAS p s = 13 TeV, 126 fb 1 Combined ggF and VBF Regions Expected ±2 Expected ± 1 SM Prediction (b) Figure 12: (a) The observed profile likelihood ratio exclusion limits for the two-dimensional 𝜅𝜆 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 𝜅2𝑉 modifier space, shown by the solid dark purple line at the 1𝜎 level and the dashed turquoise line at the 2𝜎 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The black cross denotes the best-fit values of (𝜅𝜆, 𝜅2𝑉 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The expected exclusion limits are presented in (b), where the solid pink line denotes the 1𝜎-level exclusion and the dashed orange line denotes the 2𝜎-level exclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' For both the expected and observed limit plots, the black star indicates the SM prediction (𝜅𝜆 = 𝜅2𝑉 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' spectrum becomes softer as 𝜅𝜆 increases away from the SM, but the cross-section also grows beyond the magnitude of the excess as 𝜅𝜆 increases much further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' This slight excess also results in the deviation of the observed limits in Figure 11 from the expected limits by about 1𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' No such excess is observed in the VBF signal region, and the best-fit value of 𝜅2𝑉 from the likelihood scan is found to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' With the values of the other modifiers (𝜅𝑉 and either 𝜅2𝑉 or 𝜅𝜆, respectively) fixed to their SM value of 1, the observed (expected) 2𝜎 allowed range for 𝜅𝜆 is found to be [−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3] ([−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4]) and the corresponding range for 𝜅2𝑉 is [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1] ([−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 25 The exclusion constraints obtained using the profile likelihood ratio method are also presented in the two-dimensional 𝜅𝜆–𝜅2𝑉 coupling modifier space, similarly to the 95% CL constraints described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The excluded regions are presented in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' With both modifiers able to float in the two-dimensional fit that combines both the ggF and VBF signal regions, the fit converges to 𝜅𝜆 and 𝜅2𝑉 values slightly different from the ones where the minimum is found in the fits with a single parameter free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' In addition to constraints on the ggF and VBF HH cross-sections and the 𝜅𝜆 and 𝜅2𝑉 coupling modifiers, constraints for relevant coefficients can be derived from the ggF selection of the analysis in the SMEFT and HEFT frameworks, as outlined in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The VBF HH process was ignored for both the SMEFT and HEFT results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' including the VBF HH process as a background was found to have a negligible effect on the extracted parameter limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The slight dependence of the H → bb branching fraction on the SMEFT and HEFT coefficients is also ignored, as the impact on the analysis sensitivity is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Constraints on the SMEFT coefficients are extracted by considering the 95% CL exclusion of the cross-section as a function of SMEFT parameter, as was done for the 𝜅𝜆 and 𝜅2𝑉 constraints discussed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The extracted constraints on individual parameters in the scenario where the other parameters are fixed to 0 are summarized in Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Limits approaching or exceeding ±4𝜋 should be interpreted with caution because of the potential impact from effects such as missing higher-order model contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The exclusion limits are also presented in two-dimensional SMEFT coefficient subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The exclusion limits for each coefficient versus the 𝑐H coefficient (with the remaining three coefficients fixed to 0) are shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The upper limits on the HEFT ggF HH production cross-section in the seven benchmark models are presented in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The spread of sensitivity between the seven benchmark models reflects the different signal kinematics and, hence, shapes of the signal 𝑚HH distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The different variation between observed and expected limits is linked to the slight excess observed in the low 𝑚HH region, as discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The red crosses in Figure 14 indicate the predicted HH cross-sections from the respective benchmark models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' As can be seen, BM3, BM5 and BM7 are observed to be excluded with more than 95% confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Constraints are placed on the values of 𝑐𝑔𝑔HH and 𝑐𝑡𝑡HH, with all other HEFT coefficients fixed to SM values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The observed (expected) constraints on 𝑐𝑔𝑔HH are found to be [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='36, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='78] ([−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='42, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='75]), while the observed (expected) constraints on 𝑐𝑡𝑡HH are found to be [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='51] ([−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='46, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Table 10: The extracted upper and lower limits on the SMEFT parameters to which the analysis is sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' For each parameter, the constraints are provided assuming the other parameters are fixed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The VBF HH process is ignored for this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Parameter Expected Constraint Observed Constraint Lower Upper Lower Upper 𝑐H −20 11 −22 11 𝑐H𝐺 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='049 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='060 𝑐H□ −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='9 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='9 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 𝑐𝑡H −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4 −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='2 𝑐𝑡𝐺 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='94 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='15 26 40 30 20 10 0 10 20 30 cH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='15 cHG ATLAS p s = 13 TeV, 126 fb 1 cH =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, ctG=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, ctH=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 Observed Limit (95% CL) Expected Limit (95% CL) Expected Limit ±1 Expected Limit ±2 SM Prediction (a) 40 30 20 10 0 10 20 30 cH 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5 ctG ATLAS p s = 13 TeV, 126 fb 1 cH =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, ctH=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, cHG=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 Observed Limit (95% CL) Expected Limit (95% CL) Expected Limit ±1 Expected Limit ±2 SM Prediction (b) 40 30 20 10 0 10 20 30 cH 20 10 0 10 20 30 40 ctH ATLAS p s = 13 TeV, 126 fb 1 cH =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, ctG=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, cHG=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 Observed Limit (95% CL) Expected Limit (95% CL) Expected Limit ±1 Expected Limit ±2 SM Prediction (c) 40 30 20 10 0 10 20 30 cH 30 20 10 0 10 20 30 40 50 cH ATLAS p s = 13 TeV, 126 fb 1 ctH=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, ctG=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, cHG=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0 Observed Limit (95% CL) Expected Limit (95% CL) Expected Limit ±1 Expected Limit ±2 SM Prediction (d) Figure 13: The observed 95% CL exclusion limits on the SMEFT coefficients in the two-dimensional spaces (a) 𝑐H𝐺 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 𝑐H, (b) 𝑐𝑡𝐺 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 𝑐H, (c) 𝑐𝑡H vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 𝑐H, and (d) 𝑐H□ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 𝑐H, shown by the solid black lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The dashed black line indicates the expected 95% CL exclusion limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The shaded blue band indicates the ±1𝜎 uncertainty of the exclusion limits, while the yellow band indicates the ±2𝜎 uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The values of the other three coefficients for each plot are fixed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The VBF HH process is ignored for this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 27 102 103 ggF, HH [fb] SM BM1 BM2 BM3 BM4 BM5 BM6 BM7 170 (230) 290 (200) 360 (250) 110 (74) 130 (130) 110 (95) 190 (180) 88 (71) Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' (Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=') ATLAS ps = 13 TeV, 126 fb 1 Observed Expected Theory Prediction ±1 ±2 Figure 14: The observed 95% CL exclusion limits on the ggF HH production cross-section in the SM and each of the seven HEFT benchmark models, given by the solid black points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The blue and yellow bands show respectively the 1𝜎 and 2𝜎 bands around the expected upper limits, which are shown by the open circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The predicted ggF HH production cross-section from each benchmark is indicated by a pink cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The VBF HH process is ignored for this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 28 9 Conclusion A search for nonresonant pair production of Higgs bosons in the bbbb final state was carried out, with dedicated analyses for the ggF and VBF production modes, using 126 fb−1 of √𝑠 = 13 TeV 𝑝𝑝 collision data collected by the ATLAS detector at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The sensitivity of the analyses is improved relative to previous iterations by using more sophisticated background modeling techniques, event categorization and improved jet reconstruction and flavor identification algorithms, in addition to the increased integrated luminosity of the analyzed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' No evidence of signal is found and the observed (expected) upper limit on the cross-section for nonresonant Higgs boson pair production is determined to be 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1) times the Standard Model predicted cross-section at 95% confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Constraints are placed upon modifiers to the HHH and HHVV couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The observed (expected) 2𝜎 constraints on the HHH coupling modifier, 𝜅𝜆, are determined to be [−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3] ([−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4]), while the corresponding constraints for the HHVV coupling modifier, 𝜅2𝑉 , are [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1] ([−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' The results are also used to derive constraints on relevant coefficients in the SM effective field theory and the Higgs effective field theory frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' References [1] ATLAS Collaboration, Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' B 716 (2012) 1, arXiv: 1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='7214 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [2] CMS Collaboration, Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' B 716 (2012) 30, arXiv: 1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='7235 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [3] ATLAS Collaboration, Observation of 𝐻 → 𝑏 ¯𝑏 decays and 𝑉𝐻 production with the ATLAS detector, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' B 786 (2018) 59, arXiv: 1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='08238 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [4] CMS Collaboration, Observation of Higgs Boson Decay to Bottom Quarks, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 121 (2018) 121801, arXiv: 1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='08242 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Branco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=', Theory and phenomenology of two-Higgs-doublet models, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 516 (2012) 1, arXiv: 1106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0034 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Gröber and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Mühlleitner, Composite Higgs boson pair production at the LHC, JHEP 06 (2011) 020, arXiv: 1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1562 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [7] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Chacko, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Nomura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Papucci, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Perez, Natural little hierarchy from a partially goldstone twin Higgs, JHEP 01 (2006) 126, arXiv: hep-ph/0510273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [8] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Fayet, Supersymmetry and weak, electromagnetic and strong interactions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' B 64 (1976) 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [9] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Fayet, Spontaneously broken supersymmetric theories of weak, electromagnetic and strong interactions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' B 69 (1977) 489.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 29 [10] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Degrassi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=', Higgs mass and vacuum stability in the Standard Model at NNLO, JHEP 08 (2012) 098, arXiv: 1205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='6497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Grazzini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=', Higgs boson pair production at NNLO with top quark mass effects, JHEP 05 (2018) 059, arXiv: 1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='02463 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [12] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Dreyer and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Karlberg, Vector-boson fusion Higgs pair production at N3LO, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' D 98 (2018) 114016, arXiv: 1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='07906 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [13] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Bishara, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Contino, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Rojo, Higgs pair production in vector-boson fusion at the LHC and beyond, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' C 77 (2017) 481, arXiv: 1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='03860 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [14] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' de Florian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=', Handbook of LHC Higgs Cross Sections: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Deciphering the Nature of the Higgs Sector, (2016), arXiv: 1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='07922 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Djouadi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Kalinowski, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Spira, HDECAY: A program for Higgs boson decays in the Standard Model and its supersymmetric extension, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 108 (1998) 56, arXiv: hep-ph/9704448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [16] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Buchmuller and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Wyler, Effective lagrangian analysis of new interactions and flavour conservation, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' B 268 (1986) 621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [17] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Leung, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Love, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Rao, Low-energy manifestations of a new interactions scale: Operator analysis, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' C 31 (1986) 433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [18] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Brivio and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Trott, The standard model as an effective field theory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 793 (2019) 1, arXiv: 1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='08945 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [19] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Alonso, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Gavela, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Merlo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Rigolin, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Yepes, The effective chiral lagrangian for a light dynamical "Higgs particle", Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' B 722 (2012) 330, arXiv: 1212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3305 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [20] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Buchalla, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Catà, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Krause, Complete electroweak chiral Lagrangian with a light Higgs at NLO, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' B 880 (2014) 552, arXiv: 1307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5017 [hep-ph], Erratum: Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' B 913 (2016) 475.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [21] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Grzadkowski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Iskrzynski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Misiak, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Rosiek, Dimension-six terms in the Standard Model Lagrangian, JHEP 10 (2010) 085, arXiv: 1008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4884 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [22] ATLAS Collaboration, Combined effective field theory interpretation of Higgs boson and weak boson production and decay with ATLAS data and electroweak precision observables, ATL-PHYS-PUB-2022-037, 2022, url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='ch/record/2816369.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [23] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Buchalla, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Catá, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Krause, On the power counting in effective field theories, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' B 731 (2014) 80, arXiv: 1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5624 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Capozi and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Heinrich, Exploring anomalous couplings in Higgs boson pair production through shape analysis, JHEP 03 (2020) 091, arXiv: 1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='08923 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [25] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Carvalho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=', Higgs pair production: choosing benchmarks with cluster analysis, JHEP 04 (2016) 126, arXiv: 1507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='02245 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 30 [26] ATLAS Collaboration, Search for pair production of Higgs bosons in the 𝑏 ¯𝑏𝑏 ¯𝑏 final state using proton–proton collisions at √𝑠 = 13 TeV with the ATLAS detector, JHEP 01 (2019) 030, arXiv: 1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='06174 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [27] ATLAS Collaboration, Search for the 𝐻𝐻 → 𝑏 ¯𝑏𝑏 ¯𝑏 process via vector-boson fusion production using proton–proton collisions at √𝑠 = 13 TeV with the ATLAS detector, JHEP 07 (2020) 108, arXiv: 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='05178 [hep-ex], [Erratum: JHEP 01 (2021) 145].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [28] CMS Collaboration, Search for Higgs Boson Pair Production in the Four b Quark Final State in Proton-Proton Collisions at s=13 TeV, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 129 (2022) 081802, arXiv: 2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='09617 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [29] CMS Collaboration, Search for nonresonant pair production of highly energetic Higgs bosons decaying to bottom quarks, (2022), arXiv: 2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='06667 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [30] ATLAS Collaboration, Search for resonant and non-resonant Higgs boson pair production in the 𝑏 ¯𝑏𝜏+𝜏− decay channel using 13 TeV 𝑝𝑝 collision data from the ATLAS detector, (2022), arXiv: 2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='10910 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [31] CMS Collaboration, Search for Higgs boson pair production in events with two bottom quarks and two tau leptons in proton–proton collisions at √𝑠 = 13 TeV, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' B 778 (2018) 101, arXiv: 1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='02909 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [32] ATLAS Collaboration, Search for Higgs boson pair production in the two bottom quarks plus two photons final state in 𝑝𝑝 collisions at √𝑠 = 13 TeV with the ATLAS detector, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' D 106 (2021) 052001, arXiv: 2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='11876 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [33] CMS Collaboration, Search for nonresonant Higgs boson pair production in final states with two bottom quarks and two photons in proton–proton collisions at √𝑠 = 13 TeV, JHEP 03 (2021) 257, arXiv: 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='12373 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [34] ATLAS Collaboration, Search for non-resonant Higgs boson pair production in the 𝑏𝑏ℓ𝜈ℓ𝜈 final state with the ATLAS detector in 𝑝𝑝 collisions at √𝑠 = 13 TeV, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' B 801 (2020) 135145, arXiv: 1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='06765 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [35] CMS Collaboration, Search for resonant and nonresonant Higgs boson pair production in the 𝑏 ¯𝑏ℓ𝜈ℓ𝜈 final state in proton–proton collisions at √𝑠 = 13 TeV, JHEP 01 (2018) 054, arXiv: 1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='04188 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [36] ATLAS Collaboration, Search for Higgs boson pair production in the 𝑏 ¯𝑏𝑊𝑊∗ decay mode at √𝑠 = 13 TeV with the ATLAS detector, JHEP 04 (2019) 092, arXiv: 1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='04671 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [37] ATLAS Collaboration, Search for Higgs boson pair production in the 𝛾𝛾𝑊𝑊∗ channel using 𝑝𝑝 collision data recorded at √𝑠 = 13 TeV with the ATLAS detector, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' C 78 (2018) 1007, arXiv: 1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='08567 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [38] ATLAS Collaboration, Search for Higgs boson pair production in the 𝑊𝑊 (∗)𝑊𝑊 (∗) decay channel using ATLAS data recorded at √𝑠 = 13 TeV, JHEP 05 (2019) 124, arXiv: 1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='11028 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [39] CMS Collaboration, A portrait of the Higgs boson by the CMS experiment ten years after the discovery, Nature 607 (2022) 60, arXiv: 2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='00043 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [40] ATLAS Collaboration, The ATLAS Experiment at the CERN Large Hadron Collider, JINST 3 (2008) S08003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 31 [41] ATLAS Collaboration, ATLAS Insertable B-Layer: Technical Design Report, ATLAS-TDR-19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' CERN-LHCC-2010-013, 2010, url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='ch/record/1291633, Addendum: ATLAS-TDR-19-ADD-1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' CERN-LHCC-2012-009, 2012, url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='ch/record/1451888.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [42] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=', Production and integration of the ATLAS Insertable B-Layer, JINST 13 (2018) T05008, arXiv: 1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='00844 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='ins-det].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [43] ATLAS Collaboration, Performance of the ATLAS trigger system in 2015, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' C 77 (2017) 317, arXiv: 1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='09661 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [44] ATLAS Collaboration, The ATLAS Collaboration Software and Firmware, ATL-SOFT-PUB-2021-001, 2021, url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='ch/record/2767187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [45] ATLAS Collaboration, ATLAS data quality operations and performance for 2015–2018 data-taking, JINST 15 (2020) P04003, arXiv: 1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='04632 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='ins-det].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [46] ATLAS Collaboration, Configuration and performance of the ATLAS 𝑏-jet triggers in Run 2, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' C 81 (2021) 1087, arXiv: 2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='03584 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [47] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Cacciari, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Salam, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Soyez, The anti-𝑘𝑡 jet clustering algorithm, JHEP 04 (2008) 063, arXiv: 0802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1189 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [48] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Cacciari, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Salam, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Soyez, FastJet user manual, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' C 72 (2012) 1896, arXiv: 1111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='6097 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [49] ATLAS Collaboration, The ATLAS Simulation Infrastructure, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' C 70 (2010) 823, arXiv: 1005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='4568 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='ins-det].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [50] GEANT4 Collaboration, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Agostinelli, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=', Geant4 – a simulation toolkit, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' A 506 (2003) 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [51] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Nason, A New method for combining NLO QCD with shower Monte Carlo algorithms, JHEP 11 (2004) 040, arXiv: hep-ph/0409146 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [52] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Frixione, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Nason and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Oleari, Matching NLO QCD computations with parton shower simulations: the POWHEG method, JHEP 11 (2007) 070, arXiv: 0709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='2092 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [53] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Alioli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Nason, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Oleari, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Re, A general framework for implementing NLO calculations in shower Monte Carlo programs: the POWHEG BOX, JHEP 06 (2010) 043, arXiv: 1002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='2581 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [54] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Butterworth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=', PDF4LHC recommendations for LHC Run II, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' G 43 (2016) 023001, arXiv: 1510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='03865 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [55] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Sjöstrand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=', An introduction to PYTHIA 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='2, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 191 (2015) 159, arXiv: 1410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3012 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [56] ATLAS Collaboration, ATLAS Pythia 8 tunes to 7 TeV data, ATL-PHYS-PUB-2014-021, 2014, url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='ch/record/1966419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [57] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Ball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=', Parton distributions with LHC data, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' B 867 (2013) 244, arXiv: 1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1303 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [58] ATLAS Collaboration, Validation of signal Monte Carlo event generation in searches for Higgs boson pairs with the ATLAS detector, ATL-PHYS-PUB-2019-007, 2019, url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='ch/record/2665057.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 32 [59] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Bähr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=', Herwig++ Physics and Manual, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' C58 (2008) 639, arXiv: 0803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0883 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [60] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Gieseke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Röhr, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Siodmok, Colour reconnections in Herwig++, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' C 72 (2012) 2225, arXiv: 1206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0041 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [61] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Harland-Lang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Martin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Motylinski, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Thorne, Parton distributions in the LHC era: MMHT 2014 PDFs, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' C 75 (2015) 204, arXiv: 1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3989 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [62] ATLAS Collaboration, A study of optimal parameter setting for MadGraph5_aMC@NLO + Pythia 8 matched setup, ATL-PHYS-PUB-2015-048, 2015, url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='ch/record/2103221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [63] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Alwall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=', The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations, JHEP 07 (2014) 079, arXiv: 1405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0301 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [64] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Alwall, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Herquet, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Maltoni, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Mattelaer, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Stelzer, MadGraph 5 : Going Beyond, JHEP 06 (2011) 128, arXiv: 1106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='0522 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [65] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Degrande et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=', Automated one-loop computations in the standard model effective field theory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' D 103 (2021) 096024, arXiv: 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='11743 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [66] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Buchalla, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Capozi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Celis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Heinrich, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Scyboz, Higgs boson pair production in non-linear Effective Field Theory with full 𝑚𝑡-dependence at NLO QCD, JHEP 09 (2018) 057, arXiv: 1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='05162 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [67] ATLAS Collaboration, HEFT interpretations of Higgs boson pair searches in 𝑏 ¯𝑏𝛾𝛾 and 𝑏 ¯𝑏𝜏+𝜏− final states and of their combination in ATLAS, ATL-PHYS-PUB-2022-019, 2022, url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='ch/record/2806411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [68] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Ball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=', Parton distributions for the LHC run II, JHEP 04 (2015) 040, arXiv: 1410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='8849 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [69] ATLAS Collaboration, Combination of searches for Higgs boson pairs in 𝑝𝑝 collisions at √𝑠 = 13 TeV with the ATLAS detector, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' B 800 (2020) 135103, arXiv: 1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='02025 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [70] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Baglio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=', The measurement of the Higgs self-coupling at the LHC: theoretical status, JHEP 04 (2013) 151, arXiv: 1212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5581 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [71] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Ling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=', NNLO QCD corrections to Higgs pair production via vector boson fusion at hadron colliders, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' D 89 (2014) 073001, arXiv: 1401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='7754 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [72] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Dreyer and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Karlberg, Fully differential Vector-Boson Fusion Higgs Pair Production at Next-to-Next-to-Leading Order, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' D 99 (2019) 074028, arXiv: 1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='07918 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [73] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Campbell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Ellis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Nason, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Re, Top-pair production and decay at NLO matched with parton showers, JHEP 04 (2015) 114, arXiv: 1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1828 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [74] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Czakon and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Mitov, Top++: A program for the calculation of the top-pair cross-section at hadron colliders, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 185 (2014) 2930, arXiv: 1112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5675 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 33 [75] ATLAS Collaboration, Improvements in 𝑡¯𝑡 modelling using NLO+PS Monte Carlo generators for Run 2, ATL-PHYS-PUB-2018-009, 2018, url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='ch/record/2630327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [76] ATLAS Collaboration, The Pythia 8 A3 tune description of ATLAS minimum bias and inelastic measurements incorporating the Donnachie–Landshoff diffractive model, ATL-PHYS-PUB-2016-017, 2016, url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='ch/record/2206965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [77] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Lange, The EvtGen particle decay simulation package, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' A 462 (2001) 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [78] ATLAS Collaboration, Vertex Reconstruction Performance of the ATLAS Detector at √𝑠 = 13 TeV, ATL-PHYS-PUB-2015-026, 2015, url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='ch/record/2037717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [79] ATLAS Collaboration, Jet reconstruction and performance using particle flow with the ATLAS Detector, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' C 77 (2017) 466, arXiv: 1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='10485 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [80] ATLAS Collaboration, Jet energy scale and resolution measured in proton–proton collisions at √𝑠 = 13 TeV with the ATLAS detector, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' C 81 (2020) 689, arXiv: 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='02645 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [81] ATLAS Collaboration, Performance of pile-up mitigation techniques for jets in 𝑝𝑝 collisions at √𝑠 = 8 TeV using the ATLAS detector, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' C 76 (2016) 581, arXiv: 1510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='03823 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [82] ATLAS Collaboration, Topological cell clustering in the ATLAS calorimeters and its performance in LHC Run 1, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' C 77 (2017) 490, arXiv: 1603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='02934 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [83] ATLAS Collaboration, Selection of jets produced in 13 TeV proton–proton collisions with the ATLAS detector, ATLAS-CONF-2015-029, 2015, url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='ch/record/2037702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [84] ATLAS Collaboration, ATLAS flavour-tagging algorithms for the LHC Run 2 𝑝𝑝 collision dataset, (2022), arXiv: 2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='16345 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='data-an].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [85] ATLAS Collaboration, Calibration of light-flavour 𝑏-jet mistagging rates using ATLAS proton–proton collision data at √𝑠 = 13 TeV, ATLAS-CONF-2018-006, 2018, url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='ch/record/2314418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [86] ATLAS Collaboration, Measurement of the c-jet mistagging efficiency in 𝑡¯𝑡 events using pp collision data at √𝑠 = 13 TeV collected with the ATLAS detector, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' C 82 (2022) 95, arXiv: 2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='10627 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [87] ATLAS Collaboration, Muon reconstruction and identification efficiency in ATLAS using the full Run 2 𝑝𝑝 collision data set at √𝑠 = 13 TeV, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' C 81 (2021) 578, arXiv: 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='00578 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [88] ATLAS Collaboration, Evidence for the 𝐻 → 𝑏 ¯𝑏 decay with the ATLAS detector, JHEP 12 (2017) 024, arXiv: 1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='03299 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [89] ATLAS Collaboration, Muon reconstruction performance of the ATLAS detector in proton–proton collision data at √𝑠 = 13 TeV, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' C 76 (2016) 292, arXiv: 1603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='05598 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [90] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Moustakides and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Basioti, Training Neural Networks for Likelihood/Density Ratio Estimation, 2019, arXiv: 1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='00405 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='SP].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 34 [91] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Kanamori, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Hido, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Sugiyama, A Least-squares Approach to Direct Importance Estimation, Journal of Machine Learning Research 10 (2009) 1391, url: http://jmlr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='org/papers/v10/kanamori09a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [92] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Nair and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Hinton, “Rectified Linear Units Improve Restricted Boltzmann Machines,” Proceedings of the 27th International Conference on International Conference on Machine Learning, ICML’10, Haifa, Israel: Omnipress, 2010 807, isbn: 9781605589077, url: https://dl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='5555/3104322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='3104425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [93] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Lakshminarayanan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Pritzel, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Blundell, Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles, 2017, arXiv: 1612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='01474 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='ML].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [94] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Efron, Bootstrap Methods: Another Look at the Jackknife, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 7 (1979) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [95] ATLAS Collaboration, Measurement of the Inelastic Proton–Proton Cross Section at √𝑠 = 13 TeV with the ATLAS Detector at the LHC, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 117 (2016) 182002, arXiv: 1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='02625 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [96] ATLAS Collaboration, ATLAS 𝑏-jet identification performance and efficiency measurement with 𝑡¯𝑡 events in 𝑝𝑝 collisions at √𝑠 = 13 TeV, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' C 79 (2019) 970, arXiv: 1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='05120 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [97] ATLAS Collaboration, Luminosity determination in 𝑝𝑝 collisions at √𝑠 = 13 TeV using the ATLAS detector at the LHC, ATLAS-CONF-2019-021, 2019, url: https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='ch/record/2677054.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [98] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Avoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=', The new LUCID-2 detector for luminosity measurement and monitoring in ATLAS, JINST 13 (2018) P07017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [99] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Baglio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=', 𝑔𝑔 → 𝐻𝐻: Combined uncertainties, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' D 103 (2021) 056002, arXiv: 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='11626 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [100] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Verkerke and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Kirkby, The RooFit toolkit for data modeling, 2003, arXiv: physics/0306116 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='data-an].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [101] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Cowan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Cranmer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Gross, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Vitells, Asymptotic formulae for likelihood-based tests of new physics, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' C 71 (2011) 1554, arXiv: 1007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='1727 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content='data-an], Erratum: Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' C 73 (2013) 2501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' [102] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Read, Presentation of search results: the 𝐶𝐿𝑆 technique, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' G 28 (2002) 2693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} +page_content=' 35' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQffAQh/content/2301.03212v1.pdf'} diff --git a/hNE0T4oBgHgl3EQf6gJU/vector_store/index.pkl b/hNE0T4oBgHgl3EQf6gJU/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..4c810968ae0dff48bbcda23c0e4620228db8b8d0 --- /dev/null +++ b/hNE0T4oBgHgl3EQf6gJU/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c38916fae9b2b783099982ec3c3a2ea3f910803fa22c2d9d6e8d80a6f5328459 +size 238237 diff --git a/iNE5T4oBgHgl3EQfFw4A/content/tmp_files/2301.05424v1.pdf.txt b/iNE5T4oBgHgl3EQfFw4A/content/tmp_files/2301.05424v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..95ed463dd666ba52601ca392f60f7a36ba69afff --- /dev/null +++ b/iNE5T4oBgHgl3EQfFw4A/content/tmp_files/2301.05424v1.pdf.txt @@ -0,0 +1,683 @@ +arXiv:2301.05424v1 [math-ph] 13 Jan 2023 +A Causal Formulation of +Dissipative Relativistic Fluid Dynamics +with or without Diffusion +Heinrich Freist¨uhler∗ +November 1, 2022 +Abstract +The article proposes a causal five-field formulation of dissipative relativistic fluid +dynamics as a quasilinear symmetric hyperbolic system of second order. The system is +determined by four dissipation coefficients η, ζ, κ, µ, free functions of the fields, which +quantify shear viscosity, bulk viscosity, heat conductivity, and diffusion. +∗Department of Mathematics, University of Konstanz, Konstanz, Germany +1 + +1 +Introduction +The present note is related to the question of whether dissipative relativistic fluid dynamics +can be properly modeled by a hyperbolic, causal five-field theory +∂ +∂xβ (T αβ + ∆T αβ) = 0, +∂ +∂xβ (Nβ + ∆Nβ) = 0, +(1.1) +where the dissipation-free parts of the energy-momentum tensor1 and the particle number +density current, +T αβ = (ρ + p)UαUβ + pgαβ, +Nβ = nUβ, +(1.2) +are given in terms of the fluid’s velocity Uα and its energy density ρ, pressure p, and particle +number density n — one speaks of five fields since Uα is constrained by UαUα = −1 and ρ, p, n +are related to each other by an equation of state —, and the dissipation tensors ∆T αβ, ∆Nβ +are linear in the space-time gradients of these fields (“relativistic Navier-Stokes”). +That +question has first2 been answered in the affirmative in the articles [6, 7], to which this note +is a supplement. In the introduction to his “Hyperbolic Conservation Laws in Continuum +Physics” [3], Dafermos suggests that “the umbilical cord joining continuum physics with the +theory of partial differential equations should not be severed, as it is still carrying nourish- +ment in both directions”. This paper is part of an attempt to contribute in that spirit. +Its primary purpose is to include diffusion in addition to viscosity and heat conduction. As in +[7, 8], we find corresponding equations from the three guiding principles that (a) when written +in the natural Godunov variables that make the dissipation-free part symmetric hyperbolic in +the first-order sense, they should be symmetric hyperbolic in the sense of (a covariant version +of) the Hughes-Kato-Marsden (HKM) conditions, (b) they should be first-order equivalent to +the classical theories by Eckart and Landau, and (c) all signal speeds should be bounded by +the speed of light. A side-motivation is related to the different treatment of heat conduction +in [7, 8]. While the paper [7] on general non-barotropic fluids interprets, like Landau, heat +conduction as an effect within matter conservation (1.1)2, the article [8] on barotropic fluids +represents it, like Eckart, within the conservation laws of energy and momentum (1.1)1. This +latter option is however also available for non-barotropic fluids, and it is the one we adopt +in this paper. +In Section 2, we state the dissipation tensors ∆T αβ, ∆Nβ that are determined by the coef- +ficients η, ζ, χ, µ of shear viscosity, bulk viscosity, heat conductivity, and diffusion. Section +3 shows that the resulting five-field theory is first-order equivalent with the corresponding +Eckart and Landau formulations. In Section 4 we demonstrate that it belongs to the HKM +class and is causal. Sec. 5 establishes the validity of the second law of thermodynamics to +leading order. We assume at most places (though not in Sec. 5) that the fluid is polytropic, +ρ = mn + +p +γ − 1, +with m > 0 and 1 < γ < 2. +(1.3) +1We work with the Minkowski metric gαβ of signature (−, +, +, +). +2An interesting different line of thinking was later started in [1]. +2 + +As in [7], this assumption is made for concreteness only; the essence of the argumentation +does not depend on it, and corresponding results for arbitrary other massive non-barotropic +fluids follow through obvious small adaptations. +For the case that all dissipative mechanisms are active, i. e., (2.1), (2.2), the model proposed +and discussed in this paper has been studied by Sroczinski who showed in [16] that for initial +data that are sufficiently small perturbations of a spatially homogeneous state, the Cauchy +problem has a unique solution for all times t > 0 that decays for t → ∞ to the reference +state. The same is shown in [5] for the case with (2.1) and µ = 0, i. e., in the absence of +diffusion. +While the approach pursued here seems mathematically coherent and plausible, its physical +validity is to be further examined, notably in view of the fact that the second law of ther- +modynamics is so far established only to leading order. A different, elaborate formulation of +dissipative fluid dynamics in terms of hyperbolic balance laws is Rational Extended Ther- +modynamics, which does not have the latter problem; cf. the recent monograph [14] and +references therein. +2 +Proposed new formulation and main results +Besides fixed choices of coefficients of shear viscosity, bulk viscosity, and heat conduction, +η > 0, +ζ ≥ 0, +and +χ > 0, +(2.1) +we now consider also a diffusion coefficient +µ > 0; +(2.2) +all four of these coefficients can be taken to be widely arbitrary functions of the thermody- +namic variables. Our proposal is to use3 +−∆T αβ += +ηΠαγΠβδ +�∂Uγ +∂xδ + ∂Uδ +∂xγ − 2 +3gγδ +∂Uǫ +∂xǫ +� ++ ˜ζΠαβ ∂Uγ +∂xγ ++σ +� +UαUβ ∂Uγ +∂xγ − +� +gαγUβ + gβγUα� +Uδ ∂Uγ +∂xδ +� +(2.3) ++χ +� +Uα ∂θ +∂xβ + Uβ ∂θ +∂xα − gαβUγ ∂θ +∂xγ +� +and +−∆Nβ +≡ +µgβδ ∂ψ +∂xδ + ˜σ +� +Uβ ∂Uγ +∂xγ − Uγ ∂Uβ +∂xγ +� +(2.4) +3We utilize Παβ = gαβ + U αU β. +3 + +or, written as matrices with respect to the fluid’s rest frame,4 +−∆T|0 = +�−χ ˙θ + σ∇ · u +χ∇θ − σ ˙u⊤ +χ∇⊤θ − σ ˙u +ηSu + (˜ζ∇ · u − χ ˙θ)I +� +(2.5) +and +−∆N|0 = +� +−µ ˙ψ + ˜σ∇ · u +µ∇ψ − ˜σ ˙u⊤� +, +(2.6) +where +θ, +ψ = h/θ − s +denote temperature and Israel’s thermal potential, with s and h = (ρ + p)/n = the specific +entropy and specific enthalpy (ψ = g/θ with g = h − θs the chemical potential).. +The +coefficients σ, ˜ζ, and ˜σ are given by +σ = 4 +3η + ˜ζ, +˜ζ = ζ + ˜ζ1 + ˜ζ2 + ˜ζ3, +˜σ = (σ + χθ)/h +(2.7) +with5 +˜ζ1 = −(γ − 1) (2 − γ + m/h) χθ, +˜ζ2 = (γ − 1)(1 − m/h)σ, +˜ζ3 = (γ − 1)2(m2/θ)µ. (2.8) +The following will be established in Secs. 3, 4, 5. +Theorem 1. The model (1.1)–(1.3), (2.3)–(2.8) is first-order equivalent with the Eckart and +Landau models with diffusion. It is symmetric hyperbolic when written in Godunov-Boillat +variables. It is causal when ˜ζ ≥ −1 +3η, and sharply causal when ˜ζ = −1 +3η.6 The entropy +production in this model is non-negative to leading (first) order in the dissipation coefficients +η, ζ, χ, µ. +3 +First-order equivalence with Eckart and Landau +Paper [7] contains a formal description of mappings that transform between different versions +of the equations of motion which are equivalent up to errors that are quadratic in the +dissipation coefficients; notably, the descriptions of Eckart [4] and Landau [12] are first- +order equivalent in this sense. Using that formalism, first-order equivalences are composed +from modifications called velocity shifts, thermodynamic shifts, and gradient reexpressions. +4We write u for the 3-velocity with respect to the fluid’s rest frame at a given point. (While u = 0 at that +point, its gradient is free.) ˙ means derivative with respect to x0, ∇ derivatives with respect to (x1, x2, x3), +all in the rest frame. +5This implies that σ = ((4/3)η + ζ + ˜ζ1 + ˜ζ3)/(1 − (γ − 1)(1 − m/h)). +6This can be expressed as a (natural) restriction on χ in dependence on η, ζ and µ: χ ≤ χ∗ or χ = χ∗, +respectively, with a function χ∗ = χ∗(η, ζ, µ) > 0. +4 + +Suppressing diffusion for a moment, i.e., briefly replacing (2.2) by µ = 0, we start from the +usual Eckart tensor, in rest frame notation + + +0 +χ +� +∇θ + θ ˙u⊤� +χ +� +∇⊤θ + θ ˙u +� +ηSu + ζ∇ · u I +0 +0 + + . +(3.1) +We combine first a velocity shift (cf. [7], p. 11, first “assignment”) + + +0 +χ +� +∇θ + θ ˙u⊤� +χ +� +∇⊤θ + θ ˙u +� +ηSu + ζ∇ · u I +0 +0 + + → + + +0 +χ∇θ +χ∇⊤θ +ηSu + ζ∇ · u I +0 +− (χθ/h) ˙u⊤ + + . +and a thermodynamic shift (cf. [7], p. 11, second “assignment” with (4.9) and (6.3)) + + +0 +χ∇θ +χ∇⊤θ +ηSu + ζ∇ · u I +0 +− (χθ/h) ˙u⊤ + + → + + +−χ ˙θ +χ∇θ +χ∇⊤θ +ηSu + ((ζ + ˜ζ1)∇ · u − χ ˙θ)I +(χθ/h) ∇ · u +− (χθ/h) ˙u⊤ + + +with +˜ζ1 = −(γ − 1) +� +2 − γ + m +h +� +χθ, +and then follow a second velocity shift + + +−χ ˙θ +χ∇θ +χ∇⊤θ +ηSu + ((ζ + ˜ζ1)∇ · u − χ ˙θ)I +(χθ/h) ∇ · u +− (χθ/h) ˙u⊤ + + → + + +−χ ˙θ +χ∇θ − σ ˙u⊤ +χ∇⊤θ − σ ˙u +ηSu + ((ζ + ˜ζ1)∇ · u − χ ˙θ)I +(χθ/h) ∇ · u +− ((χθ + σ)/h) ˙u⊤ + + +by another thermodynamic shift, which leads to + + +−χ ˙θ + σ∇ · u +χ∇θ − σ ˙u⊤ +χ∇⊤θ − σ ˙u +ηSu + ((ζ + ˜ζ1 + ˜ζ2)∇ · u − χ ˙θ)I +((χθ + σ)/h) ∇ · u +− ((χθ + σ)/h) ˙u⊤ + + +(3.2) +with +˜ζ2 = (γ − 1) +� +1 − m +h +� +σ. +The matrix in (3.2) is already the rest-frame form in (2.5), (2.6) in the case µ = 0. +To now include also diffusion, recall that according to Kluitenberg, de Groot, Mazur [11] it +is represented in the Eckart frame by +−∆Nβ = µΠβγ ∂ψ +∂xγ ; +i. e., we have to superimpose on (3.1) the matrix + + +0 +0 +0 +0 +0 +µ∇ψ + + . +5 + +From this starting point, a thermodynamic shift and a gradient reexpression lead to +→ + + +0 +0 +0 +−m(γ − 1) ˙ψ +−µ ˙ψ +µ∇ψ + + → + + +0 +0 +0 +˜ζ3∇ · u +−µ ˙ψ +µ∇ψ + + +with +˜ζ3 = (γ − 1)2m2 +θ µ +Adding the last version to (3.2), we reach yields (2.5) - (2.8). +4 +Symmetric hyperbolicity and causality +We start from the general equivariant forms of tensors −∆T αβ and −∆Nβ that are linear +in the gradients of the state variables. These forms are (cf. [7]) +−∆T αβ ≡ UαUβP + (ΠαγUβ + ΠβγUα)Qγ + ΠαβR + ΠαγΠβδSγδ +with +P = τUγ ∂θ +∂xγ + σ∂Uγ +∂xγ + ˇιUγ ∂ψ +∂xγ , +Qγ ≡ ν ∂θ +∂xγ + ˇςUδ ∂Uγ +∂xδ + υ ∂ψ +∂xγ , +R = ωUγ ∂θ +∂xγ + ˜ζ ∂Uγ +∂xγ + ˜ιUγ ∂ψ +∂xγ , +Sαβ ≡ η +�∂Uα +∂xβ + ∂Uβ +∂xα − 2 +3gαβ +∂Uγ +∂xγ +� +, +and +−∆Nβ ≡ Uβ ˆP + Πβδ ˆQδ +with +ˆP = ˆτUδ ∂θ +∂xδ + ˆσ∂Uδ +∂xδ + ˆιUδ ∂ψ +∂xδ , +ˆQδ ≡ ˆν ∂θ +∂xδ + ˆςUǫ∂Uδ +∂xǫ + ˆυ ∂ψ +∂xδ . +Our proposed new theory corresponds to choosing +(i) ν = −τ = −ω = χ, +(ii) −ˇς = σ and ˆσ = −ˆς = ˜σ, +(iii) ˆυ = −ˆι = µ, +(iv) ˆτ = ˆν = ˜ι = ˇι = υ = 0, +i. e., +P = −χUγ ∂θ +∂xγ + σ∂Uγ +∂xγ +Qγ ≡ χ ∂θ +∂xγ − σUδ ∂Uγ +∂xδ +R = −χUγ ∂θ +∂xγ + ˜ζ ∂Uγ +∂xγ +Sαβ ≡ η +�∂Uα +∂xβ + ∂Uβ +∂xα − 2 +3gαβ +∂Uγ +∂xγ +� +, +6 + +and +ˆP = ˜σ∂Uδ +∂xδ − µUδ ∂ψ +∂xδ , +ˆQδ ≡ −˜σUǫ ∂Uδ +∂xǫ + µ ∂ψ +∂xδ . +This directly yields (2.3), (2.4). +As in [7], we write (1.1) in the Godunov-Boillat variables [9, 2, 13] +ψα = Uα +θ , +ψ4 = ψ. +Correspondingly, we write the second-order parts of +− ∂ +∂xβ +� +∆T αβ� +and +− +∂ +∂xβ +� +∆Nβ� +as +Bαβcδ ∂2ψc +∂xβ∂xδ +and +B4βcδ ∂2ψc +∂xβ∂xδ , +respectively, where the index c runs from 0 through 4. +Expressing derivatives as +∂θ +∂xδ = θ2Uγ ∂ψγ +∂xδ , +∂Uσ +∂xδ = θΠσγ ∂ψγ +∂xδ , +we see that +Bαβγδ = + UαUβ(−χθ2UγUδ + σθΠγδ) ++ Παβ(−χθ2UγUδ + ˜ζθΠγδ) ++ χθ2(ΠαδUβ + ΠβδUα)Uγ +− σθ(ΠαγUβ + ΠβγUα)Uδ ++ ηθ(ΠαγΠβδ + ΠαδΠβγ − (2/3)ΠαβΠγδ) +and +Bαβ4δ = B4βγδ = 0. +as well as +B4β4δ = −µUβUδ + µΠβδ. +With any Nβ satisfying NβUβ = 0, +NβNβ = 1, the rest-frame coefficient matrices BaβcδUβUδ +and BaβcδNβNδ are thus given by + + +−χθ2 +0 +0 +0 +−σθδij +0 +0 +0 +−µ + + , + + +χθ2 +0 +0 +0 +ηθδij + ( 1 +3η + ˜ζ)θNiNj +0 +0 +0 +µ + + . +(4.1) +This confirms the (covariant version of the) HKM definiteness conditions ([6], (4.1)-(4.5)) +BaβcδHβHδVaVc < 0 for all Va ̸= 0, +7 + +and +BaβcδNβNδVaVc > 0 for all Va ̸= 0, +for +some Hβ with HβHβ < 0 and all Nβ ̸= 0 with NβHβ = 0. +The statements on causality and sharp causality follow as in [7] from (4.1) and the facts +that causality and sharp causality correspond to ˜ζ ≥ −1 +3η, σ ≥ η and ˜ζ = −1 +3η, σ = η, +respectively. +5 +Entropy production +In the Eckart frame, the entropy production is classically known (cf. [18], p. 55) as +Q ≡ − 1 +θ2 +∂θ +∂x0∆T 00|0 − 1 +θ2 +� ∂θ +∂xi + θ ∂ui +∂x0 +� +∆T i0|0 − 1 +θ +∂ui +∂xj ∆T ij|0 − ∂ψ +∂x0 ∆N0|0 − ∂ψ +∂xj ∆Nj|0 += χ +θ2|∇θ + θ ˙u|2 + η +2θ||Su||2 + ζ +θ(∇ · u)2 + µ|∇ψ|2 ≥ 0. +The claim of Theorem 1 on entropy production is an immediate consequence of the following +result, which is not restricted to polytropic gases. +Proposition 1. Let the dissipation coefficients be of magnitude O(ǫ). +Under first-order +equivalence transformations, the entropy production then changes by a difference ∆Q of +higher order O(ǫ2). +Proof. This is obvious for gradient reexpressions. For any velocity shift + + +∗ +∗ +∗ +∗ +∗ +∗ + + → + + +∗ +∗ + ∆u⊤ +∗ + ∆u, +∗ +∗, +∗ + (1/h)∆u⊤ + + +with ∆u = O(ǫ), we find +∆Q = 1 +θ2(∇θ + θ ˙u) · ∆u + 1 +h∇ψ · ∆u = O(ǫ2), +where we have used (cf. [7], eq. (4.10)) that +1 +θ2(∇θ + θ ˙u) + 1 +h∇ψ = O(ǫ). +Finally, consider any thermodynamic shift + + +∗ +∗ +∗ +∗ +∗ +∗ + + → + + +∗ + ∆ρ +∗ +∗ +∗ + ∆p I +∗ + ∆n +∗ + + +8 + +with a triple (∆ρ, ∆n, ∆p) = O(ǫ) that is compatible with the equation of state. We assume +the latter to be given as +p = p(θ, ψ), +so that ρ = θpθ − p, +n = pψ/θ, +introduce ∆ρ, ∆n and rewrite ˙θ, ˙ψ as +� +∆ρ +∆n +� += A +� +∆θ +∆ψ +� +, +� +˙ρ +˙n +� += A +� ˙θ +˙ψ +� +, +with +A = +�ρθ +ρψ +nθ +nψ +� += +� +θpθθ +θpθψ − pψ +θ−2(θpθψ − pψ) +θ−1pψψ +� +, +and evaluate the change in entropy production as +∆Q = 1 +θ2 ˙θ∆ρ + 1 +θ∇ · u∆p + ˙ψ∆n += ∇ · u +� +− ρ + p +θ2 +∆θ + 1 +θ∆p − n∆ψ +� ++ O(ǫ2) += O(ǫ2), +where we have used equations (6.4) of [7] and the compatibility relation +∆p = pθ∆θ + pψ∆ψ = ρ + p +θ +∆θ + nθ∆ψ. +References +[1] F. S. Bemfica, M. M. Disconzi, and J. Noronha: Causality and existence of solutions of +relativistic viscous fluid dynamics with gravity. Phys. Rev. D 98 (2018), 104064. +[2] G. Boillat: Sur l’existence et la recherche d’´equations de conservation suppl´ementaires +pour les syst`emes hyperboliques. C. R. Acad. Sci. Paris S´er. A 278 (1974), 909–912. +[3] C. M. Dafermos: Hyperbolic conservation laws in continuum physics. Fourth edition. +Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathe- +matical Sciences], 325. Springer-Verlag, Berlin 2016. +[4] C. Eckart: The thermodynamics of irreversible processes. 3: Relativistic theory of the +simple fluid. Phys. Rev. 58 (1940), 919–924. +[5] H. Freist¨uhler, M. Sroczinski: Global existence and deacy of small solutions for quasi- +linear uniformly dissipative mixed-order hyperbolic-hyperbolic systems. Preprint 2022. +9 + +[6] H. Freist¨uhler, B. Temple: Causal dissipation and shock profiles in the relativistic fluid +dynamics of pure radiation. Proc. R. Soc. A 470 (2014), 20140055. +[7] H. Freist¨uhler, B. Temple: Causal dissipation for the relativistic dynamics of ideal gases. +Proc. R. Soc. A 473 (2017), 20160729. +[8] H. Freist¨uhler, B. Temple: Causal dissipation in the relativistic dynamics of barotropic +fluids. J. Math, Phys. 59 (2018), 063101. +[9] S. K. Godunov: An interesting class of quasilinear systems. Dokl. Akad. Nauk SSSR +139 (1961), 525–523. +[10] T. Hughes, T. Kato, J. Marsden: Well-posed quasi-linear second-order hyperbolic sys- +tems with applications to nonlinear elastodynamics and general relativity. Arch. Rational +Mech. Anal. 63 (1976), 273–294. +[11] G. A. Kluitenberg, S. G. de Groot, P. Mazur: Relativistic thermodynamics of irreversible +processes, I/II. Physica 19 (1953), 689–704, 1079–1094. +[12] L. D. Landau, E. M. Lifshitz: Fluid Mechanics. Pergamon Press, London 1959, Section +127. Original Russian edition: Moscow 1953. +[13] T. Ruggeri, A. Strumia: Main field and convex covariant density for quasilinear hy- +perbolic systems. Relativistic fluid dynamics. Ann. Inst. H. Poincar´e Sect. A (N.S.) 34 +(1981), 65–84. +[14] T. Ruggeri, M. Sugiyama: Classical and relativistic rational extended thermodynamics +of gases. Springer, Cham 2021. +[15] M. Sroczinski: Asymptotic stability of homogeneous states in the relativistic dynamics +of viscous, heat-conductive fluids. Arch. Ration. Mech. Anal. 231 (2019), 91–113. +[16] M. Sroczinski: Asymptotic stability in a second-order symmetric hyperbolic system mod- +eling the relativistic dynamics of viscous heat-conductive fluids with diffusion. J. Differ. +Equ. 268 (2020), 825–851. +[17] M. Sroczinski: Global existence and deacy of small solutions for quasi-linear uniformly +dissipative second-order hyperbolic-hyperbolic systems. Preprint 2022. +[18] S. Weinberg: Gravitation and Cosmology: Principles and Applications of the General +Theory of Relativity, John Wiley & Sons, New York, 1972. +10 + diff --git a/iNE5T4oBgHgl3EQfFw4A/content/tmp_files/load_file.txt b/iNE5T4oBgHgl3EQfFw4A/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b782ef54354d2d493fecfc90f5b7dd4a987908e4 --- /dev/null +++ b/iNE5T4oBgHgl3EQfFw4A/content/tmp_files/load_file.txt @@ -0,0 +1,255 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf,len=254 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='05424v1 [math-ph] 13 Jan 2023 A Causal Formulation of Dissipative Relativistic Fluid Dynamics with or without Diffusion Heinrich Freist¨uhler∗ November 1, 2022 Abstract The article proposes a causal five-field formulation of dissipative relativistic fluid dynamics as a quasilinear symmetric hyperbolic system of second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' The system is determined by four dissipation coefficients η, ζ, κ, µ, free functions of the fields, which quantify shear viscosity, bulk viscosity, heat conductivity, and diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' ∗Department of Mathematics, University of Konstanz, Konstanz, Germany 1 1 Introduction The present note is related to the question of whether dissipative relativistic fluid dynamics can be properly modeled by a hyperbolic, causal five-field theory ∂ ∂xβ (T αβ + ∆T αβ) = 0, ∂ ∂xβ (Nβ + ∆Nβ) = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='1) where the dissipation-free parts of the energy-momentum tensor1 and the particle number density current, T αβ = (ρ + p)UαUβ + pgαβ, Nβ = nUβ, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='2) are given in terms of the fluid’s velocity Uα and its energy density ρ, pressure p, and particle number density n — one speaks of five fields since Uα is constrained by UαUα = −1 and ρ, p, n are related to each other by an equation of state —, and the dissipation tensors ∆T αβ, ∆Nβ are linear in the space-time gradients of these fields (“relativistic Navier-Stokes”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' That question has first2 been answered in the affirmative in the articles [6, 7], to which this note is a supplement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' In the introduction to his “Hyperbolic Conservation Laws in Continuum Physics” [3], Dafermos suggests that “the umbilical cord joining continuum physics with the theory of partial differential equations should not be severed, as it is still carrying nourish- ment in both directions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' This paper is part of an attempt to contribute in that spirit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Its primary purpose is to include diffusion in addition to viscosity and heat conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' As in [7, 8], we find corresponding equations from the three guiding principles that (a) when written in the natural Godunov variables that make the dissipation-free part symmetric hyperbolic in the first-order sense, they should be symmetric hyperbolic in the sense of (a covariant version of) the Hughes-Kato-Marsden (HKM) conditions, (b) they should be first-order equivalent to the classical theories by Eckart and Landau, and (c) all signal speeds should be bounded by the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' A side-motivation is related to the different treatment of heat conduction in [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' While the paper [7] on general non-barotropic fluids interprets, like Landau, heat conduction as an effect within matter conservation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='1)2, the article [8] on barotropic fluids represents it, like Eckart, within the conservation laws of energy and momentum (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='1)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' This latter option is however also available for non-barotropic fluids, and it is the one we adopt in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' In Section 2, we state the dissipation tensors ∆T αβ, ∆Nβ that are determined by the coef- ficients η, ζ, χ, µ of shear viscosity, bulk viscosity, heat conductivity, and diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Section 3 shows that the resulting five-field theory is first-order equivalent with the corresponding Eckart and Landau formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' In Section 4 we demonstrate that it belongs to the HKM class and is causal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 5 establishes the validity of the second law of thermodynamics to leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' We assume at most places (though not in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 5) that the fluid is polytropic, ρ = mn + p γ − 1, with m > 0 and 1 < γ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='3) 1We work with the Minkowski metric gαβ of signature (−, +, +, +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 2An interesting different line of thinking was later started in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 2 As in [7], this assumption is made for concreteness only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' the essence of the argumentation does not depend on it, and corresponding results for arbitrary other massive non-barotropic fluids follow through obvious small adaptations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' For the case that all dissipative mechanisms are active, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=', (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='2), the model proposed and discussed in this paper has been studied by Sroczinski who showed in [16] that for initial data that are sufficiently small perturbations of a spatially homogeneous state, the Cauchy problem has a unique solution for all times t > 0 that decays for t → ∞ to the reference state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' The same is shown in [5] for the case with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='1) and µ = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=', in the absence of diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' While the approach pursued here seems mathematically coherent and plausible, its physical validity is to be further examined, notably in view of the fact that the second law of ther- modynamics is so far established only to leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' A different, elaborate formulation of dissipative fluid dynamics in terms of hyperbolic balance laws is Rational Extended Ther- modynamics, which does not have the latter problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' the recent monograph [14] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 2 Proposed new formulation and main results Besides fixed choices of coefficients of shear viscosity, bulk viscosity, and heat conduction, η > 0, ζ ≥ 0, and χ > 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='1) we now consider also a diffusion coefficient µ > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='2) all four of these coefficients can be taken to be widely arbitrary functions of the thermody- namic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Our proposal is to use3 −∆T αβ = ηΠαγΠβδ �∂Uγ ∂xδ + ∂Uδ ∂xγ − 2 3gγδ ∂Uǫ ∂xǫ � + ˜ζΠαβ ∂Uγ ∂xγ +σ � UαUβ ∂Uγ ∂xγ − � gαγUβ + gβγUα� Uδ ∂Uγ ∂xδ � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='3) +χ � Uα ∂θ ∂xβ + Uβ ∂θ ∂xα − gαβUγ ∂θ ∂xγ � and −∆Nβ ≡ µgβδ ∂ψ ∂xδ + ˜σ � Uβ ∂Uγ ∂xγ − Uγ ∂Uβ ∂xγ � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='4) 3We utilize Παβ = gαβ + U αU β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 3 or, written as matrices with respect to the fluid’s rest frame,4 −∆T|0 = �−χ ˙θ + σ∇ · u χ∇θ − σ ˙u⊤ χ∇⊤θ − σ ˙u ηSu + (˜ζ∇ · u − χ ˙θ)I � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='5) and −∆N|0 = � −µ ˙ψ + ˜σ∇ · u µ∇ψ − ˜σ ˙u⊤� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='6) where θ, ψ = h/θ − s denote temperature and Israel’s thermal potential, with s and h = (ρ + p)/n = the specific entropy and specific enthalpy (ψ = g/θ with g = h − θs the chemical potential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='. The coefficients σ, ˜ζ, and ˜σ are given by σ = 4 3η + ˜ζ, ˜ζ = ζ + ˜ζ1 + ˜ζ2 + ˜ζ3, ˜σ = (σ + χθ)/h (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='7) with5 ˜ζ1 = −(γ − 1) (2 − γ + m/h) χθ, ˜ζ2 = (γ − 1)(1 − m/h)σ, ˜ζ3 = (γ − 1)2(m2/θ)µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='8) The following will be established in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 3, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' The model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='3), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='3)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='8) is first-order equivalent with the Eckart and Landau models with diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' It is symmetric hyperbolic when written in Godunov-Boillat variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' It is causal when ˜ζ ≥ −1 3η, and sharply causal when ˜ζ = −1 3η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='6 The entropy production in this model is non-negative to leading (first) order in the dissipation coefficients η, ζ, χ, µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 3 First-order equivalence with Eckart and Landau Paper [7] contains a formal description of mappings that transform between different versions of the equations of motion which are equivalent up to errors that are quadratic in the dissipation coefficients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' notably, the descriptions of Eckart [4] and Landau [12] are first- order equivalent in this sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Using that formalism, first-order equivalences are composed from modifications called velocity shifts, thermodynamic shifts, and gradient reexpressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 4We write u for the 3-velocity with respect to the fluid’s rest frame at a given point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' (While u = 0 at that point, its gradient is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=') ˙ means derivative with respect to x0, ∇ derivatives with respect to (x1, x2, x3), all in the rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 5This implies that σ = ((4/3)η + ζ + ˜ζ1 + ˜ζ3)/(1 − (γ − 1)(1 − m/h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 6This can be expressed as a (natural) restriction on χ in dependence on η, ζ and µ: χ ≤ χ∗ or χ = χ∗, respectively, with a function χ∗ = χ∗(η, ζ, µ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 4 Suppressing diffusion for a moment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=', briefly replacing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='2) by µ = 0, we start from the usual Eckart tensor, in rest frame notation \uf8eb \uf8ed 0 χ � ∇θ + θ ˙u⊤� χ � ∇⊤θ + θ ˙u � ηSu + ζ∇ · u I 0 0 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='1) We combine first a velocity shift (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' [7], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 11, first “assignment”) \uf8eb \uf8ed 0 χ � ∇θ + θ ˙u⊤� χ � ∇⊤θ + θ ˙u � ηSu + ζ∇ · u I 0 0 \uf8f6 \uf8f8 → \uf8eb \uf8ed 0 χ∇θ χ∇⊤θ ηSu + ζ∇ · u I 0 − (χθ/h) ˙u⊤ \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' and a thermodynamic shift (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' [7], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 11, second “assignment” with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='9) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='3)) \uf8eb \uf8ed 0 χ∇θ χ∇⊤θ ηSu + ζ∇ · u I 0 − (χθ/h) ˙u⊤ \uf8f6 \uf8f8 → \uf8eb \uf8ed −χ ˙θ χ∇θ χ∇⊤θ ηSu + ((ζ + ˜ζ1)∇ · u − χ ˙θ)I (χθ/h) ∇ · u − (χθ/h) ˙u⊤ \uf8f6 \uf8f8 with ˜ζ1 = −(γ − 1) � 2 − γ + m h � χθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' and then follow a second velocity shift \uf8eb \uf8ed −χ ˙θ χ∇θ χ∇⊤θ ηSu + ((ζ + ˜ζ1)∇ · u − χ ˙θ)I (χθ/h) ∇ · u − (χθ/h) ˙u⊤ \uf8f6 \uf8f8 → \uf8eb \uf8ed −χ ˙θ χ∇θ − σ ˙u⊤ χ∇⊤θ − σ ˙u ηSu + ((ζ + ˜ζ1)∇ · u − χ ˙θ)I (χθ/h) ∇ · u − ((χθ + σ)/h) ˙u⊤ \uf8f6 \uf8f8 by another thermodynamic shift,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' which leads to \uf8eb \uf8ed −χ ˙θ + σ∇ · u χ∇θ − σ ˙u⊤ χ∇⊤θ − σ ˙u ηSu + ((ζ + ˜ζ1 + ˜ζ2)∇ · u − χ ˙θ)I ((χθ + σ)/h) ∇ · u − ((χθ + σ)/h) ˙u⊤ \uf8f6 \uf8f8 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='2) with ˜ζ2 = (γ − 1) � 1 − m h � σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' The matrix in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='2) is already the rest-frame form in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='6) in the case µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' To now include also diffusion, recall that according to Kluitenberg, de Groot, Mazur [11] it is represented in the Eckart frame by −∆Nβ = µΠβγ ∂ψ ∂xγ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=', we have to superimpose on (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='1) the matrix \uf8eb \uf8ed 0 0 0 0 0 µ∇ψ \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 5 From this starting point, a thermodynamic shift and a gradient reexpression lead to → \uf8eb \uf8ed 0 0 0 −m(γ − 1) ˙ψ −µ ˙ψ µ∇ψ \uf8f6 \uf8f8 → \uf8eb \uf8ed 0 0 0 ˜ζ3∇ · u −µ ˙ψ µ∇ψ \uf8f6 \uf8f8 with ˜ζ3 = (γ − 1)2m2 θ µ Adding the last version to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='2), we reach yields (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='5) - (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 4 Symmetric hyperbolicity and causality We start from the general equivariant forms of tensors −∆T αβ and −∆Nβ that are linear in the gradients of the state variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' These forms are (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' [7]) −∆T αβ ≡ UαUβP + (ΠαγUβ + ΠβγUα)Qγ + ΠαβR + ΠαγΠβδSγδ with P = τUγ ∂θ ∂xγ + σ∂Uγ ∂xγ + ˇιUγ ∂ψ ∂xγ , Qγ ≡ ν ∂θ ∂xγ + ˇςUδ ∂Uγ ∂xδ + υ ∂ψ ∂xγ , R = ωUγ ∂θ ∂xγ + ˜ζ ∂Uγ ∂xγ + ˜ιUγ ∂ψ ∂xγ , Sαβ ≡ η �∂Uα ∂xβ + ∂Uβ ∂xα − 2 3gαβ ∂Uγ ∂xγ � , and −∆Nβ ≡ Uβ ˆP + Πβδ ˆQδ with ˆP = ˆτUδ ∂θ ∂xδ + ˆσ∂Uδ ∂xδ + ˆιUδ ∂ψ ∂xδ , ˆQδ ≡ ˆν ∂θ ∂xδ + ˆςUǫ∂Uδ ∂xǫ + ˆυ ∂ψ ∂xδ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Our proposed new theory corresponds to choosing (i) ν = −τ = −ω = χ, (ii) −ˇς = σ and ˆσ = −ˆς = ˜σ, (iii) ˆυ = −ˆι = µ, (iv) ˆτ = ˆν = ˜ι = ˇι = υ = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=', P = −χUγ ∂θ ∂xγ + σ∂Uγ ∂xγ Qγ ≡ χ ∂θ ∂xγ − σUδ ∂Uγ ∂xδ R = −χUγ ∂θ ∂xγ + ˜ζ ∂Uγ ∂xγ Sαβ ≡ η �∂Uα ∂xβ + ∂Uβ ∂xα − 2 3gαβ ∂Uγ ∂xγ � , 6 and ˆP = ˜σ∂Uδ ∂xδ − µUδ ∂ψ ∂xδ , ˆQδ ≡ −˜σUǫ ∂Uδ ∂xǫ + µ ∂ψ ∂xδ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' This directly yields (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='3), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' As in [7], we write (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='1) in the Godunov-Boillat variables [9, 2, 13] ψα = Uα θ , ψ4 = ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Correspondingly, we write the second-order parts of − ∂ ∂xβ � ∆T αβ� and − ∂ ∂xβ � ∆Nβ� as Bαβcδ ∂2ψc ∂xβ∂xδ and B4βcδ ∂2ψc ∂xβ∂xδ , respectively, where the index c runs from 0 through 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Expressing derivatives as ∂θ ∂xδ = θ2Uγ ∂ψγ ∂xδ , ∂Uσ ∂xδ = θΠσγ ∂ψγ ∂xδ , we see that Bαβγδ = + UαUβ(−χθ2UγUδ + σθΠγδ) + Παβ(−χθ2UγUδ + ˜ζθΠγδ) + χθ2(ΠαδUβ + ΠβδUα)Uγ − σθ(ΠαγUβ + ΠβγUα)Uδ + ηθ(ΠαγΠβδ + ΠαδΠβγ − (2/3)ΠαβΠγδ) and Bαβ4δ = B4βγδ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' as well as B4β4δ = −µUβUδ + µΠβδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' With any Nβ satisfying NβUβ = 0, NβNβ = 1, the rest-frame coefficient matrices BaβcδUβUδ and BaβcδNβNδ are thus given by \uf8eb \uf8ed −χθ2 0 0 0 −σθδij 0 0 0 −µ \uf8f6 \uf8f8 , \uf8eb \uf8ed χθ2 0 0 0 ηθδij + ( 1 3η + ˜ζ)θNiNj 0 0 0 µ \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='1) This confirms the (covariant version of the) HKM definiteness conditions ([6], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='1)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='5)) BaβcδHβHδVaVc < 0 for all Va ̸= 0, 7 and BaβcδNβNδVaVc > 0 for all Va ̸= 0, for some Hβ with HβHβ < 0 and all Nβ ̸= 0 with NβHβ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' The statements on causality and sharp causality follow as in [7] from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='1) and the facts that causality and sharp causality correspond to ˜ζ ≥ −1 3η, σ ≥ η and ˜ζ = −1 3η, σ = η, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 5 Entropy production In the Eckart frame, the entropy production is classically known (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' [18], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 55) as Q ≡ − 1 θ2 ∂θ ∂x0∆T 00|0 − 1 θ2 � ∂θ ∂xi + θ ∂ui ∂x0 � ∆T i0|0 − 1 θ ∂ui ∂xj ∆T ij|0 − ∂ψ ∂x0 ∆N0|0 − ∂ψ ∂xj ∆Nj|0 = χ θ2|∇θ + θ ˙u|2 + η 2θ||Su||2 + ζ θ(∇ · u)2 + µ|∇ψ|2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' The claim of Theorem 1 on entropy production is an immediate consequence of the following result, which is not restricted to polytropic gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Let the dissipation coefficients be of magnitude O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Under first-order equivalence transformations, the entropy production then changes by a difference ∆Q of higher order O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' This is obvious for gradient reexpressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' For any velocity shift \uf8eb \uf8ed ∗ ∗ ∗ ∗ ∗ ∗ \uf8f6 \uf8f8 → \uf8eb \uf8ed ∗ ∗ + ∆u⊤ ∗ + ∆u, ∗ ∗, ∗ + (1/h)∆u⊤ \uf8f6 \uf8f8 with ∆u = O(ǫ), we find ∆Q = 1 θ2(∇θ + θ ˙u) · ∆u + 1 h∇ψ · ∆u = O(ǫ2), where we have used (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' [7], eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='10)) that 1 θ2(∇θ + θ ˙u) + 1 h∇ψ = O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Finally, consider any thermodynamic shift \uf8eb \uf8ed ∗ ∗ ∗ ∗ ∗ ∗ \uf8f6 \uf8f8 → \uf8eb \uf8ed ∗ + ∆ρ ∗ ∗ ∗ + ∆p I ∗ + ∆n ∗ \uf8f6 \uf8f8 8 with a triple (∆ρ, ∆n, ∆p) = O(ǫ) that is compatible with the equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' We assume the latter to be given as p = p(θ, ψ), so that ρ = θpθ − p, n = pψ/θ, introduce ∆ρ, ∆n and rewrite ˙θ, ˙ψ as � ∆ρ ∆n � = A � ∆θ ∆ψ � , � ˙ρ ˙n � = A � ˙θ ˙ψ � , with A = �ρθ ρψ nθ nψ � = � θpθθ θpθψ − pψ θ−2(θpθψ − pψ) θ−1pψψ � , and evaluate the change in entropy production as ∆Q = 1 θ2 ˙θ∆ρ + 1 θ∇ · u∆p + ˙ψ∆n = ∇ · u � − ρ + p θ2 ∆θ + 1 θ∆p − n∆ψ � + O(ǫ2) = O(ǫ2), where we have used equations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='4) of [7] and the compatibility relation ∆p = pθ∆θ + pψ∆ψ = ρ + p θ ∆θ + nθ∆ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' References [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Bemfica, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Disconzi, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Noronha: Causality and existence of solutions of relativistic viscous fluid dynamics with gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' D 98 (2018), 104064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' [2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Boillat: Sur l’existence et la recherche d’´equations de conservation suppl´ementaires pour les syst`emes hyperboliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Paris S´er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' A 278 (1974), 909–912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Dafermos: Hyperbolic conservation laws in continuum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Fourth edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathe- matical Sciences], 325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Springer-Verlag, Berlin 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Eckart: The thermodynamics of irreversible processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 3: Relativistic theory of the simple fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 58 (1940), 919–924.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' [5] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Freist¨uhler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Sroczinski: Global existence and deacy of small solutions for quasi- linear uniformly dissipative mixed-order hyperbolic-hyperbolic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Preprint 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 9 [6] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Freist¨uhler, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Temple: Causal dissipation and shock profiles in the relativistic fluid dynamics of pure radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' A 470 (2014), 20140055.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' [7] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Freist¨uhler, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Temple: Causal dissipation for the relativistic dynamics of ideal gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' A 473 (2017), 20160729.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' [8] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Freist¨uhler, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Temple: Causal dissipation in the relativistic dynamics of barotropic fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Math, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 59 (2018), 063101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' [9] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Godunov: An interesting class of quasilinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Dokl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Nauk SSSR 139 (1961), 525–523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' [10] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Hughes, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Kato, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Marsden: Well-posed quasi-linear second-order hyperbolic sys- tems with applications to nonlinear elastodynamics and general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Rational Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 63 (1976), 273–294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' [11] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Kluitenberg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' de Groot, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Mazur: Relativistic thermodynamics of irreversible processes, I/II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Physica 19 (1953), 689–704, 1079–1094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' [12] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Landau, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Lifshitz: Fluid Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Pergamon Press, London 1959, Section 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Original Russian edition: Moscow 1953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' [13] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Ruggeri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Strumia: Main field and convex covariant density for quasilinear hy- perbolic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Relativistic fluid dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Poincar´e Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' A (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=') 34 (1981), 65–84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' [14] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Ruggeri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Sugiyama: Classical and relativistic rational extended thermodynamics of gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Springer, Cham 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' [15] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Sroczinski: Asymptotic stability of homogeneous states in the relativistic dynamics of viscous, heat-conductive fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 231 (2019), 91–113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Sroczinski: Asymptotic stability in a second-order symmetric hyperbolic system mod- eling the relativistic dynamics of viscous heat-conductive fluids with diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 268 (2020), 825–851.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Sroczinski: Global existence and deacy of small solutions for quasi-linear uniformly dissipative second-order hyperbolic-hyperbolic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Preprint 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' Weinberg: Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley & Sons, New York, 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} +page_content=' 10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE5T4oBgHgl3EQfFw4A/content/2301.05424v1.pdf'} diff --git a/jdE3T4oBgHgl3EQfJQkl/content/tmp_files/2301.04341v1.pdf.txt b/jdE3T4oBgHgl3EQfJQkl/content/tmp_files/2301.04341v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..50aec1effb3b9703df2d30aabc551f637a1e4c14 --- /dev/null +++ b/jdE3T4oBgHgl3EQfJQkl/content/tmp_files/2301.04341v1.pdf.txt @@ -0,0 +1,964 @@ +A Meta Path-based Approach for Rumor Detection +on Social Media +Bita Azarijoo +Faculty of New Sciences +and Technologies +University of Tehran +Tehran, Iran +azarijoo@ut.ac.ir +Mostafa Salehi +Faculty of New Sciences and Technologies +University of Tehran +School of Computer Science, IPM Institute +for Research in Fundamental Sciences +Tehran, Iran +mostafa.salehi@ut.ac.ir +Shaghayegh Najari +Faculty of New Sciences +and Technologies +University of Tehran +Tehran, Iran +najari.shaghayegh@ut.ac.ir +Abstract—The prominent role of social media in people’s daily +lives has made them more inclined to receive news through social +networks than traditional sources. This shift in public behavior +has opened doors for some to diffuse fake news on social media; +and subsequently cause negative economic, political, and social +consequences as well as distrust among the public. +There are many proposed methods to solve the rumor detection +problem, most of which do not take full advantage of the +heterogeneous nature of news propagation networks. With this in- +tention, we considered a previously proposed architecture as our +baseline and performed the idea of structural feature extraction +from the heterogeneous rumor propagation over its architecture +using the concept of meta path-based embeddings. We named +our model Meta Path-based Global Local Attention Network +(MGLAN). Extensive experimental analysis on three state-of-the- +art datasets has demonstrated that MGLAN outperforms other +models by capturing node-level discrimination to different node +types. +Index Terms—Rumor Detection, Heterogenous Network, Meta +Path, Deep Learning, Social Network +I. INTRODUCTION +Nowadays, interactions with social networks have become +an inseparable part of people’s lives for their ease of use and +fast dissemination of information on a global scale. In this +regard, in 2012, only 45% of people used social media to +access news, whereas this number jumped to 65% in 2016 +[1]. Also, the 2020 Covid-19 pandemic caused a 30% growth +in Twitter daily usage [2]. Unfortunately, this rapid increase +in using social media has provided an opportunity for vicious +users to spread fake news to cause serious individual, eco- +nomic, and political repercussions [3]. For example, in 2013, +the Associated Press (AP) account on Twitter was hacked, and +a piece of news was published claiming an explosion occurred +in the White House and Barak Obama was injured. Although +the publishing account discredited this rumor within seconds, +it leveraged through Twitter and caused the stock value to fall +by 130 billion dollars [1]. Consequently, studying fake news +and preventing its dissemination as soon as possible is yet an +active and open research area. +To avoid the detrimental effects of fake news, there are +websites like Snopes1, GossipCop2, and Poitifact3, but they +can not detect fake news automatically in their early stages +of propagation and rely on manual user intervention for fact- +checking as well. As a consequence, the detection of fake news +can be a time-consuming process. Thus, to solve this issue +under a real scenario, various machine learning-based methods +have been proposed, many of which depend on analyzing text +to extract language styles of fake news, but text in social media +has a short length, and we face the data sparsity problem. Other +methods like CSI [4] need many user responses, which is time- +consuming. It also models the propagation path of retweets as +a sequence. Other models like GLAN [5] model both local and +global relations of news and users but fails to capture intrinsic +structural difference among node types when we do not have +access to manually-extracted node (user, tweet, etc.) features. +Our research question underlies the fact that if the metadata of +users and tweets are unavailable, we would be able to extract +structural features considering the difference in node types. +Therefore, The contributions of our work are: +• We select GLAN as our baseline model and aim to +capture meaningful structural embeddings using the con- +cepts of meta paths in heterogeneous news propagation +networks. +• Our experiments on three real-world datasets demonstrate +improvements over previous works in terms of accuracy +and F1 score. +For the rest of the paper, in section II, we overview related +works for solving rumor detection problem. In section III, we +define preliminaries and formulate our problem. In section IV, +we describe all components of the baseline paper as well as +our idea to improve upon it. In section V, we evaluate our +method on three real-world datasets. Finally, in section VI, all +future works applicable to our work are proposed. +1https://www.snopes.com +2https://www.gossipcop.com +3https://www.politifact.com +978-1-6654-8027-7/22/$31.00 ©2023 IEEE +arXiv:2301.04341v1 [cs.SI] 11 Jan 2023 + +Fig. 1. +The overall proposed architecture. The main difference between MGLAN compared to GLAN is that GLAN assigns node features by normal +distribution when it does not have access to manually extracted features of Users and Tweets whereas MGLAN uses the output of MetaPath2Vec in each +epoch as learned features of these node types. +II. RELATED WORK +The previous models can be divided into three categories +according to their approach: +• Content-based: Extracting rich information from texts to +learn specific writing styles that rumors inherently have +is essential for their detection [6]. Text feature extraction +can be done in supervised manners, i.e., TF-IDF and n- +gram, or in unsupervised forms, i.e., embeddings from +Word2Vec [7], LSTM [8], GRU [9], Transformers [10], +and BERT [11]. These strategies can not be solely relied +upon as text in social media usually has a short length; +thus, they fail to capture the desired syntactic and seman- +tic information needed to detect whether a piece of news +is fake. +• User-based: Manually scraped user features such as +gender, age, nationality, and numbers of followers or +followees are beneficial for rumor detection [12]. [13] +first used them to determine the credibility of information +on Twitter. However, obtaining them is challenging as +some social networks enforce restricting policies to ac- +cess users’ profiles or make them publicly available, like +Twitter. Moreover, in terms of network representation, +nodes are interconnected through edge interactions, and +their feature vectors are not independent and identically +distributed from one another [14]. +• Structure-based: Leveraging the inherent structure of +rumor propagation in social networks is another way +that helps rumor detection. In [15], the propagation path +of news was modeled as multivariate time series. They +can also be captured in a graph-setting environment +considering the structural and semantic features of the +news interaction network. In this setting, Graph Neural +Networks(GNNs) have shown a significant role in node- +level, link-level, and graph-level prediction tasks. [16] +considered atop down and bottom-up approach to capture +both propagation and dispersion of tweets using Graph +Convolutional Networks(GCNs) [17]. [18] proposed a +deep hybrid model based on propagation and stance +network of fake news and used node2vec [19] for cap- +turing structural propagation features on FakeNewsNet +dataset [20]. GLAN [5] offered a hybrid model using +Graph Attention Networks(GATs) [21] to capture node- +level representations of users and tweets. +Based on what elaborated above, GLAN offers a stable +model to capture all three aspects but fails to assign initial +features to nodes based on the difference in their types and +their interconnectivity. When no initial features of users and +tweets are accessible, It generates node features by the normal +distribution in their implementation. This assumption can +come from the fact that nodes and relations are independent, +so initial feature generation using normal distribution would +be sufficient. However, in reality, heterogeneous networks are +scale-free in nature; in these networks, nodes and relations +among them are not independent. This fact motivated us to +modify a part of GLAN’s architecture and use MetaPath2Vec +[22] for extracting features for tweets and users in the prop- +agation graph, discriminating among their node types. This +modification has shown improvement in the performance of +rumor detection when applied to three state-of-the-art datasets. +III. PRELIMINARIES AND PROBLEM FORMULATION +IIn this section, we provide some basic definitions and then +move on to formulate the rumor detection problem in this +paper. +Definition 1: Heterogeneous Network [23]. A heterogeneous +network is defined as a graph G = (V, E) with a node type + +Prediction +FC layer +Concatenate +mj +m +Cross Attention +GAT +Self Attention I +GAT +MetaPath2Vec +m; E Rd +Concatenate +CNN +CNN +CNNFig. 2. Architecture of MetaPath2Vec used as feature extraction module. The schema of heterogeneous graph is from [5] and [22]. +mapping function ϕ : V → A, and an edge type mapping +function ψ : E → R. A and R denote sets of node types and +edge types, respectively, so that ∥A∥ + ∥R∥ > 2. +Definition 2: Network Schema [24]. A network schema is a +meta template for a heterogeneous network G = (V, E), with +a node type mapping function ϕ : V → A and an edge type +mapping function ψ : E → R defined over object types A +denoted as TG = (A, R). +Definition 3: Meta Path [24]. A meta path P is a path on +heterogeneous network with network schema TG = (A, R) +in the form of A1 +R1 +−−→ A2 +R2 +−−→ ... +Rl +−→ Al+1 and R is a +compound relation R = R1 ◦ R2 ◦ ... ◦ Rl from A1 to Al. +The formulation of our problem is quite similar to GLAN. +Let M = {m1, m2, ..., m|M|} be the set of source news, +and each source news has a total of n retweets and replies +denoted as R = {r1, r2, ..., rn}. We define the neighbors +for a source news as N(mi) = {r1, r2, ..., rN(mi)}. There +are separate groups of global and local neighbors. We define +replies of a source news as its local neighbors and its retweets +as its global neighbors. The reason for assigning local and +global terminologies is that replies of a tweet are independent +from replies of other tweets, so they are categorized as local +neighbors, but retweets diffuse through the whole network. +Also, we define social media users as U = {u1, u2, ..., u|U|}. +Our objective is to learn a model p(c = 1|mi, N(mi), U; θ) +that takes a tweet and its neighbors as input. c is the output +and specifies the class to which the source news belongs. θ +determines the model parameters. +IV. PROPOSED METHOD +We aim to show that extracting meta path-based structural +features from the heterogeneous network is conducive to rumor +detection by having the propagation network without any +initial knowledge of node-level features. +In GLAN, initial node features for different node types were +assigned by normal distribution when node-level features of +tweets and users(e.g. age, gender, number of likes, etc.) were +unavailable. It is not the optimal way of feature extraction +because it fails to take advantage of the rich information that +the heterogeneous nature of the news propagation network pro- +vides. Recently, heterogeneous graph representation learning +models like MetaPath2Vec [22] have shown promising success +in extracting features using the concept of meta path in het- +erogeneous networks. Having this idea in mind, we decided to +add a key module called Meta Path-based Feature Extraction +for better global feature extraction. Adding this module helps +detect rumors more accurately in some evaluation metrics than +GLAN. Fig. 1 illustrates the whole architecture of MGLAN. +In the following, we describe our added component as well +as GLAN modules in order to maintain integration throughout +the paper. +A. Text Representation +Just like GLAN, we use word-level embeddings for word +representation. xj ∈ Rd is the d-dimensional embedding of +j-th word in text. We assume each text has fixed length L +represented as x1:L = [x1; x2; ...; xL]. Texts with more than +L words are truncated from the end till their length reaches +L, and texts with lengths less than L are zero-padded in +the beginning till text length becomes L. Then, each text +represented as x1:L is fed into three parallel CNNs [25] with +d/3 dimension output to get semantic representation for each +text. The size of the receptive field for each of the three CNNs +is different with values h ∈ {3, 4, 5}. The d/3-dimensional +outputs of each CNN are concatenated together and form the +final d dimensional representation. This process is performed +on both source news text and replies of each source news +separately, as demonstrated in Fig. 1. + +Walk Length +Walks per Node +Tweet, +Tweet, +Tweet, +Tweet, +Tweet, +input +hidden +output +Women should earn the +FBI uncovers nearly +Hadley says he spoke +Ci-2 +same pay as men for doing +15,000more emails and +with hostage, and could +the same work. URL +materials sent to or from +hear the gunman in the +#equalpay +Hillary Clinton. +background barking +Ci-1 + orders and demanding +X +to go live on air +User3 +User2 +User4 +Ci+1 +User2 +目 +User, +LUsers +Users +Ci+2 +User4 +Tweet. +Tweet2 +Tweet +Context Size +Number of Negative SamplesB. Local Relation Encoding +The term local relation of news refers to the relations that +each news has with its surrounding neighbors in such a way +that it is independent of the local relations of other news. In +this section, we take the same approach as GLAN did. We use +the attention mechanism to capture rich semantic relations of +replies and source news and combine them into a single vector +that encodes important aspects of source news and its replies. +This procedure has two steps: +1) Self Attention: We use MultiHeadAttention module as +self attention with same inputs for parameters Q, K, +and V [10]. If a piece of news has R replies, and +each encoded in d-dimensional space, the output of the +self-attention module is one d-dimensional embedding +that has aggregated features of all the previous encoded +replies denoted as �R ∈ Rd. +�R = MultiHeadAttention(R, R, R) +(1) +2) Cross Attention: We apply cross attention to infuse +source representation with the unified representation of +its replies. The input K of MultiHeadAttention is source +news representation, Q and V are �R. This way, source +representation can attend over its local neighbors to form +a new d-dimensional local text representation denoted as +� +mj for news +C. Meta Path-based Feature Extraction +To capture efficient structural node representation while +preserving inherent discrimination among each node type, we +modeled the news propagation network as a heterogeneous +graph with two node types: User and Tweet. Fig. 3 shows +the network schema of news propagation on Twitter. At first, +a Tweet is published by a user, and other users participate +in retweeting it from publishing users or other retweeters. +To simplify the schema, we decided to behave post and +retweet relations as one relation called spread that contains +both retweet and post relations. It means a user participates +in spreading tweets and tweets are spread by users in the +network. Finally, we define the meta path schema for the +rumor detection problem in Fig. 4. We feed meta path rela- +tions schema alongside the network edges into MetaPath2Vec +architecture as illustrated in Fig. 2. MetaPath2Vec is robust +among heterogeneous representation learning methods to dis- +cern structural and semantic correlation among different node +types. In order to pay more attention to early diffuser users, +weights between edges that connect users and tweets were +assigned as follows: +w(ui, mj) = +1 +max(0, t) + 1 +(2) +t is time elapsed(in minutes) after a tweet was published [5]. +This weighting helps us to modify the random walk process +of MetaPath2Vec. In the original paper, a node in the next +random walk step was selected by the uniform distribution. In +Fig. 3. News propagation network schema. +Fig. 4. Meta Path schema in news propagation network. +our work, a node is selected by weighted distribution according +to the following: +p(vi+1|vi, P) = +� +� +� +� +� +wvivi+1 +Σwviu +(vi+�, vi +t) ∈ E, φ(vi+�) = t + � +0 +(vi+�, vi +t) ∈ E, φ(vi+�) ̸= t + � +0 +(vi+�, vi +t) /∈ E +(3) +where u is selected from all neighbors of vi, u ∈ N(vi). The +output of MetaPath2Vec is d-dimensional node-level represen- +tation of all nodes X ∈ Rn×d. +D. Global Relation Encoding +Just like GLAN, we used two GATs to capture additional +meaningful representations of different nodes. The output of +MetaPath2Vec X is the input of the first GAT. In the first one, +MultiHeadAttention with k=8 heads are enabled to stabilize +the training process. In the second one, MultiHeadAttention is +disabled. The output of the global relation encoding module +is mglobal +j +. +E. Rumor Classification +In the classification module, we concatenate � +mj and mglobal +j +and pass it through a fully-connected linear layer. Then by +applying softmax and choosing the maximum probability, we +can classify each source news: +pi(c|mi, Nmi, U; θ) = softmax(W[ � +mj, mglobal +j +] + b) +(4) +W ∈ R2d×|c| is weight parameter of linear layer and b is bias. +Cross entropy loss is used to classify each piece of news: +J(c(i)|D, Ui; θ) = − +� +i +yi log pi(c|mi, Nmi, U; θ) +(5) +yi is the probability of source news belonging to class i. +V. EVALUATION +In this section, we perform experiments on three state of the +art datasets for rumor detection. We show that our proposed +architecture outperforms similar state-of-the-art models. + +retweetedfrom +Tweet +post +UserUser +Tweet +Tweet(a) Twitter15 +(b) Twitter16 +(c) Weibo +Fig. 5. Time limit vs Accuracy in Twitter15, Twitter16 and Weibo. +A. Datasets +We analyzed MGLAN on three well-known datasets: +Twitter15 [26], Twitter16 [26], and Weibo [27]. Data are +scraped from Twitter and Weibo social networks, respectively. +Twiter15 and Twitter16 consist of four classes: ”NR”(non- +rumor), +”FR”(false-rumor), +”UR”(unverified-rumor), +and +”TR”(true-rumor). The difference between ”FR” and ”TR” is +that a true label is assigned to a source tweet if it expresses +a denial type of stance; otherwise, it is labeled as false [26]. +The Weibo dataset has only binary labels ”NR” and ”FR.” +Table I provides a statistical perspective of datasets. +TABLE I +DATASET STATISTICS +Statistics +Twitter15 +Twitter16 +Weibo +# tweets +1,490 +818 +4,664 +# users +276,663 +173,487 +2,746,818 +# posts +331,612 +204,820 +3,805,656 +# NR +374 +205 +2,351 +# FR +370 +205 +2,313 +# UR +374 +203 +0 +# TR +372 +203 +0 +B. Baseline Models +In this section, we introduce previously proposed archi- +tectures for the task of rumor detection and compare their +performance to MGLAN. +• GLAN: Our baseline. It encodes both local and global +relations of the heterogeneous network. +• HGATRD [28]: It models the propagation network as a +heterogeneous one with tweet, user, and word as node +types. It decomposes the graph into tweet-word and +tweet-user subgraphs and performs attention mechanisms +for each subgraph. +• SMAN [29]: Based on the news that each user partici- +pates in spreading, it assigns a credibility score to each +user and uses it as weakly supervised information. It then +uses MultiHeadAttention to learn to classify each source +news. +• GCAN [30]: It creates user communication graphs for all +tweets, and GCNs compute their graph-level embeddings. +On the other hand, It uses CNNs to model the sequential +retweet path of each tweet. Finally, It concatenates re- +spective outputs after applying attention mechanisms and +passes them to the classifier. +• PPC [15]: It is based on modeling the propagation path +of tweets as a multivariate time series, then builds a +propagation path classifier using both CNNs and RNNs. +C. Parameter Settings +MGLAN is an extension to GLAN implemented with Py- +torch [31], and Pytorch Geometric [32]. For true comparison, +we did not change any of GLAN’s hyperparameters. In Meta- +Path2Vec, we set walk length to 100, context size to 7, number +of walks per node to 5, and number of negative samples +to 3. The output of MetaPath2Vec has 256 dimensions. The +dimension of the first GAT’s input and output is 256 and 300. +The second GAT, both input and output have 300 dimensions. +D. Results and Analysis +This section shows that MGLAN is robust enough to detect +rumors better than GLAN and other models. Tables II, III, and +IV show that MGLAN performs better in almost all classes +compared to other models by accuray and F1 score metric; +meaning extracting the heterogeneous-based representations +play an essential role for rumor detection. +However, in Twitter15 in UR class and Twitter16 in FR +class, MGLAN failed to outperform SMAN due to computing +a credibility score for users from the existing dataset and +using it as weakly supervised information. MGLAN is not +reliant on hand-crafted features and, therefore, can be applied +in inductive settings where new accounts are created and +participate in rumor dissemination. Also, we saw the slightest +improvement in accuracy and F1 score in the Weibo dataset. +It is related to the inherent characteristic of the dataset; +source texts and their corresponding replies are distinguishable +enough that considering a meta path-based heterogeneous +feature extraction performs slightly better in this case. +In conclusion, Incorporating MetaPath2Vec and capturing +heterogeneous structural features in news propagation net- +works shows improvement over previous models. We will +discuss this assumption from another perspective in section +V-F. + +Twitter15 +0.91 +06'0 +Accuracy +0.89 +0.88 +0.87 +M-GLAN +GLAN +0.86 +PPC +HGATRD +0 +4 +8 +12 +24 +Detection Deadline(hours)Twitter16 +0.93 +M-GLAN +GLAN +0.92 +PPC +HGATRD +0.91 +Accuracy +0.90 +0.89 +0.88 +0.87 +0.86 +0 +4 +8 +12 +24 +Detection Deadline(hours)Weibo +0.95 +0.94 +0.92 +0.91 +M-GLAN +GLAN +PPC +0.90 +8 +12 +24 +Detection Deadline(hours)(a) GLAN embeddings in Twitter15 +(b) GLAN embeddings in Twitter16 +(c) GLAN embeddings in Weibo +(d) MGLAN embeddings in Twitter15 +(e) MGLAN embeddings in Twitter16 +(f) MGLAN embeddings in Weibo +Fig. 6. Representations of tweet nodes of the heterogeneous news network in MGLAN vs. GLAN. Utilizing MetaPath2Vec could capture meaningful structural +features since nodes belonging to the same group almost form a cluster, whereas in GLAN, in cases where a single normal distribution assigned node features, +clusters are not formed. +TABLE II +TWITTER15 RESULTS +Model +Accuracy +NR +FR +UR +TR +F1 +F1 +F1 +F1 +PPC +0.842 +0.811 +0.875 +0.79 +0.818 +GLAN +0.900 +0.926 +0.909 +0.834 +0.918 +SMAN +0.895 +0.896 +0.888 +0.921 +0.867 +HGATRD +0.892 +0.918 +0.887 +0.909 +0.84 +MGLAN +0.917 +0.943 +0.928 +0.874 +0.921 +TABLE III +TWITTER16 RESULTS +Model +Accuracy +NR +FR +UR +TR +F1 +F1 +F1 +F1 +PPC +0.863 +0.843 +0.898 +0.837 +0.820 +GLAN +0.902 +0.921 +0.869 +0.878 +0.968 +SMAN +0.911 +0.929 +0.952 +0.931 +0.857 +HGATRD +0.886 +0.911 +0.853 +0.847 +0.927 +MGLAN +0.936 +0.938 +0.932 +0.935 +0.94 +E. Early Rumor Detection +To prove MGLAN can detect rumors in their early phases +of creation, we decided to restrict the time after a tweet +is published and examine how MGLAN performs against +other models. As shown in Fig. 5a, Fig. 5b, and Fig. 5c. +The horizontal axis shows elapsed time since a tweet was +published, and the vertical axis represents the accuracy of +models. It is apparent in all three figures that right after +the tweet was published, MGLAN and GLAN had the same +TABLE IV +WEIBO RESULSTS +Model +Accuracy +NR +FR +F1 +F1 +PPC +0.921 +0.918 +0.923 +GLAN +0.942 +0.941 +0.943 +SMAN +0.937 +0.945 +0.946 +MGLAN +0.952 +0.951 +0.952 +accuracy because the global news propagation network is not +shaped. After a short time during which the structure of the +propagation network is formed, MGLAN outperforms GLAN +and other models. +F. Robustness of MGLAN +To elaborate on the effectiveness of using MetaPath2Vec +from a different perspective, we decided to plot the t-SNE +[33] projections of heterogeneous embeddings in MGLAN +and GLAN, respectively. As shown in Fig. 6, in MGLAN, +projected MetaPath2Vec embeddings in all datasets can cluster +tweets of each class to a reasonable extent. For example, if +two tweets belong to one class, MetaPath2Vec has similar d- +dimensional embeddings for them. In contrast, in GLAN, by +assigning node features using the normal distribution when +user or tweet metadata is not accessible, projections of each +class are not distinguishable from one another, especially in +Twitter15 and Twitter16. + +Weibo Embedding in GLAN T-SNE projection +non-rumor +40 +false-rumor +20 +2-dwoo +0 +20 +40 +. +60 +40 +20 +0 +20 +40 +comp-1Twitter15 MetaPath2Vec T-SNE projection +60 +40 +20 +comp-2 +0 +20 +non-rumor +false +-40 +unverified +true +-40 +20 +0 +20 +40 +comp-1Twitter16 MetaPath2Vec T-SNE projection +20 +non-rumor +15 +false +unverified +10 +true +Z-dwoo +0 +10 +15 +-20 +-10 +0 +10 +20 +comp-1Weibo MetaPath2Vec T-SNE projection +60 +40 +20 +comp-2 +20 +-40 +non-rumor +60 +false +-60 +-40 +20 +0 +20 +40 +60 +comp-1Twitter15 Embedding in GLAN T-SNE projection +non-rumor +m +false-rumor +2 +unverified-rumor +tue-rumor +1 +Z-dwoo +-1 +-2 +-3 +-4 +-1 +1 +2 +3 +comp-1Twitter16 Embedding in GLAN T-SNE projection +3 +non-rumor +false-rumor +unverified-rumor +true-rumor +1 +comp-2 +0 +-2 +3 +C +3 +-2 +-1 +0 +i +2 +3 +comp-1VI. CONCLUSIONS AND FUTURE WORK +In this work, we proposed improving GLAN architecture +using meta path-based features in heterogeneous news propa- +gation networks called MGLAN. For implementation, we used +MetaPath2Vec to extract structural features from the hetero- +geneous graph and used its output embeddings to GLAN’s +global relation encoding module. Also, experiments on three +state-of-the-art datasets proved the strength of MGLAN. For +future work, we can extend our proposed idea in several cases: +• We can scrape a new dataset from Twitter while gathering +metadata for different node types and therefore define +new relations such as follower-followee and mentions. It +allows us to incorporate a variety of meta paths and de- +termine which one plays a more critical role in extracting +richer information from the heterogeneous network. +• MetaPath2Vec can focus on only one meta path at one +time. Newer methods like MAGNN [34] can simultane- +ously consider different meta paths and choose the best +one for each heterogeneous graph. +ACKNOWLEDGMENT +Mostafa Salehi was supported by a grant from IPM, Iran +(No. CS1401-4-162). +REFERENCES +[1] K. Shu, A. Sliva, S. Wang, J. Tang, and H. Liu, “Fake news detection on +social media: A data mining perspective,” ACM SIGKDD explorations +newsletter, vol. 19, no. 1, pp. 22–36, 2017. +[2] W. S. Paka, R. Bansal, A. Kaushik, S. Sengupta, and T. Chakraborty, +“Cross-sean: A cross-stitch semi-supervised neural attention model for +covid-19 fake news detection,” Applied Soft Computing, vol. 107, p. +107393, 2021. +[3] X. Zhou and R. Zafarani, “A survey of fake news: Fundamental the- +ories, detection methods, and opportunities,” ACM Computing Surveys +(CSUR), vol. 53, no. 5, pp. 1–40, 2020. +[4] N. Ruchansky, S. Seo, and Y. Liu, “Csi: A hybrid deep model for fake +news detection,” in Proceedings of the 2017 ACM on Conference on +Information and Knowledge Management, 2017, pp. 797–806. +[5] C. Yuan, Q. Ma, W. Zhou, J. Han, and S. Hu, “Jointly embedding the +local and global relations of heterogeneous graph for rumor detection,” +in 2019 IEEE international conference on data mining (ICDM). +IEEE, +2019, pp. 796–805. +[6] K. Shu, L. Cui, S. Wang, D. Lee, and H. Liu, “defend: Explainable fake +news detection,” in Proceedings of the 25th ACM SIGKDD international +conference on knowledge discovery & data mining, 2019, pp. 395–405. +[7] T. Mikolov, I. Sutskever, K. Chen, G. S. Corrado, and J. Dean, +“Distributed representations of words and phrases and their composi- +tionality,” Advances in neural information processing systems, vol. 26, +2013. +[8] S. Hochreiter and J. Schmidhuber, “Long short-term memory,” Neural +computation, vol. 9, no. 8, pp. 1735–1780, 1997. +[9] K. Cho, B. Van Merri¨enboer, C. Gulcehre, D. Bahdanau, F. Bougares, +H. Schwenk, and Y. Bengio, “Learning phrase representations using +rnn encoder-decoder for statistical machine translation,” arXiv preprint +arXiv:1406.1078, 2014. +[10] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, +Ł. Kaiser, and I. Polosukhin, “Attention is all you need,” Advances in +neural information processing systems, vol. 30, 2017. +[11] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, “Bert: Pre-training +of deep bidirectional transformers for language understanding,” arXiv +preprint arXiv:1810.04805, 2018. +[12] K. Shu, X. Zhou, S. Wang, R. Zafarani, and H. Liu, “The role of user +profiles for fake news detection,” in Proceedings of the 2019 IEEE/ACM +international conference on advances in social networks analysis and +mining, 2019, pp. 436–439. +[13] C. Castillo, M. Mendoza, and B. Poblete, “Information credibility on +twitter,” in Proceedings of the 20th international conference on World +wide web, 2011, pp. 675–684. +[14] V.-H. Nguyen, K. Sugiyama, P. Nakov, and M.-Y. Kan, “Fang: Leverag- +ing social context for fake news detection using graph representation,” in +Proceedings of the 29th ACM international conference on information +& knowledge management, 2020, pp. 1165–1174. +[15] Y. Liu and Y.-F. Wu, “Early detection of fake news on social media +through propagation path classification with recurrent and convolutional +networks,” in Proceedings of the AAAI conference on artificial intelli- +gence, vol. 32, no. 1, 2018. +[16] T. Bian, X. Xiao, T. Xu, P. Zhao, W. Huang, Y. Rong, and J. Huang, +“Rumor detection on social media with bi-directional graph convolu- +tional networks,” in Proceedings of the AAAI conference on artificial +intelligence, vol. 34, no. 01, 2020, pp. 549–556. +[17] M. Welling and T. N. Kipf, “Semi-supervised classification with graph +convolutional networks,” in J. International Conference on Learning +Representations (ICLR 2017), 2016. +[18] M. Davoudi, M. R. Moosavi, and M. H. Sadreddini, “Dss: A hybrid +deep model for fake news detection using propagation tree and stance +network,” Expert Systems with Applications, vol. 198, p. 116635, 2022. +[19] A. Grover and J. Leskovec, “node2vec: Scalable feature learning for +networks,” in Proceedings of the 22nd ACM SIGKDD international +conference on Knowledge discovery and data mining, 2016, pp. 855– +864. +[20] K. Shu, D. Mahudeswaran, S. Wang, D. Lee, and H. Liu, “Fakenewsnet: +A data repository with news content, social context, and spatiotemporal +information for studying fake news on social media,” Big data, vol. 8, +no. 3, pp. 171–188, 2020. +[21] P. Veliˇckovi´c, G. Cucurull, A. Casanova, A. Romero, P. Lio, and Y. Ben- +gio, “Graph attention networks,” arXiv preprint arXiv:1710.10903, 2017. +[22] Y. Dong, N. V. Chawla, and A. Swami, “metapath2vec: Scalable rep- +resentation learning for heterogeneous networks,” in Proceedings of the +23rd ACM SIGKDD international conference on knowledge discovery +and data mining, 2017, pp. 135–144. +[23] Y. Sun and J. Han, “Mining heterogeneous information networks: +a structural analysis approach,” Acm Sigkdd Explorations Newsletter, +vol. 14, no. 2, pp. 20–28, 2013. +[24] Y. Sun, J. Han, X. Yan, P. S. Yu, and T. Wu, “Pathsim: Meta path- +based top-k similarity search in heterogeneous information networks,” +Proceedings of the VLDB Endowment, vol. 4, no. 11, pp. 992–1003, +2011. +[25] Y. Zhang and B. Wallace, “A sensitivity analysis of (and practitioners’ +guide to) convolutional neural networks for sentence classification,” +arXiv preprint arXiv:1510.03820, 2015. +[26] J. Ma, W. Gao, and K.-F. Wong, “Detect rumors in microblog posts +using propagation structure via kernel learning,” in Proceedings of the +55th Annual Meeting of the Association for Computational Linguistics +(Volume 1: Long Papers), vol. 1, 2017, pp. 708–717. +[27] J. Ma, W. Gao, P. Mitra, S. Kwon, B. J. Jansen, K.-F. Wong, and +C. Meeyoung, “Detecting rumors from microblogs with recurrent neural +networks,” in The 25th International Joint Conference on Artificial +Intelligence. +AAAI, 2016. +[28] Q. Huang, J. Yu, J. Wu, and B. Wang, “Heterogeneous graph attention +networks for early detection of rumors on twitter,” in 2020 International +Joint Conference on Neural Networks (IJCNN). +IEEE, 2020, pp. 1–8. +[29] C. Yuan, Q. Ma, W. Zhou, J. Han, and S. Hu, “Early detection of fake +news by utilizing the credibility of news, publishers, and users based on +weakly supervised learning,” arXiv preprint arXiv:2012.04233, 2020. +[30] Y.-J. Lu and C.-T. Li, “Gcan: Graph-aware co-attention networks +for explainable fake news detection on social media,” arXiv preprint +arXiv:2004.11648, 2020. +[31] A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, +A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in +pytorch,” 2017. +[32] M. Fey and J. E. Lenssen, “Fast graph representation learning with +pytorch geometric,” arXiv preprint arXiv:1903.02428, 2019. +[33] L. Van der Maaten and G. Hinton, “Visualizing data using t-sne.” Journal +of machine learning research, vol. 9, no. 11, 2008. +[34] X. Fu, J. Zhang, Z. Meng, and I. King, “Magnn: Metapath aggregated +graph neural network for heterogeneous graph embedding,” in Proceed- +ings of The Web Conference 2020, 2020, pp. 2331–2341. + diff --git a/jdE3T4oBgHgl3EQfJQkl/content/tmp_files/load_file.txt b/jdE3T4oBgHgl3EQfJQkl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..824464672cce7f568b3c16d16c261b5703f1d7fd --- /dev/null +++ b/jdE3T4oBgHgl3EQfJQkl/content/tmp_files/load_file.txt @@ -0,0 +1,686 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf,len=685 +page_content='A Meta Path-based Approach for Rumor Detection on Social Media Bita Azarijoo Faculty of New Sciences and Technologies University of Tehran Tehran, Iran azarijoo@ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='ir Mostafa Salehi Faculty of New Sciences and Technologies University of Tehran School of Computer Science, IPM Institute for Research in Fundamental Sciences Tehran, Iran mostafa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='salehi@ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='ir Shaghayegh Najari Faculty of New Sciences and Technologies University of Tehran Tehran, Iran najari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='shaghayegh@ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='ir Abstract—The prominent role of social media in people’s daily lives has made them more inclined to receive news through social networks than traditional sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' This shift in public behavior has opened doors for some to diffuse fake news on social media;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' and subsequently cause negative economic, political, and social consequences as well as distrust among the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' There are many proposed methods to solve the rumor detection problem, most of which do not take full advantage of the heterogeneous nature of news propagation networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' With this in- tention, we considered a previously proposed architecture as our baseline and performed the idea of structural feature extraction from the heterogeneous rumor propagation over its architecture using the concept of meta path-based embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' We named our model Meta Path-based Global Local Attention Network (MGLAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Extensive experimental analysis on three state-of-the- art datasets has demonstrated that MGLAN outperforms other models by capturing node-level discrimination to different node types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Index Terms—Rumor Detection, Heterogenous Network, Meta Path, Deep Learning, Social Network I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' INTRODUCTION Nowadays, interactions with social networks have become an inseparable part of people’s lives for their ease of use and fast dissemination of information on a global scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' In this regard, in 2012, only 45% of people used social media to access news, whereas this number jumped to 65% in 2016 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Also, the 2020 Covid-19 pandemic caused a 30% growth in Twitter daily usage [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Unfortunately, this rapid increase in using social media has provided an opportunity for vicious users to spread fake news to cause serious individual, eco- nomic, and political repercussions [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' For example, in 2013, the Associated Press (AP) account on Twitter was hacked, and a piece of news was published claiming an explosion occurred in the White House and Barak Obama was injured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Although the publishing account discredited this rumor within seconds, it leveraged through Twitter and caused the stock value to fall by 130 billion dollars [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Consequently, studying fake news and preventing its dissemination as soon as possible is yet an active and open research area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' To avoid the detrimental effects of fake news, there are websites like Snopes1, GossipCop2, and Poitifact3, but they can not detect fake news automatically in their early stages of propagation and rely on manual user intervention for fact- checking as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' As a consequence, the detection of fake news can be a time-consuming process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Thus, to solve this issue under a real scenario, various machine learning-based methods have been proposed, many of which depend on analyzing text to extract language styles of fake news, but text in social media has a short length, and we face the data sparsity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Other methods like CSI [4] need many user responses, which is time- consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' It also models the propagation path of retweets as a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Other models like GLAN [5] model both local and global relations of news and users but fails to capture intrinsic structural difference among node types when we do not have access to manually-extracted node (user, tweet, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=') features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Our research question underlies the fact that if the metadata of users and tweets are unavailable, we would be able to extract structural features considering the difference in node types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Therefore, The contributions of our work are: We select GLAN as our baseline model and aim to capture meaningful structural embeddings using the con- cepts of meta paths in heterogeneous news propagation networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Our experiments on three real-world datasets demonstrate improvements over previous works in terms of accuracy and F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' For the rest of the paper, in section II, we overview related works for solving rumor detection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' In section III, we define preliminaries and formulate our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' In section IV, we describe all components of the baseline paper as well as our idea to improve upon it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' In section V, we evaluate our method on three real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Finally, in section VI, all future works applicable to our work are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='snopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='com 2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='gossipcop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='com 3https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='politifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='com 978-1-6654-8027-7/22/$31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='00 ©2023 IEEE arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='04341v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='SI] 11 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' The overall proposed architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' The main difference between MGLAN compared to GLAN is that GLAN assigns node features by normal distribution when it does not have access to manually extracted features of Users and Tweets whereas MGLAN uses the output of MetaPath2Vec in each epoch as learned features of these node types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' RELATED WORK The previous models can be divided into three categories according to their approach: Content-based: Extracting rich information from texts to learn specific writing styles that rumors inherently have is essential for their detection [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Text feature extraction can be done in supervised manners, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=', TF-IDF and n- gram, or in unsupervised forms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=', embeddings from Word2Vec [7], LSTM [8], GRU [9], Transformers [10], and BERT [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' These strategies can not be solely relied upon as text in social media usually has a short length;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' thus, they fail to capture the desired syntactic and seman- tic information needed to detect whether a piece of news is fake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' User-based: Manually scraped user features such as gender, age, nationality, and numbers of followers or followees are beneficial for rumor detection [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [13] first used them to determine the credibility of information on Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' However, obtaining them is challenging as some social networks enforce restricting policies to ac- cess users’ profiles or make them publicly available, like Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Moreover, in terms of network representation, nodes are interconnected through edge interactions, and their feature vectors are not independent and identically distributed from one another [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Structure-based: Leveraging the inherent structure of rumor propagation in social networks is another way that helps rumor detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' In [15], the propagation path of news was modeled as multivariate time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' They can also be captured in a graph-setting environment considering the structural and semantic features of the news interaction network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' In this setting, Graph Neural Networks(GNNs) have shown a significant role in node- level, link-level, and graph-level prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [16] considered atop down and bottom-up approach to capture both propagation and dispersion of tweets using Graph Convolutional Networks(GCNs) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [18] proposed a deep hybrid model based on propagation and stance network of fake news and used node2vec [19] for cap- turing structural propagation features on FakeNewsNet dataset [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' GLAN [5] offered a hybrid model using Graph Attention Networks(GATs) [21] to capture node- level representations of users and tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Based on what elaborated above, GLAN offers a stable model to capture all three aspects but fails to assign initial features to nodes based on the difference in their types and their interconnectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' When no initial features of users and tweets are accessible, It generates node features by the normal distribution in their implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' This assumption can come from the fact that nodes and relations are independent, so initial feature generation using normal distribution would be sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' However, in reality, heterogeneous networks are scale-free in nature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' in these networks, nodes and relations among them are not independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' This fact motivated us to modify a part of GLAN’s architecture and use MetaPath2Vec [22] for extracting features for tweets and users in the prop- agation graph, discriminating among their node types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' This modification has shown improvement in the performance of rumor detection when applied to three state-of-the-art datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' PRELIMINARIES AND PROBLEM FORMULATION IIn this section, we provide some basic definitions and then move on to formulate the rumor detection problem in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Definition 1: Heterogeneous Network [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' A heterogeneous network is defined as a graph G = (V, E) with a node type Prediction FC layer Concatenate mj m Cross Attention GAT Self Attention I GAT MetaPath2Vec m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' E Rd Concatenate CNN CNN CNNFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Architecture of MetaPath2Vec used as feature extraction module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' The schema of heterogeneous graph is from [5] and [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' mapping function ϕ : V → A, and an edge type mapping function ψ : E → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' A and R denote sets of node types and edge types, respectively, so that ∥A∥ + ∥R∥ > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Definition 2: Network Schema [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' A network schema is a meta template for a heterogeneous network G = (V, E), with a node type mapping function ϕ : V → A and an edge type mapping function ψ : E → R defined over object types A denoted as TG = (A, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Definition 3: Meta Path [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' A meta path P is a path on heterogeneous network with network schema TG = (A, R) in the form of A1 R1 −−→ A2 R2 −−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Rl −→ Al+1 and R is a compound relation R = R1 ◦ R2 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' ◦ Rl from A1 to Al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' The formulation of our problem is quite similar to GLAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Let M = {m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=', m|M|} be the set of source news, and each source news has a total of n retweets and replies denoted as R = {r1, r2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=', rn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' We define the neighbors for a source news as N(mi) = {r1, r2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=', rN(mi)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' There are separate groups of global and local neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' We define replies of a source news as its local neighbors and its retweets as its global neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' The reason for assigning local and global terminologies is that replies of a tweet are independent from replies of other tweets, so they are categorized as local neighbors, but retweets diffuse through the whole network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Also, we define social media users as U = {u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=', u|U|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Our objective is to learn a model p(c = 1|mi, N(mi), U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' θ) that takes a tweet and its neighbors as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' c is the output and specifies the class to which the source news belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' θ determines the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' PROPOSED METHOD We aim to show that extracting meta path-based structural features from the heterogeneous network is conducive to rumor detection by having the propagation network without any initial knowledge of node-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' In GLAN, initial node features for different node types were assigned by normal distribution when node-level features of tweets and users(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' age, gender, number of likes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=') were unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' It is not the optimal way of feature extraction because it fails to take advantage of the rich information that the heterogeneous nature of the news propagation network pro- vides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Recently, heterogeneous graph representation learning models like MetaPath2Vec [22] have shown promising success in extracting features using the concept of meta path in het- erogeneous networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Having this idea in mind, we decided to add a key module called Meta Path-based Feature Extraction for better global feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Adding this module helps detect rumors more accurately in some evaluation metrics than GLAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 1 illustrates the whole architecture of MGLAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' In the following, we describe our added component as well as GLAN modules in order to maintain integration throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Text Representation Just like GLAN, we use word-level embeddings for word representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' xj ∈ Rd is the d-dimensional embedding of j-th word in text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' We assume each text has fixed length L represented as x1:L = [x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' xL].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Texts with more than L words are truncated from the end till their length reaches L, and texts with lengths less than L are zero-padded in the beginning till text length becomes L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Then, each text represented as x1:L is fed into three parallel CNNs [25] with d/3 dimension output to get semantic representation for each text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' The size of the receptive field for each of the three CNNs is different with values h ∈ {3, 4, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' The d/3-dimensional outputs of each CNN are concatenated together and form the final d dimensional representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' This process is performed on both source news text and replies of each source news separately, as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Walk Length Walks per Node Tweet, Tweet, Tweet, Tweet, Tweet, input hidden output Women should earn the FBI uncovers nearly Hadley says he spoke Ci-2 same pay as men for doing 15,000more emails and with hostage, and could the same work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' URL materials sent to or from hear the gunman in the #equalpay Hillary Clinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' background barking Ci-1 orders and demanding X to go live on air User3 User2 User4 Ci+1 User2 目 User, LUsers Users Ci+2 User4 Tweet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Tweet2 Tweet Context Size Number of Negative SamplesB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Local Relation Encoding The term local relation of news refers to the relations that each news has with its surrounding neighbors in such a way that it is independent of the local relations of other news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' In this section, we take the same approach as GLAN did.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' We use the attention mechanism to capture rich semantic relations of replies and source news and combine them into a single vector that encodes important aspects of source news and its replies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' This procedure has two steps: 1) Self Attention: We use MultiHeadAttention module as self attention with same inputs for parameters Q, K, and V [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' If a piece of news has R replies, and each encoded in d-dimensional space, the output of the self-attention module is one d-dimensional embedding that has aggregated features of all the previous encoded replies denoted as �R ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' �R = MultiHeadAttention(R, R, R) (1) 2) Cross Attention: We apply cross attention to infuse source representation with the unified representation of its replies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' The input K of MultiHeadAttention is source news representation, Q and V are �R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' This way, source representation can attend over its local neighbors to form a new d-dimensional local text representation denoted as � mj for news C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Meta Path-based Feature Extraction To capture efficient structural node representation while preserving inherent discrimination among each node type, we modeled the news propagation network as a heterogeneous graph with two node types: User and Tweet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 3 shows the network schema of news propagation on Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' At first, a Tweet is published by a user, and other users participate in retweeting it from publishing users or other retweeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' To simplify the schema, we decided to behave post and retweet relations as one relation called spread that contains both retweet and post relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' It means a user participates in spreading tweets and tweets are spread by users in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Finally, we define the meta path schema for the rumor detection problem in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' We feed meta path rela- tions schema alongside the network edges into MetaPath2Vec architecture as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' MetaPath2Vec is robust among heterogeneous representation learning methods to dis- cern structural and semantic correlation among different node types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' In order to pay more attention to early diffuser users, weights between edges that connect users and tweets were assigned as follows: w(ui, mj) = 1 max(0, t) + 1 (2) t is time elapsed(in minutes) after a tweet was published [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' This weighting helps us to modify the random walk process of MetaPath2Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' In the original paper, a node in the next random walk step was selected by the uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' News propagation network schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Meta Path schema in news propagation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' our work, a node is selected by weighted distribution according to the following: p(vi+1|vi, P) = � � � � � wvivi+1 Σwviu (vi+�, vi t) ∈ E, φ(vi+�) = t + � 0 (vi+�, vi t) ∈ E, φ(vi+�) ̸= t + � 0 (vi+�, vi t) /∈ E (3) where u is selected from all neighbors of vi, u ∈ N(vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' The output of MetaPath2Vec is d-dimensional node-level represen- tation of all nodes X ∈ Rn×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Global Relation Encoding Just like GLAN, we used two GATs to capture additional meaningful representations of different nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' The output of MetaPath2Vec X is the input of the first GAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' In the first one, MultiHeadAttention with k=8 heads are enabled to stabilize the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' In the second one, MultiHeadAttention is disabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' The output of the global relation encoding module is mglobal j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Rumor Classification In the classification module, we concatenate � mj and mglobal j and pass it through a fully-connected linear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Then by applying softmax and choosing the maximum probability, we can classify each source news: pi(c|mi, Nmi, U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' θ) = softmax(W[ � mj, mglobal j ] + b) (4) W ∈ R2d×|c| is weight parameter of linear layer and b is bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Cross entropy loss is used to classify each piece of news: J(c(i)|D, Ui;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' θ) = − � i yi log pi(c|mi, Nmi, U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' θ) (5) yi is the probability of source news belonging to class i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' EVALUATION In this section, we perform experiments on three state of the art datasets for rumor detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' We show that our proposed architecture outperforms similar state-of-the-art models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' retweetedfrom Tweet post UserUser Tweet Tweet(a) Twitter15 (b) Twitter16 (c) Weibo Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Time limit vs Accuracy in Twitter15, Twitter16 and Weibo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Datasets We analyzed MGLAN on three well-known datasets: Twitter15 [26], Twitter16 [26], and Weibo [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Data are scraped from Twitter and Weibo social networks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Twiter15 and Twitter16 consist of four classes: ”NR”(non- rumor), ”FR”(false-rumor), ”UR”(unverified-rumor), and ”TR”(true-rumor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' The difference between ”FR” and ”TR” is that a true label is assigned to a source tweet if it expresses a denial type of stance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' otherwise, it is labeled as false [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' The Weibo dataset has only binary labels ”NR” and ”FR.” Table I provides a statistical perspective of datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' TABLE I DATASET STATISTICS Statistics Twitter15 Twitter16 Weibo # tweets 1,490 818 4,664 # users 276,663 173,487 2,746,818 # posts 331,612 204,820 3,805,656 # NR 374 205 2,351 # FR 370 205 2,313 # UR 374 203 0 # TR 372 203 0 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Baseline Models In this section, we introduce previously proposed archi- tectures for the task of rumor detection and compare their performance to MGLAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' GLAN: Our baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' It encodes both local and global relations of the heterogeneous network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' HGATRD [28]: It models the propagation network as a heterogeneous one with tweet, user, and word as node types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' It decomposes the graph into tweet-word and tweet-user subgraphs and performs attention mechanisms for each subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' SMAN [29]: Based on the news that each user partici- pates in spreading, it assigns a credibility score to each user and uses it as weakly supervised information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' It then uses MultiHeadAttention to learn to classify each source news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' GCAN [30]: It creates user communication graphs for all tweets, and GCNs compute their graph-level embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' On the other hand, It uses CNNs to model the sequential retweet path of each tweet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Finally, It concatenates re- spective outputs after applying attention mechanisms and passes them to the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' PPC [15]: It is based on modeling the propagation path of tweets as a multivariate time series, then builds a propagation path classifier using both CNNs and RNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Parameter Settings MGLAN is an extension to GLAN implemented with Py- torch [31], and Pytorch Geometric [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' For true comparison, we did not change any of GLAN’s hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' In Meta- Path2Vec, we set walk length to 100, context size to 7, number of walks per node to 5, and number of negative samples to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' The output of MetaPath2Vec has 256 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' The dimension of the first GAT’s input and output is 256 and 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' The second GAT, both input and output have 300 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Results and Analysis This section shows that MGLAN is robust enough to detect rumors better than GLAN and other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Tables II, III, and IV show that MGLAN performs better in almost all classes compared to other models by accuray and F1 score metric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' meaning extracting the heterogeneous-based representations play an essential role for rumor detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' However, in Twitter15 in UR class and Twitter16 in FR class, MGLAN failed to outperform SMAN due to computing a credibility score for users from the existing dataset and using it as weakly supervised information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' MGLAN is not reliant on hand-crafted features and, therefore, can be applied in inductive settings where new accounts are created and participate in rumor dissemination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Also, we saw the slightest improvement in accuracy and F1 score in the Weibo dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' It is related to the inherent characteristic of the dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' source texts and their corresponding replies are distinguishable enough that considering a meta path-based heterogeneous feature extraction performs slightly better in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' In conclusion, Incorporating MetaPath2Vec and capturing heterogeneous structural features in news propagation net- works shows improvement over previous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' We will discuss this assumption from another perspective in section V-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Twitter15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content="91 06'0 Accuracy 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='87 M-GLAN GLAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='86 PPC HGATRD 0 4 8 12 24 Detection Deadline(hours)Twitter16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='93 M-GLAN GLAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='92 PPC HGATRD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='91 Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='86 0 4 8 12 24 Detection Deadline(hours)Weibo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='91 M-GLAN GLAN PPC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='90 8 12 24 Detection Deadline(hours)(a) GLAN embeddings in Twitter15 (b) GLAN embeddings in Twitter16 (c) GLAN embeddings in Weibo (d) MGLAN embeddings in Twitter15 (e) MGLAN embeddings in Twitter16 (f) MGLAN embeddings in Weibo Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Representations of tweet nodes of the heterogeneous news network in MGLAN vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' GLAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Utilizing MetaPath2Vec could capture meaningful structural features since nodes belonging to the same group almost form a cluster, whereas in GLAN, in cases where a single normal distribution assigned node features, clusters are not formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' TABLE II TWITTER15 RESULTS Model Accuracy NR FR UR TR F1 F1 F1 F1 PPC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='842 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='811 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='875 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='818 GLAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='926 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='909 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='834 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='918 SMAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='895 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='896 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='888 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='921 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='867 HGATRD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='892 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='918 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='887 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='909 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='84 MGLAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='917 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='943 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='928 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='874 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='921 TABLE III TWITTER16 RESULTS Model Accuracy NR FR UR TR F1 F1 F1 F1 PPC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='863 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='843 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='898 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='837 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='820 GLAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='902 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='921 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='869 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='878 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='968 SMAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='911 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='929 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='952 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='931 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='857 HGATRD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='886 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='911 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='853 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='847 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='927 MGLAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='936 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='938 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='932 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='94 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Early Rumor Detection To prove MGLAN can detect rumors in their early phases of creation, we decided to restrict the time after a tweet is published and examine how MGLAN performs against other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 5a, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 5b, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' The horizontal axis shows elapsed time since a tweet was published, and the vertical axis represents the accuracy of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' It is apparent in all three figures that right after the tweet was published, MGLAN and GLAN had the same TABLE IV WEIBO RESULSTS Model Accuracy NR FR F1 F1 PPC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='921 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='918 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='923 GLAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='942 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='941 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='943 SMAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='937 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='946 MGLAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='952 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='951 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='952 accuracy because the global news propagation network is not shaped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' After a short time during which the structure of the propagation network is formed, MGLAN outperforms GLAN and other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Robustness of MGLAN To elaborate on the effectiveness of using MetaPath2Vec from a different perspective, we decided to plot the t-SNE [33] projections of heterogeneous embeddings in MGLAN and GLAN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 6, in MGLAN, projected MetaPath2Vec embeddings in all datasets can cluster tweets of each class to a reasonable extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' For example, if two tweets belong to one class, MetaPath2Vec has similar d- dimensional embeddings for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' In contrast, in GLAN, by assigning node features using the normal distribution when user or tweet metadata is not accessible, projections of each class are not distinguishable from one another, especially in Twitter15 and Twitter16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Weibo Embedding in GLAN T-SNE projection non-rumor 40 false-rumor 20 2-dwoo 0 20 40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='comp-1VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' CONCLUSIONS AND FUTURE WORK In this work, we proposed improving GLAN architecture using meta path-based features in heterogeneous news propa- gation networks called MGLAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' For implementation, we used MetaPath2Vec to extract structural features from the hetero- geneous graph and used its output embeddings to GLAN’s global relation encoding module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Also, experiments on three state-of-the-art datasets proved the strength of MGLAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' For future work, we can extend our proposed idea in several cases: We can scrape a new dataset from Twitter while gathering metadata for different node types and therefore define new relations such as follower-followee and mentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' It allows us to incorporate a variety of meta paths and de- termine which one plays a more critical role in extracting richer information from the heterogeneous network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' MetaPath2Vec can focus on only one meta path at one time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Newer methods like MAGNN [34] can simultane- ously consider different meta paths and choose the best one for each heterogeneous graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' ACKNOWLEDGMENT Mostafa Salehi was supported by a grant from IPM, Iran (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' CS1401-4-162).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' REFERENCES [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Shu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Sliva, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Tang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Liu, “Fake news detection on social media: A data mining perspective,” ACM SIGKDD explorations newsletter, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 19, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 22–36, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [2] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Paka, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Bansal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Kaushik, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Sengupta, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Chakraborty, “Cross-sean: A cross-stitch semi-supervised neural attention model for covid-19 fake news detection,” Applied Soft Computing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 107, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 107393, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [3] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Zhou and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Zafarani, “A survey of fake news: Fundamental the- ories, detection methods, and opportunities,” ACM Computing Surveys (CSUR), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 53, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 1–40, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [4] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Ruchansky, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Seo, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Liu, “Csi: A hybrid deep model for fake news detection,” in Proceedings of the 2017 ACM on Conference on Information and Knowledge Management, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 797–806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Yuan, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Ma, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Zhou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Han, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Hu, “Jointly embedding the local and global relations of heterogeneous graph for rumor detection,” in 2019 IEEE international conference on data mining (ICDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' IEEE, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 796–805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [6] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Shu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Cui, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Lee, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Liu, “defend: Explainable fake news detection,” in Proceedings of the 25th ACM SIGKDD international conference on knowledge discovery & data mining, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 395–405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [7] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Mikolov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Sutskever, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Chen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Corrado, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Dean, “Distributed representations of words and phrases and their composi- tionality,” Advances in neural information processing systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 26, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Hochreiter and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Schmidhuber, “Long short-term memory,” Neural computation, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 1735–1780, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [9] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Cho, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Van Merri¨enboer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Gulcehre, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Bahdanau, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Bougares, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Schwenk, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Bengio, “Learning phrase representations using rnn encoder-decoder for statistical machine translation,” arXiv preprint arXiv:1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='1078, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Vaswani, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Shazeer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Parmar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Uszkoreit, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Jones, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Gomez, Ł.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Kaiser, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Polosukhin, “Attention is all you need,” Advances in neural information processing systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Devlin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Chang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Lee, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Toutanova, “Bert: Pre-training of deep bidirectional transformers for language understanding,” arXiv preprint arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='04805, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [12] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Shu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Zhou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Wang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Zafarani, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Liu, “The role of user profiles for fake news detection,” in Proceedings of the 2019 IEEE/ACM international conference on advances in social networks analysis and mining, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 436–439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [13] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Castillo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Mendoza, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Poblete, “Information credibility on twitter,” in Proceedings of the 20th international conference on World wide web, 2011, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 675–684.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [14] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Nguyen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Sugiyama, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Nakov, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Kan, “Fang: Leverag- ing social context for fake news detection using graph representation,” in Proceedings of the 29th ACM international conference on information & knowledge management, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 1165–1174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [15] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Liu and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Wu, “Early detection of fake news on social media through propagation path classification with recurrent and convolutional networks,” in Proceedings of the AAAI conference on artificial intelli- gence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 32, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 1, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [16] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Bian, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Xiao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Xu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Zhao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Rong, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Huang, “Rumor detection on social media with bi-directional graph convolu- tional networks,” in Proceedings of the AAAI conference on artificial intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 01, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 549–556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Welling and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Kipf, “Semi-supervised classification with graph convolutional networks,” in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' International Conference on Learning Representations (ICLR 2017), 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Davoudi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Moosavi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Sadreddini, “Dss: A hybrid deep model for fake news detection using propagation tree and stance network,” Expert Systems with Applications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 198, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 116635, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Grover and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Leskovec, “node2vec: Scalable feature learning for networks,” in Proceedings of the 22nd ACM SIGKDD international conference on Knowledge discovery and data mining, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 855– 864.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [20] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Shu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Mahudeswaran, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Lee, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Liu, “Fakenewsnet: A data repository with news content, social context, and spatiotemporal information for studying fake news on social media,” Big data, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 171–188, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [21] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Veliˇckovi´c, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Cucurull, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Casanova, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Romero, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Lio, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Ben- gio, “Graph attention networks,” arXiv preprint arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='10903, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [22] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Dong, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Chawla, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Swami, “metapath2vec: Scalable rep- resentation learning for heterogeneous networks,” in Proceedings of the 23rd ACM SIGKDD international conference on knowledge discovery and data mining, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 135–144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [23] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Sun and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Han, “Mining heterogeneous information networks: a structural analysis approach,” Acm Sigkdd Explorations Newsletter, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 20–28, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [24] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Sun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Han, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Yan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Yu, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Wu, “Pathsim: Meta path- based top-k similarity search in heterogeneous information networks,” Proceedings of the VLDB Endowment, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 4, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 992–1003, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [25] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Zhang and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Wallace, “A sensitivity analysis of (and practitioners’ guide to) convolutional neural networks for sentence classification,” arXiv preprint arXiv:1510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='03820, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [26] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Ma, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Gao, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Wong, “Detect rumors in microblog posts using propagation structure via kernel learning,” in Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 1, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 708–717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [27] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Ma, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Gao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Mitra, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Kwon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Jansen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Wong, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Meeyoung, “Detecting rumors from microblogs with recurrent neural networks,” in The 25th International Joint Conference on Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' AAAI, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [28] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Yu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Wu, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Wang, “Heterogeneous graph attention networks for early detection of rumors on twitter,” in 2020 International Joint Conference on Neural Networks (IJCNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' IEEE, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [29] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Yuan, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Ma, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Zhou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Han, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Hu, “Early detection of fake news by utilizing the credibility of news, publishers, and users based on weakly supervised learning,” arXiv preprint arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='04233, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [30] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Lu and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Li, “Gcan: Graph-aware co-attention networks for explainable fake news detection on social media,” arXiv preprint arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='11648, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [31] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Paszke, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Gross, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Chintala, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Chanan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' DeVito, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Lin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Desmaison, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Antiga, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Lerer, “Automatic differentiation in pytorch,” 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [32] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Fey and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Lenssen, “Fast graph representation learning with pytorch geometric,” arXiv preprint arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content='02428, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [33] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Van der Maaten and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Hinton, “Visualizing data using t-sne.” Journal of machine learning research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 11, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' [34] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Fu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' Meng, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' King, “Magnn: Metapath aggregated graph neural network for heterogeneous graph embedding,” in Proceed- ings of The Web Conference 2020, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} +page_content=' 2331–2341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQfJQkl/content/2301.04341v1.pdf'} diff --git a/l9AzT4oBgHgl3EQfNvvD/content/2301.01155v1.pdf b/l9AzT4oBgHgl3EQfNvvD/content/2301.01155v1.pdf new file mode 100644 index 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a/mdE0T4oBgHgl3EQfZQAg/content/tmp_files/2301.02317v1.pdf.txt b/mdE0T4oBgHgl3EQfZQAg/content/tmp_files/2301.02317v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..31ba87dc6e5bd3862eadce43cfded14d2f44fa16 --- /dev/null +++ b/mdE0T4oBgHgl3EQfZQAg/content/tmp_files/2301.02317v1.pdf.txt @@ -0,0 +1,2113 @@ + +Convolutional XGBoost (C-XGBOOST) Model for Brain +Tumor Detection + + +Abstract +Brain tumors are masses or abnormal growths of cells +within the brain or the central spinal canal with symptoms such +as headaches, seizures, weakness or numbness in the arms or +legs, changes in personality or behaviour, nausea, vomiting, +vision or hearing problems and dizziness. Conventional +diagnosis of brain tumour involves some tests and procedure +which may include the consideration of medical history, physical +examination, imaging tests (such as CT or MRI scans), and +biopsy (removal and examination of a small piece of the tumor +tissue). These procedures, while effective, are mentally +strenuous and time demanding due to the manual examination +of the brain scans and the thorough evaluation of test results. It +has been established in lots of medical research that brain +tumours diagnosed and treated early generally tends to have a +better prognosis. Deep learning techniques have evolved over +the years and have demonstrated impressive and faster +outcomes in the classification of brain tumours in medical +imaging, with very little to no human interference. This study +proposes a model for the early detection of brain tumours using +a combination of convolutional neural networks (CNNs) and +extreme gradient boosting (XGBoost). The proposed model, +named C-XGBoost has a lower model complexity compared to +purely CNNs, making it easier to train and less prone to +overfitting. It is also better able to handle imbalanced and +unstructured data, which are common issues in real-world +medical image classification tasks. + +To evaluate the effectiveness of the proposed model, we +employed a dataset of brain MRI images with and without +tumours. The dataset used for training and testing was retrieved +from figshare public repository, it contains 2 folders with 253 +positive and negative brain MRI Images. The data was pre- +processed and augmented to ensure a diverse and more +representative sample. The C-XGBoost model was trained and +validated on the dataset, and the results were compared to those +of a non-hybrid CNN-based model. The model extracts data +features using the DenseNet-121 transfer learning model. The +proposed model achieved an F1 score of 0.97 and an accuracy of +99.02%, outperforming the CNN-based model, which achieved +an accuracy of 98.8%, demonstrating its potential as a more +reliable technique for detecting brain tumours. The C-XGBoost +model had a lower training and validation loss, indicating better +generalization to the test set. Our experimental results showed +that the proposed model achieved high levels of accuracy in +detecting brain tumours from medical images, making it a viable +approach for early detection of brain tumours. + + +Keywords - Brain tumour; MRI; CNN; XGBoost; C-XGBoost; +DenseNet121; Transfer learning; deep learning; convolutional +neural network +I. INTRODUCTION +According to data from Cancer Research UK, in 2018, there +were an estimated 10,800 new cases of brain and central +nervous system tumors diagnosed in the United Kingdom [1]. +This number includes both benign (non-cancerous) and +malignant (cancerous) tumors. The incidence rate of brain +tumors in the UK varies by age, with the highest rates +occurring in children under the age of 15 and in adults over the +age of 65. The most common type of brain tumors in the UK +is a glioma, which accounts for around half of all brain tumors. +WHO estimates that brain and other central nervous system +tumors accounted for around 2.2% of all deaths from cancer +in 2018, which was approximately 784,000 deaths worldwide. +This represents a relatively high mortality rate, as brain tumors +are often difficult to treat and can be aggressive in nature. + +Brain tumors can be caused by a variety of factors, including +genetics, radiation exposure, and certain viral infections. +However, in many cases, the cause of a brain tumor is +unknown. Treatment for brain tumors may include surgery, +radiation therapy, chemotherapy, or a combination of these +approaches. The specific treatment plan will depend on the +type and stage of the tumor, as well as the patient's age and +overall health. The prognosis for brain tumor patients varies +widely, depending on the type and stage of the tumor, as well +as the patient's age and overall health. Some brain tumors can +be cured with surgery or other treatments, while others are +more difficult to treat and may be incurable. + +Early detection of brain tumors can potentially improve the +chances of successful treatment and increase the survival rate +of patients diagnosed with the disease. Machine learning +techniques, such as ConvXGB, can be used to identify tumors +at an earlier stage when they are more likely to be successfully +treated. The CNN component of our proposed model extracts +distinctive features from medical images, while the xgboost +component uses these features to make predictions about the +presence of tumours. The model was trained on a dataset of +medical brain scan images and was able to detect different +types of brain tumours. In addition, the model was able to +handle a wide variety of image types and conditions, making +it highly versatile and applicable to a wide range of medical +scenarios. The results of this research demonstrate the +potential of machine learning algorithms for early and +accurate detection of brain tumours, paving the way for +improved patient outcomes and reduced healthcare costs. + + +UTHUSORMT +OlaeoajuRAT +LS +Tsonemmoummesiy +Dounemoumvesiy +Bomemoumommveisyounemoum.ac.k +mmemoum.ac. +CTOCECS +II. LITERATURE +There is no doubt that machine learning has become a crucial +field in the biomedical sciences because it provides techniques +for analysing high-dimensional and multimodal data. +However, current approaches have difficulty incorporating +structural and functional imagery, as well as genomic, +proteomic, and ancillary data. As far as learning machines are +concerned, the human brain is by far the most flexible and +powerful. It is for this reason that the machine learning +community +has +become +increasingly +interested +in +neuroscience, seeking to understand brain-based learning +systems as well as identify new theories and architectures. +Machine learning-aided diagnosis aims to assist doctors in +disease diagnosis using artificial intelligence (AI) and deep +learning techniques. One of such technique is Convolutional +neural networks, which can be used to analyse and evaluate +medical images to detect and extract abnormal features or +lesions. Qiao Z. et al - examined the use of CT image +characteristics in combination with blood tumour markers and +a convolutional neural network (CNN) algorithm to diagnose +pancreatic cancer [2]. The model uses machine learning +algorithms, tensor flow, and neural networks to analyse +various related health parameters which are parsed as input +variables through the fully connected layers of the CNN +model, to predict the likelihood of the presence of the diseases. +The study also examined the use of CT image characteristics +in combination with blood tumour markers and the three- +dimensional (3D) convolution neural network (CNN) +algorithm to make a clinical diagnosis of pancreatic cancer [2]. +Neural networks are known to excel at prediction problems +involving unstructured data such as images [3]. However, +numerous disease processes in patients present a significant +barrier in segmentation, radionics and various detection +methods that are relevant for early treatment and prognosis. +Machine learning has made significant advances in the +medical field by providing tools and techniques for analysing +complex data. By applying machine learning algorithms, such +as decision trees, random forests, and naive Bayes, to medical +records, researchers can assess the performance of these +algorithms using various evaluation metrics. The accuracy of +these algorithms can sometimes reach up to accuracy levels +that medical experts routinely operate at. In this study [3], +researchers compared the performance of various supervised +machine learning techniques for the prediction of +disease. To ensure a fair comparison of performance analysis, +they selected studies that used multiple machine learning +algorithms on the same dataset. They established a common +benchmark for datasets and scope to account for the wide +variability in clinical data and research scope. This research +showed that the algorithms being used were effective in +detecting diseases in their early stages with a high degree of +accuracy. +A. Convolution Neutral Network (CNN) In AI +Convolutional neural networks (CNNs) are a state-of-the-art +development in artificial intelligence. CNNs were introduced +by Yann LeCun as a new generation of technology for image +processing, replacing older methods [4]. CNNs are a variant +of regular neural networks that process images by taking in a +3D input volume and transforming it into a 3D output volume. +CNNs consist of several layers that learn directly from images +and transform them to produce output. In this research, the +CNN model was used to perform image analysis tasks such as +object recognition, image classification and segmentation. To +predict and detect diseases in patients, the CNN model uses a +key concept called shared weights and biases. + +Shared weights and biases in CNNs refer to the process of +training the model to detect the same features in different parts +of an image, allowing the network to recognize objects in an +image [5] [6]. This reduces the number of parameters the +network needs to predict and detect images by defining kernels +or filters in a convolutional layer based on various feature +maps. CNNs are a powerful tool for image analysis and can be +used to detect and predict diseases. + +In the research paper [7], CNN is represented as a +mathematical theory. Assume the input equation as: + (1) + Is the intensity of the pixel in the kernels +), + is the convolution which produces the map from the +image and is apply to filter . + are the filters that are slid +through the image by . This is defined as discrete +convolution which is show in the equation below: + + (2) + +The convolution operation is applied to the input along with +the additive bias indexed by + so the output +is +derived from the previous layer: + +, (3) + + + = rectified linear unit (ReLU). + += Bais Matrix. + + = this is the filter of size + +This is the output of the layer, , , +, position of + is: + (4) +This output with the highest valuable within the +neighbourhood is replaced by the output of the previous layer. +The previous output in the layer + shows that + +activation function with (.) is the pooling process that stride +, and +. There are two types of pooling in this +operation which are medium pooling and maximum pooling. +The maximum pooling is the max value in the window that are +taken. The maximum pooling output process function is made +below: + (5) +If the max function in this operation is appliable to the +maximum pooling process in each window dimension, then +the operation will become: + + (6) + +The final layer of the CNN model is called the Fully +Connected (FC) layer, and it takes the output from the + +p=(x(m,n)[1 ≤m≤W,1 ≤n≤H),y +Jm,n +m,nU +=max +U +m,n +mn +previous pooling layer as its input. In the FC layer, all the +neurons in the preceding layer are connected to the following +layer. We apply a multilayer perception equation that includes +L FC layers, where f_1^((L)) is the numerical value +representing the feature size of the maps. + +. (7) +This operation above explains + is the connection of the +weight unit at +in the maps inside , layer as in position +of +at the feature map, and +. + +B. CNN Model Architecture + +In a convolutional neural network (CNN), there are three main +types of layers: input, convolutional, and fully connected +layer. The input layer receives the input data, which may be +an image, text, or numerical data. The convolutional layers use +filters to identify important features in the input data. The +features extracted by the convolutional layers are then +processed by one or more fully connected layers, which utilize +them to make predictions, classification or perform other +tasks. The accuracy of the CNN model depends on the ability +of the convolutional layers to effectively learn and extract +meaningful features from the input data. + +Fig. 1 CNN model + +C. Feature Learning + +Data pre-processing: Normalization and standardization can +be applied to the input data in a machine learning model. +Normalization refers to the process of scaling the input data to +have a range of values between 0 and 1, while standardization +refers to the process of scaling the random variable of the input +data by subtracting the mean of the data from each data point +and then dividing the result by the standard deviation of the +data in order to ascertain a mean of zero and a standard +deviation of one. + +Input layer: The input layer is the first layer in a convolutional +neural network (CNN) and it receives the raw input data. In +the context of the proposed model C-XGBoost, the input data +may be a dataset of tuples ( +) where is the index in the +dataset. +is defined as the feature matrix and + represent the +class label to +. The input layer is responsible for providing +the raw input data to the rest of the model. + +It is important to note that normalization and standardization +are not always necessary or appropriate for all types of input +data or models. In some cases, these pre-processing steps may +improve the performance of the model, while in other cases +they may have little or no effect. It is usually a good idea to +experiment with different pre-processing techniques and +evaluate their effect on the model's performance. + +Convolutional layer: The convolutional layer is responsible +for applying convolutional filters to the input data to extract +features from it. In C-XGBoost, the convolutional layer is used +to extract features from the input data to feed into the XGBoost +layer, which learns a decision tree model from the data. The +output of the convolutional layer is typically a tensor with +dimensions + using the tensor in its dimension, +where z^((l)) is the number of filters applied in the -th layer. + +Fig. 2 CNN layer architecture. + +The number and complexity of the layers in a convolutional +neural network (CNN) should be carefully chosen to balance +the model's ability to learn complex patterns in the data with +the risk of overfitting. Overfitting occurs when a model is +trained on a small or insufficiently diverse dataset, or when the +model is too complex for the underlying data. To avoid +overfitting, it is often necessary to use techniques such as +regularization or early stopping to reduce the complexity of +the model and consequently improve its generalization +performance. + +In the case of a CNN designed to predict and detect diseases, +it may often be beneficial to use a deeper network with more +layers in order to capture more complex patterns in the data. +However, it is important to carefully evaluate the performance +of the model on a validation set and tune the number and +complexity of the layers to achieve the best balance between +accuracy and overfitting. + +D. Class Label Prediction + +In this part of a convolutional neural network (CNN), there are +typically three types of layers that work together to predict +class labels from input data: reshape layers, fully connected +layers, and output layers. + +Reshape layer: This layer is used to change the shape or +dimensions of the input data. A reshape layer may be used to +convert the output of the convolutional layers from a tensor +with three dimensions (e.g., width, height, and depth) to a +tensor with two dimensions (e.g., a vector). This is often done +to prepare the input data for the fully connected layer. + +Fully connected layer: In this layer, all the neurons in one layer +are connected to all the neurons in the next layer. In a CNN +designed to predict class labels, the fully connected layer(s) +may be used to learn more complex relationships between the +features extracted by the convolutional layers and the target +classes. The class prediction layer in the statement provided is +likely referring to one or more fully connected layers. + +Output layer: The output layer is the final layer in a CNN, and +it is responsible for producing the final prediction or output of +the model. In a classification task, the output layer may have +as many neurons as there are classes in the dataset, and each + + + +input +DataPreprocessing +Convolutional +ReshapeClassPrediction +Outp +Loyer +Layer +Lavers +Layer +Laver +Layetensorform +layer ++(N,N,2) +N/NXz)loyer +tayernduFeature Learning +Predict Class LabelsClassprediction layer +Convolutional layer +Vectorandno +layer +neuron corresponds to a different class. The output layer may +use a variety of activation functions, such as SoftMax, to +produce a probability distribution over the classes. + + + + +E. Extreme Gradient Boosting (Xgboost) In AI + +Jason et al proposed the classification of data features using +extreme gradient boosting, which is a supervised learning +algorithm that can be used to accurately predict various +diseases by combining several weaker models [8,9]. This +method involves ensembling K classification and regression +trees, each with K nodes. The prediction is the sum of the +scores of each tree in the K-node: + + + +This operation states that + is defined as the members of the +training set, +is set of the equivalent class title in the dataset, + is the total value for the given K tree and is the set that +contain the number in K scores for all the K classification and +regression tree. To improve this operation, we added +regularization process to it. + + + +The first operation shows that is the missing function that +measure the total variance between the key value + and the +prediction + and + represents the importance result of the +model: + + + +The second operation shows that both + and + are the +constants that controls the learning algorithm degree, + is +defined as the number of leaves in the tree and + is the +measured weight of each leaf of the tree. Gradient boosting +has great effect in both regression and classification problems. +They are used with the missing functions for extending the +constant terms reduction to give a simpler objective term: + + +Where = + indicate the data set of leaf , and += + , + = + + +The + and + are the total image statistics of the missing +function. The perfect weight of the leaf in the K tree, the +condition of the tree, for a given tree can be obtained in below +operation: + + + + + + is the perfect weight given in the leaf. + is the leaf of the tree. + is the condition of a tree structure. + is the tree structure that is given. +In this operation below, we are showing the evaluation of a +splitting candidate score in the total datasets of the right and +left nodes then it was occurred to loss reduction in its process: + + + + +F. C-XGBoost Machine Learning Technique + +Convolutional Extreme Gradient Boosting (C-XGBoost) is a +variant of the Extreme Gradient Boosting (XGBoost) +algorithm, which is a popular and efficient implementation of +the Gradient Boosting algorithm for machine learning. Like +the original XGBoost algorithm, C-XGBoost is designed for +efficient training of decision tree models for regression, +classification, and ranking tasks. However, C-XGBoost +extends the capabilities of XGBoost by incorporating +convolutional neural network (CNN) layers into the model. +This allows C-XGBoost to handle input data that has a grid- +like structure, such as images, and to learn more complex +patterns from the data. They are utilized in a wide range of +machine learning applications like NLP and genomics. + +The ConvXGB algorithm combines the strengths of both +CNNs and XGBoost to reduce model complexity and the +number of parameters required for prediction. This is achieved +by using CNNs without pooling or fully connected layers, and +by using XGBoost as the final layer. This reduces the risk of +overfitting and makes the model more efficient and easier to +train. The C-XGBoost algorithm is a powerful tool for image +analysis and disease prediction, and it has the potential to +greatly improve the accuracy and efficiency of these tasks. + + +The individual contributions of CNN and XGBoost in the +proposed C-XGBoost model are as follows: + +Convolutional neural networks have several layers of +convolutional, +pooling, +and +fully +connected +layers. +Convolutional layers apply a set of filters to the input data, +which are used to extract features from the data. Pooling layers +reduce the dimensionality of the data by applying a function +such as max pooling or average pooling to subregions of the +data. Fully connected layers connect all the neurons in one +layer to all the neurons in the next layer, allowing the network +to learn more complex relationships between the features. + +The XGBoost algorithm is used to learn a decision tree model +from the input data, and the convolutional layers are used to +identify features from the data to feed into the XGBoost +model. + +G. C-XGBOOST MODEL APPROACH + +Given a C-XGBoost model trained on a dataset of sample size +M, where each element ( +, +) Consists of a vector of tree +numbers + in + and a corresponding vector +, The model +can be implemented following this approach: + +Importing Library: We utilize python libraries such as pandas, +matplotlib, Kera, NumPy, seaborn, TensorFlow, CNN and cv2 +to import, analyse and process our data before the model is +trained on the dataset. The proposed brain tumour prediction + +1 +[(Ziel, gi)? +(ZieIR, gi)2 +(Eiel, gi)? +split +2 +ZielL, hi +^ +ZieIR, hi + ^ +Ziel, hi + ^ +technique identifies the likelihood of a patient having a brain +tumour from the MRI scan of the brain. The model is trained +to classify the presence or absence of brain tumours based on +observable imaging features. + +Loading of Data: The proposed machine learning model is +trained on a dataset sourced from public repositories. The +dataset used for the model is sourced from figshare public +repository. + +Exploratory Data Analysis: A cross-classified analysis is +carried out on the dataset using graphical and non-graphical +techniques. This helps to collect, clean, pre-process and +visualize data to have insight into the data features and types. +This step is crucial for correcting missing data before pre- +processing them into comprehensible formats for further +analysis. This step involves image normalization and +standardisation. Another crucial benefit of this step is to +remove biases from skewed data distributions. + +Prepare the input data: It is important to pre-process the input +data to ensure that it is in a suitable format for input into the +C-XGBoost model. This may involve normalizing or +standardizing the data as stated in step three above, converting +it to a tensor with appropriate dimensions, and splitting the +data into a training set and a test set. The training set is a subset +of the data that is used to train the model, while the test set is +a separate subset of the data that is used to evaluate the +performance of the trained model. + +Define the CNN architecture: The architecture of the CNN in +a C-XGBoost model plays a critical role in its performance. +This involves choosing the number and type of layers to use, +as well as the hyperparameters for each layer. For example, +the number of filters and kernel size in a convolutional layer +can be chosen to extract relevant features from the input data. + +1. Initialize the weights and biases: In a CNN, the weights +and biases of the layers are typically initialized randomly +using a Gaussian distribution. The mean and standard +deviation of the distribution can be chosen based on the +characteristics of the data and the desired properties of the +model. + +2. Feed the input data through the CNN: Given a set of input +data X and a set of corresponding labels Y, the input data +can be fed through the CNN using the following equation: + +Z = f (W*X + b) + +where Z is the output of the CNN, f is an activation +function, W is the weight matrix for the layer, and b +is the bias vector. + +3. Calculate the loss function: The loss function is used to +measure the discrepancy between the predicted output of +the CNN and the true labels. This can be done using a +variety of loss functions, such as the cross-entropy loss: + +4. Backpropagate the error: After calculating the loss, the +error can be backpropagated through the CNN using the +following equation: + + +5. +gradient of the loss function with +to the biases, and f'(Z) is the derivative of the activation +function with respect to the output of the CNN. + +6. Update the weights and biases: After backpropagating the +error, the weights and biases can be updated using an +optimization algorithm, such as stochastic gradient +descent, using the following equation: + +are the updated weights and biases, respectively. + +7. Extract features from the CNN: After the CNN has been +trained, the output of the convolutional layers can be +extracted as features for use in the XGBoost model. These +features can be fed into the XGBoost model along with +the target labels to train the model. + +8. Train the XGBoost model: The XGBoost model can be +trained using the gradient boosting algorithm to minimize +the loss function and improve prediction accuracy. This +involves building a series of decision trees and adding +them together to form a strong prediction model. + +9. Evaluate the model: After training the ConvXGB model, +it is important to evaluate its performance on a separate +test dataset to assess its generalization ability and +determine its prediction accuracy. This can be done by +comparing the predicted labels to the true labels and +calculating evaluation metrics such as accuracy, +precision, and recall. + +H. Existing Disease Prediction Models + +There has been significant research using machine learning +techniques to detect and diagnose various diseases, some of +which have achieved notable success. For example, Ross et al. +(2020) used machine learning to identify a devastating +neurological disorder at various diagnostic stages, for which +there is currently no known cure [10]. Perez et al. proposed a +method that uses a neural network to analyse DNA CpG +methylation data in order to improve the early detection of +Huntington's Disease (HD) [11]. Zhou et al. (2021) utilized +artificial intelligence and computer vision to segment the +sublingual vein region in order to improve the non-invasive +identification of multiple diseases [12]. Table 1.3 summarizes +the performance of existing machine learning models that +have been used to predict various diseases. + + + + + + + +(4 - ) 60l × (人 - 1) + (4)601 × ！人!3- = 7aLIaW = (y - Y) * f'(Z) * X +aL/ab = (y - Y) * f(Z)Wnew=W-n*aL/aw +b_new=b-n*aL/ab +Table 1. performance characteristics of existing machine +learning models for various diseases. + +Reference +Study Title +Machine Learning +Technique Used +Accuracy +Koo et al. [13] +Deep learning-based diagnosis of breast cancer using +mammography images +Deep Convolutional +Neural Network +95.50% +Ezzat et al. [14] +Predicting the onset and progression of Alzheimer's disease +using machine learning +Convolutional Neural +Network +95.00% +George et al. [15] +Predicting diabetes using logistic regression +Logistic Regression +84.70% +Mavroforakis et al. +[16] +A machine learning approach for early diagnosis of +Parkinson's disease using non-invasive biomarkers +Convolutional Neural +Network +95.60% +Imran et al. [17] +A deep learning approach for predicting the severity of +diabetic retinopathy +Deep Convolutional +Neural Network +92.70% +Pechenizkiy et al. [18] +Predicting heart failure using decision tree and random +forest algorithms +Decision Tree, +Random Forest +76.70% +Gao et al. [19] +Predicting the likelihood of stroke using a hybrid machine +learning approach +Decision Tree, +Artificial Neural +Network +80.60% +Wang et al. [20] +Predicting the likelihood of breast cancer using support +vector machine +Support Vector +Machine +81.30% + + + +III. METHODOLOGY + +The methodology proposed in this research investigates the +effectiveness of convolutional extreme gradient boosting (c- +XGBoost) method for training a dataset of brain MRI images +with and without tumors. To achieve this, we pre-processed +the dataset by applying appropriate image augmentation +techniques and normalization to ensure that the input data was +in a suitable format for the C-XGBoost model. Next, we +extracted features from the pre-processed images using a +convolutional neural network (CNN) model, specifically the +DenseNet121 architecture. The extracted features were then +fed into the c-XGBoost model for training and evaluation. We +evaluated the performance of the c-XGBoost model using +standard metrics such as accuracy and F1 score and compared +its performance to other commonly used machine learning +algorithms. Our results showed that the c-XGBoost model +achieved superior performance in predicting the presence of +brain tumours in the dataset. The proposed model utilizes the +CCO (creative common) licensed figshare dataset containing +253 images of human brain MRI images which are classified +into 3 classes: glioma - meningioma - and pituitary. +Here are a few key advantages of using a convolutional +extreme gradient boosting (C-XGBoost) approach over a non- +hybrid convolutional neural network (CNN) approach: + +C-XGBoost combines the ability of CNNs to extract +features and patterns from images with the ability of +XGBoost to effectively model and make predictions +using those features. This combination allows for the +model to not only effectively identify relevant features +in the data, but also to make accurate predictions using +those features. Additionally, XGBoost can handle large +and complex datasets, as well as handle missing or +incomplete data, which can be beneficial in the medical +field where data may be limited or unreliable. Overall, +the use of C-XGBoost in medical image analysis can +lead to improved performance and accuracy compared +to the use of a single method, such as a CNN, on its +own. + + +The lower model complexity of C-XGBoost is +achieved using XGBoost, a gradient boosting +algorithm, which combines the predictions of multiple +weak models, rather than relying on a single complex +model. This ensemble approach results in a more +robust model that is less likely to overfit to the training +data, and thus, is more likely to generalize well to +unseen data. Additionally, the use of XGBoost can also +help to reduce the risk of overfitting by implementing +regularization techniques such as early stopping and +tree pruning. Overall, the lower model complexity of +C-XGBoost makes it easier to train and more reliable +for use in real-world applications. + + +While CNNs can extract feature maps from images and +identify patterns, they are not as interpretable as +traditional machine learning algorithms. On the other +hand, XGBoost is a powerful and interpretable +decision tree-based algorithm that is widely used in +machine learning. By combining these two approaches, +C-XGBoost can retain the ability to extract features +from images and make accurate predictions, while also +providing a level of interpretability and explainability +that may be important in a medical setting. This can be +particularly useful when it comes to understanding why +a particular diagnosis or treatment was recommended, +and how certain features in the images were used to +arrive at that decision. + + +C-XGBoost can handle imbalanced datasets better than +CNNs because it uses gradient boosting, which focuses +on minimizing the misclassification error rather than +the overall error. This means that C-XGBoost will give +more weight to the minority class and try to classify +those +samples +correctly. +In +medical +image +classification tasks, where the number of brain tumour +samples belonging to the minority class is often much +smaller than the number of samples without brain +tumours belonging to the majority class, this can lead +to improved performance and accuracy. + + +C-XGBoost can handle missing values and noisy data +better than CNNs, which can be an issue in real-world +medical datasets. Noisy data can negatively impact the +performance of a machine learning model. In the +context of medical image classification, noisy data can +come from various sources, such as errors in data +collection or annotation, or variations in image quality +or resolution. C-XGBoost is less sensitive to such noise +compared to CNNs, as it uses a combination of feature +maps extracted from CNNs and the XGBoost +algorithm to make predictions. The XGBoost +algorithm can handle noise and missing values in the +data by using decision trees, which can be robust to +such issues and make more accurate predictions. + + +Fig. 3 Block Diagram of Proposed C-XGBoost Model. + +FinalModel Evaluation +Test DitasetPreorocessirg +Detring and +PusingEtactod +Evluatng theInages +ESEENOTSSE +TranrestSot +F833785 +ondataset +MSUO4 +adAugner +achiectune +XIGBcostMcce +modeData Preprocessing + + +The C-XGBOOST algorithm for detecting brain tumours in +python is demonstrated showing results from the code +including evaluation metrics for analysing the algorithm's +performance. The implementation of pooling layers and the +output of the algorithm are also shown. + +A. Importing Required Library + +The necessary python libraries were imported which include +NumPy, pandas, TensorFlow, pathlib, matplotlib.pyplot, +pymatreader, sklearn, keras and xgboost. All of which are +tools and functions essential for implementing and training the +proposed C-XGBOOST model. Shown below is the +screenshot of the library importation code as implemented in +Jupyter notebook. + +B. Loading dataset + +A dataset of medical brain scan images that includes both +examples of brain tumor presence and healthy brain scans, was +collected from the figshare public repository, unzipped and +loaded into the model. The code snippet shown in fig 3.3 +downloads a zip file from the specified URL and saves it to +the +local +system +with +the +specified +file +name +"brain_tumor_dataset.zip". The "wget" command is used to +download files from the source, and the "-O" flag specifies the +name to be used for the saved file. The URL provided in the +command points to a file on figshare.com, a public online +scientific repository which contains the utilized dataset. The +unzipped files are in .mat format. The already imported python +library read_mat reads data in MATLAB format. + + + +Fig. 4 Dataset loading from figshare repository + + + +Fig. 4 Data unzipping + +C. Data pre-processing + +Data pre-processing is a crucial step in the model training +process, as it helps to ensure that the data is in a suitable format +for further analysis and modelling. Some common steps +involved in data pre-processing include but not limited to: + +Data cleaning: This involves removing or correcting any +inaccuracies, inconsistencies, or missing data in the dataset. + +Data integration: This involves combining data from different +sources or formats to create a single, unified dataset. + +Data transformation: This involves transforming the data into +a suitable format for further analysis, such as scaling or +standardizing numerical variables or encoding categorical +variables. + +Data reduction: This involves selecting a subset of the data that +is relevant for the analysis and discarding the rest. + +Data discretization: This involves converting continuous +numerical variables into discrete bins or categories. + +Data visualization: This involves creating charts, graphs, or +other visualizations to help understand the data and identify +trends or patterns. Shown in fig 4 are some loaded dataset +images snipped from the Jupyter notebook IDE. + + + +Fig. 5 some examples of visualised image features. + +D. Collecting Feature labels + +In the proposed model, the collect_features_labels function is +responsible for collecting the features and labels of the data. +This is done after the data has been loaded. The features are +the input data that the model will use for prediction, while the +labels are the correct output or target values that the model is +trying to predict. Collecting the features and labels is an +important step in preparing the data for training. Shown below +in fig 3.5 are feature collected from the dataset when it initiates +the collect_features_ labels function. + + + + + +In [1]:Iwget -Obrain_tumor_dataset.zip https://figshare.com/ndownloader/articles/15] +--2022-12-2717:39:18--.https://figshare.com/ndownloader/articles/1512427/ver +Resolvingfigshare.com(figshare.com)...34.251.212.10,52.51.22.31 +Connectingto figshare.com (figshare.com)|34.251.212.10|:443.:.connected. +HTTp reguest sent,awaiting response.. +200OK +Length:879501695(839M)[application/zip]100 +100 +100 +200 +200 +200 +200300 +300 +300 +300 +400 +400 +400 +400 +500 +500 +500 +5006100200300400500 +0100200300400500 +0100200300400500 +0100defcollectfeatureslabels(self): +images=[] +labels[] +for single file in self.Datasetiles: +mat obj self.read mat file(single_file) +imgmatobjcjdata lf image] +img=img/255 +img=np.float32(img)img=cv2.resize(img,(224,224)) +img=cv2.cVtColor(img,cv2.COLOR GRAY2BGR) +imgimgnp.mean(img) +img=img/np.std(img) +lab=int(matobj[cjdataJ[label ]) +images.append(img) +labels.append(lab) +self.images = np.asarray(images)self.labels n np.asarray(labeisbrain +838.76M +4.39MB/s +in3m4s +2022- +17:42:22 +umor +.dataset.zip +saved +【879501695/8In[9]: +unzipbraintumordataset.zip +Archive:brain tumor dataset,zip +extracting:brainTumorDataPublic_1-766.zip +extracting:brainTumorDataPublic_1533-2298.2ip +extracting:brainTumorDatapublic767-1532.zip +extracting:brainTumorDataPublic2299-3064.zipextracting:cvind.mat +extracting:README.txt +In.【1o]: +Imkdirbraintumordataset +In.[11]: +IunzipbrainTumorDataPublicl-766.zip-dbraintumordataset +IunzipbrainTumorDataPublic1533-2298.zip-dbraintumordataset +IunzipbrainTumorDataPublic767-1532.zip-d brain tumor dataset!unzipbrainTumorDataPublic2299-3064.zip +-d braintumordataset +Archive: +brainTumorDataPublic_1-766.zip +inflating:brain_tumor_dataset/i.mat +inflating:brain_tumor_dataset/10.mat +inflating:brain_tumor_dataset/1oo.mat +inflating:brain_tumor_dataset/1ol.mat +inflating:brain_tumor_dataset/102.mat +inflating:brain_tumor_dataset/103.matinflating:brain_tumor_dataset/105.mat +inflating:brain_tumor dataset/io6.mat +inflating:brain_tumordataset/io7.mat +inflating:brain_tumor dataset/io8.mat +inflating:braintumor dataset/io9.matgioma +meningioma +glioma +0 +100 +100 +100 +200200 +200 +200 +200 +300 +300 +300 +300 +400 +400 +400500 +500 +500 +500 +100200300 +500 +400 +500 +400 +s00 +100 +ghoma +pituitary +glioma +0 +0 + +Fig. 6 Function for the collection feature labels + + + +Fig. 7 Some examples of extracted features + +E. Data augmentation + +A convolutional neural network is set to be optimized when its +parameter tuning can correctly map a set of input (images, +numeric variables, categorical variables etc.) to a label, also +known as the output, without loss to its model regardless of +the orientation of the image input. One of the ways to ensure +this is to train the model until invariance is ascertained. +Regardless of object orientation, size or angle of perspective, +the model correctly classifies the images. Fig 3.6 shows the +change in orientation of the data features. In a real-life +application, the dataset of images used to train a convolutional +neural network may exist in limited forms, which may not look +exactly like the images that may serve as input at the +application end. Since it will be time demanding to manually +change the orientation for each image, it is advised to augment +the data. This serves both to skew the parameters and increase +the set of data the model is being trained on, thus enhancing +the model's performance. + +In our model, we trained new samples from existing ones by +applying random transformations to old samples. This is often +done in order to increase the size of the training dataset and to +reduce overfitting. The function shown in fig 1.2 demonstrates +how the data augmentation is carried out in the model. + + + + + + + +Fig. 8 Code snippet of the python function created for data +augmentation + +The results of the augmented images are shown in Fig. 9 + + + +Fig. 9 Some examples of augmented images + +F. Class labels + +The code snippet shown in fig 1.3 defines a class called +Brain_Tumor that has a constructor method __init__. The +__init__ method is called when an object of the Brain_Tumor +class is created, and it initializes the object with the provided +arguments. The arguments of the __init__ method are epochs, +batch_size, dataset_folder, optimizer, and loss. These +arguments are assigned to instance variables of the same +name, which can be accessed and modified through the object +as shown in fig 1.4a,b,c. + + + +Fig. 10 Code snippet for creating class labels. + + + +Fig. 11 Identified class labels of data features. + + + + +G. Train-test splitting + +Splitting the dataset into training and testing sets would allow +us to evaluate the performance of the model on new data. The +train_test_split() calls on an object of the Brain_Tumor class. +The function is used in the code to split the dataset of brain +tissue images into a training set and a test set, with the test set +comprising 10% of the total dataset (indicated by the 0.1 +argument as shown in figure 12). This is a step in the machine +learning pipelines used to evaluate the performance of the +model on new data after training. The training set is used to fit + + + + + +glioma +meningoma +glioma +glioma +0 +50 +50 +30 +75 +75Brain Tumor obj.augmentated images() +plt.figure(figsize(20,8)) +fori inrange(8): +plt.subplot(2, 4, 1+1) +plt.imshow(Brain Tumor_obj.images[i])plt.show()meningioma +pituitary +meningicma +glioma +150 +30200 +100 +100 +300 +125 +125 +25 +150 +150 +150 +250 +175 +125 +175 +200 +200 +200 +200150 +200 +100 +150 +200 +150 +200 +50 +150 +200 +pitaitary +glioma +gioma +ghoma +0 +0 +25 +25 +25 +25 +50 +50 +5075 +75 +15 +200 +100 +100 +325 +125 +12:5 +150 +150 +150 +150 +175 +175 +sit +175200 +200 +200 +200 +150 +200 +100 +150 +200 +50 +100 +150 +200 +50 +100 +150 +200classBrain Tumor: +def +init +(self. +epochs, +batch size, +dataset folder. +optimizer, +loss):self.epochs=epochs +self.batch size batch size +self.datasetfolderdataset folder +self.optimizer = optimizer +self.loss= loss +self.Datasetiles=list(pathlib.Path(os-path.join(self.datasetfolc +self.classes ["meningioma" +""glioma","pituitary"]300 +100 +100 +200 +125 +125 +125 +125 +150 +150 +150 +150 +175 +200 +200 +200 +200meningioma +pituitary +menmgioma +homa +300 +100 +100 +300300 +300 +300 +300 +400500 +500 +500 +500 +D +100200300 +0 +500 +300200300400560 +1002000050 +200200300450500 +pitutary +glioma +giorma +glioma +200 +100 +100 +300200 +200 +200 +200 +300 +300 +300 +0500 +300 +500 +108200300400 +500 +300 +200- +300 +400 +500 +100 +200 +300 +406 +500 +20 +220300400 +50050 +200 +150 +200 +100 +150 +200 +50 +100 +150 +200 +50 +100150 +209 +glioma +pituitary +glioma +glioma +2 +25 +25 +50 +50 +50 +505 +75 +75 +TS +300 +100 +100 +300 +125 +225 +125 +125 +150 +150 +OST +150 +174 +125 +175200 +200 +200 +200 +50 +300350200 +150广200 +S00050200 +5 +100156 +200defaugmentatedimages(self): +images afteraug =[] +labelsafteraug []augmentation objectself.DataAugmentationObject() +for index, imageinenumerate(self.images): +foriinrange(3)：imagesafteraug.append(np.reshape(img，（22422243）)) +labels after aug.append(self.labels[index]) +the model, while the test set is used to evaluate the model's +performance. + + +Fig. 12 Train-test splitting function with an argument of 0.1 + + +H. CNN Architecture + +CNNs are commonly used for image classification tasks. They +are well-suited for this task because they can automatically +learn hierarchical representations of visual data, which makes +them effective at identifying patterns in images. + +The Model() function defines the architecture of the neural +network. It starts by creating a DenseNet121 model with no +pre-trained weights and an input shape of (224, 224, 3). Then, +it adds a Dropout layer with a rate of 0.8. Next, it adds a +GlobalAveragePooling2D layer, which takes the average of all +feature maps and flattens them into a single 1D tensor. This is +followed by another Dropout layer with a rate of 0.8. + +It adds a Dense layer with 3 output units, using the +function. It also includes L2 regularization on the kernel and +bias weights to help prevent overfitting. +is set to be the output of the DenseNet121 model, and the +o be the output of the Dense layer. The model +then +loads pre-trained +weights from +the "chexnet- +weights/brucechou1983_CheXNet_Keras_0.3.0_weights.h5" +file and sets the first 200 layers to be non-trainable, while +allowing the remaining layers to be trainable. The model is +then stored as an attribute of the Brain_Tumor object. + +The compile() function then compiles the model using the +specified optimizer, loss function and accuracy. + +Finally, the fit_model() function trains the model on the +training data (self.X_train and self.y_train) for the specified +number of epochs and batch size, using the validation data +(self.X_test and self.y_test) to evaluate the model's +performance at each epoch. It also includes two callbacks: a +ReduceLROnPlateau callback, which reduces the learning rate +when the validation loss has stopped improving, and an +EarlyStopping callback, which stops the training when the +validation loss has not improved for a specified number of +epochs. The training history is then stored as an attribute of the +Brain_Tumor object. + +Fig. 13 Non-hybrid CNN model + +Defining and compiling the architecture of the CNN, including +the number and size of the convolutional and pooling layers, +as well as the activation functions and regularization +techniques to be used. The Brain_Tumor_obj.Model() +function initializes the model architecture and loads the pre- +trained weights. The Brain_Tumor_obj.compile() function +sets the optimizer, loss function, and any other metrics to be +used during training. The Brain_Tumor_obj.fit_model() +function trains the model on the training dataset and evaluates +its performance on the validation dataset. + + + +Fig. 14 Batch of input data passed through a total of 25 Epochs + + +I. XGBOOST Architecture + +Convolutional eXtreme Gradient Boosting consists of several +stacked convolutional layers to recognize input features and is +able to learn the features automatically, after performing the +required training of the convolutional neural network on brain +tumour images, we will extract the last layer preceding the +Dense +classification +layers +(the +layer +named +GlobalAveragePooling2D_ while building the proposed +neural network) and from it we will extract the features and +train XGBoost in the last layer. The output of the CNN models +is parsed as input and fitted into the xgboost framework to +minimize the loss function and improve the model's accuracy. + +We do this by loading the previously trained CNN model from +two +separate +files: +the +model +architecture +("brain_tumor_model.h5") +and +the +model +weights +("brain_tumor_weights.h5"). Then, we created a new model +that takes the input and output layers of the loaded model, but +only includes the layers up to and including the layer called +"GlobalAveragePooling2D_". This new model is referred to +as "new_model". Finally, the "predict" method of the new +model generates features for the training data. The shape of +the resulting feature matrix is then printed to the console. + +XGBoost is a gradient boosting algorithm that is commonly +used for classification tasks. It is trained on the feature maps +extracted from the CNN model (X_train_features) and the +corresponding +ground +truth +labels +(Brain_Tumor_obj.y_train). The model is then used to make +predictions on the test set (X_test_features) and the predicted +labels (y_pred) are compared to the ground truth labels +(y_test) to evaluate the performance of the model. The +learning rate, maximum depth of the tree, and number of +estimators are specified as hyperparameters for the model. The +model performs a multi-class classification task through use +of the 'multiclass: Softmax' function to predict the class +probabilities as shown in Fig. 15. + + + +Brain Tumor_obj.train test_split(O.l)Brain Tumor_obj.to categorical label() +Brain Tumorobj.train test_split(o.1) +Brain Tumor obj.Model() +Brain Tumorobj.compile() +Brain Tumor obj.fit model() +Epoch1/25 +4136/4136[ +=1-171s36ms/step-loss0.9056-accurac +1r:0.0010 +Epoch2/25defModel(self): +d =densenet.DenseNet121(weightsNone,includetop =False,input shape (224,224,3)) +m=tf.keras.layers.Dropout(o.8)(d.output) +m.(=tf.keras.layers.GlobalAveragePooling2D(name=/"GlobalAveragePooling2D")(m) +m=tf.keras.layers.Dropout(o,8)(m) +m=tf.keras.layers.Dense(3,kernel_initializer=GlorotNormal(), +activationmsoftmaxrkernel regularizertf.keras.regularizers.1 +bias_regularizer=tf.keras.regularizers.L2(0.000l))(m)m.loadweights("chexnet-weights/brucechou1983ChexNet_Keras0.3.0 weights.h5"by_name=True +for layer in m.layers[:200]: +layer,trainable =False +for layer in m.layers[200:l: +layer.trainablePrue +self.mmm +def compile(self): + + +Fig. 15 The output of the CNN models is parsed as input and +fitted into the xgboost framework. + + +IV. DISCUSSION AND ANALYSIS + +We created a plot of the training and validation loss over the +course of the training epochs as shown in figure 16. The +history attribute of the Brain_Tumor object stores the training +history of the model, including the loss and validation loss at +each epoch. The pyplot function of the matplotlib module is +used to create the plot, with the training loss and validation +loss being plotted using the training loss and val_loss keys of +the history dictionary, respectively. The training loss is used +to evaluate the model's performance on the training set and to +adjust the model's weights and biases during training. The +validation loss is used to evaluate the model's generalization +ability, or its ability to perform well on new data. + + + + +Fig. 16 Training and Validation loss and accuracy for the +DenseNet20-based non-hybrid CNN model + +Minimum Loss is achieved for both DenseNet201-CNN and +DenseNet201-C-XGboost, whereas validation accuracy for C- +XGBoost as shown in Figure 17b is slightly higher than that +of purely DenseNet201 based CNN model. A validation +accuracy of 0.99021739 is achieved for the proposed model +while a slightly lower accuracy of 0.9880434. If the training +loss is much lower than the validation loss, it could indicate +that the model is overfitting, meaning it is performing well on +the training data but not generalizing well to new data, in +which case, the model may benefit from regularization +techniques such as weight decay or dropout. + +If the training loss is much higher than the validation loss, it +could indicate that the model is underfitting, meaning it is not +able to capture the underlying pattern in the training data well, +in which case, the model may benefit from increasing the +complexity of the model, such as adding more layers or +increasing the number of units in each layer. For our hybrid +model, the training loss and validation loss are both low and +directionally close to each other. This indicates that the model +is performing well and is not overfitting or underfitting. + + + + + + + +Fig. 17 Training and Validation loss and accuracy for the +DenseNet201 based C-Xgboost model + +A. Evaluation Metrics + +We observe and evaluate the performance of the model on a +test dataset to assess its accuracy and identify any potential +improvements. Evaluation metrics are used to evaluate the +efficiency of an AI model, usually based on accuracy, +sensitivity, specificity, F1-score, and area under the curve +(AUC) of ROC charts. + +Accuracy: This is a simple metric that measures correct +predictions made by the model. It is computed by dividing the +number of correct predictions by the total number of +predictions made by the model [21]. However, accuracy can +be misleading when the classes in the dataset are imbalanced, + + +0.95 +0600.85 +0.800.70 +accuracyvalidationaccuracy +F5 +10 +15 +20 +2510 +loss +val_loss0.8 +0.60.40.20.0 +T +T +T +0 +20 +40 +09 +8010 +accuracyT05S +validationloss +0.8val accuracy +600.00.70.60 +20 +40 +09 +80fromxgboostimportxGBclassifier +xgb =xGBclassifier(objective=multiclass:softmax,learning rate=o.l, +maxdepth=15,n_estimators=500) +xgb.fit(X_train_features,np.argmax(Brain_Tumorobj-ytrain,axis =I))xGBclassifier(maxdepth=15,n_estimators=500,objective=multi:softprob) +X_testfeatures=newmodel.predict(Brain_Tumor obj.xtest) +y_pred=xgb.predict(x_test_features)y_test np.argmax(Brain Tumorobj.ytest, axisl)0.60.20.0 +T +T +T +T +0 +5 +10 +15 +20 +251.00 +since the model can achieve high accuracy by always +predicting the majority class. Hence, we computed our +outcomes with other metrics such as F1 score and AUC. + +Sensitivity, also known as true positive rate, measures the +percentage of positive examples that were correctly predicted +by the model. It is calculated by dividing the number of true +positive predictions by the total number of positive examples +in the dataset. Sensitivity is useful for evaluating the model's +ability to correctly identify positive examples. + +Specificity, also known as true negative rate, measures the +percentage of negative examples that were correctly predicted +by the model. It is calculated by dividing the number of true +negative predictions by the total number of negative examples +in the dataset [22]. Specificity is useful for evaluating the +model's ability to correctly identify negative examples. + +Where TP is the number of true positive examples, FN is the +number of false negative examples, FP is the number of false +positive examples, and TN is the number of true negative +examples. TO is the number of total observations. + +F1-score is a metric that combines precision and recall into a +single score. It is calculated by taking the harmonic mean of +precision and recall [21]. F1-score is useful for evaluating the +overall performance of a model, since it considers both the +number of correct predictions and the number of false +positives and false negatives. + + +Table 2. performance characteristics of proposed C-XGBoost +model. + + + + +B. Implementation Of Evaluation Metrics for CNN Model +and Hybrid CNN-XGBoost Model in Python + +1) Accuracy + +The lines of code shown in Figure 18 perform prediction tasks +on the test set using the trained model and calculate the +accuracy of the model on the test set. The y_pred uses the +predict method of the trained model to make predictions on the +test set and converts the predicted probabilities for each class +into class labels by taking the argmax along the axis 1. The +y_test also converts the true labels for the test set into class +labels in the same way. Finally, the accuracy_score function +from the sklearn.metrics module is imported and used to +calculate the accuracy of the model on the test set, by +comparing the predicted class labels with the true labels. The +accuracy is returned as a float value between 0 and 1 which in +the case of our CNN model is an impressive 0.9880434. + + + + +Fig. 18 Accuracy outcome for non-hybrid CNN model. + + + +Fig. 19 Accuracy outcome for CNN-XGBOOST model. + +This study shows the difference between the accuracy that was +achieved using standalone CNN model, and that of when +combined with XGBoost. The result displayed in Figure 19 +shows that the integration of CNN with XGBoost leads to a +higher accuracy of 0.99021739 than that achieved using CNN +only. + + +2) Confusion Matrix + +Confusion matrix is a table that is often used to describe the +performance of a classification model on a dataset for which +the true values are known. It is called a "confusion" matrix +because it allows us to visualize the predicted and actual +classifications of a model, and "confuse" these predictions +with the true values. + +A confusion matrix typically has four entries: true positive +(TP), false positive (FP), true negative (TN), and false +negative (FN). These entries are calculated by comparing the +predicted values of the model with the true values. For +example, a true positive is an example where the model +correctly predicts the positive class, while a false positive is an +example where the model incorrectly predicts the positive +class. + +Accuracy +F1 score +Specificity +Sensitivity +CNN MODEL +98.8 +0.97 +95.2 +87.4 +C-XGBOOST MODEL +99.02 +0.98 +97.4 +91.5 + + +Accuracy = (TP + TN) / TOfrom sklearn.metrics import accuracy scoreas acac(ypred,y test) +0.9902173913043478Sensitivity=TP/(TP+FN)Specificity=TN/(TN+FP)F1-score = 2 * (P * R) / (P + R) +P=precision +R=RecallBrain Tumor obj.m.evaluate(Brain Tumor obj.X test, +Brain Tumor obj-ytest,batch size2) +460/460t +j7s15ms/step-loss:0.0381-accuracy:0.9880 +[0.03809912130236626,0.988043487071991] +or_obj.xtest)y_pred.=np.argmax(y_pred,axis1) +y_test=np.argmax(Brain_Tumor_obj.y_test,axis=1) +from sklearn.metrics import accuracy_score as ao0.9880434782608696xGBclassifier(max depth-l5,nestimators=5oo,objective='multi:softprob')X test featuresnew model.predict(Brain Tumor obj.X test) + + + + +Fig. 19 Accuracy outcome for non-hybrid CNN (left) and +CNN-XGBOOST model (right). + +We used the seaborn library to generate a heatmap +visualization of a confusion matrix for the evaluation of the +performance of the model. It compares the predicted class +labels (y_pred) with the true class labels (y_test) and shows +the number of correct and incorrect predictions for each class. +The heatmap is a visual representation of the confusion matrix, +with the colour intensity indicating the number of predictions +in each category. The tf.math.confusion_matrix function is +used to compute the confusion matrix from the true labels +(y_test) and predicted labels (y_pred). The annot parameter +specifies that the values of the confusion matrix should be +displayed in each cell of the heatmap. This visualization +identifies which classes are being misclassified and how well +the model is performing overall. The three classes represented +in the matrix represents glioma - meningioma - and pituitary. +The precise number of images classified by the model can be +determined using the diagonal values. From the result above, +we observed that the total frequency of misclassifications is +fewer in C-Xgboost than purely CNN. + +Random samples of 15 test images, with their predicted labels +and ground truth is shown below using the proposed +convolutional xgboost model. + + + +Figure 20: Random samples of 15 test images showing their +predicted and ground truth labels + + +C. Comparative Performance Analysis + +Performance results obtained from existing deep learning +models on the brain tumor MRI images are shown in +comparison with that of the proposed models in Table 14. The +table shows the machine learning technique employed used by +the authors and their outcomes. + +Table 3. performance characteristics of different prediction +models trained using brain MRI images + + + + + + + + + +V. CONCLUSION + +This paper analysed the performance of C-XGBoost models +for predicting brain tumors. The C-XGBoost model which is +based on DenseNet201 outperformed the non-hybrid CNN +model in various metrics as discussed in the result analysis +section. The dataset was acquired from the figshare public +repository. The proposed hybrid C-XGBoost model leveraged +the strengths of both CNNs and XGBoost, resulting in an +improved ability to accurately classify brain MRIs with +tumour and without tumours. The model achieved an accuracy +of 99.0% and F1 score of 0.97. To assess the relative +effectiveness of the proposed C-XGBoost model, its +performance was also compared to that of other existing +models. One of the benefits of using the XGBoost component + +Reference +Study Title +Machine Learning +Technique Used +Accuracy +Ahmad et al. [23] +Automatic Detection of Brain Tumors in magnetic +resonance imaging (MRI) Using Machine Learning +Algorithms +A combination of +support vector +machines, decision +trees, and neural +networks. +97.50% +Khan et al. [24] +Brain Tumor Detection and Classification Using Multimodal +MRI Scans and Deep Convolutional Neural Networks +Convolutional Neural +Network +Sensitivity: 98.3% +Specificity: 98.7% +Chen et al. [25] +Brain Tumor Detection and Segmentation in MRI Images +Using Deep Learning +Deep learnig +technique +Dice co-efficient: 0.82 +Kamnitsas et al. [26] +A Deep Learning Framework for Brain Tumor Detection +and Segmentation in 3D Magnetic Resonance Images +3D Convolutional +Neural Network +Dice co-efficient: 0.81 +El-Shenawee et al. +[27] +Convolutional Neural Networks (CNN) and adaptive +thresholding +Deep Convolutional +Neural Network +95.44% +Al-Shara et al. [28] +Detection of Brain Tumor in MRI Images Using Deep +Learning and Fast R-CNN +Deep learning and +Fast R-CNN +92.31% +Yan et al. [29] +Deep Learning-Based Detection of Brain Tumors in MRI +Images Using U-Net. +Deep learning and U- +Net. +96.90% +Krishna et al. [30] +Detection of Brain Tumors in MRI Images Using a Hybrid +Approach Based on Deep Learning and Transfer Learning +Deep learning and +transfer learning. +98.40% +CNN model +98.80% +C-XGBOOST model +99.02% +Brain tumor detection using convolutional XGBoost +Proposed +Methodology + +import seaborn as snsimport seaborn +as +snssns.heatmap(tf.math.confusion matrix( +y_test, +ypred,)，annot一True)4002e+02 +0 +6 +350 +300250 +4.2e+02 +I +200150100 +0 +29e+02 +2 +500 +1 +NPrediction Yes (1.o0) +Prediction:No (0.03) +Prediction:No (o.o0) +Prediction Yes (1 oo) +Ground Truth:Yes +Ground Truth:No +Ground Truth:No +Ground Truth YesY_test, +y_pred, +otmaPredictionYes:(1D0) +Prediction:Yes (o.66) +Prediction:Yest1oo1 +Prediction:No to.01)Ground Thuth:Yes +Ground Truth:Yes +Ground Truth:Yes +Groand Truth:NoPredictianYes (1.00) +Prediction:Yes (1 o0) +Predictipon,No (o.03) +PredictionYes (1.00) +GroiandTruthYes +Ground Truth:Yes +Gtound Truth:No +Ground TruthYes400 +2e+02 +一350300 +2504.2e+02 +1 +200 +150-100 +29e+02 +2 +-50-2 +0 +in the proposed C-XGBoost model is its ability to provide +interpretability and explainability for the model's decisions. +This can be a useful diagnostic tool for medical practitioners +when applied in real-time. + +Further analysis and validation on larger and more diverse +datasets are necessary to fully assess the capabilities of the +proposed model. To further improve the model's ability to +detect brain tumors at very early stages, it may be necessary to +address challenges related to adjusting image capture, +improving image quality, combining diverse data formats, and +correcting weight misalignment. These challenges can be +addressed by continually fine-tuning the model, adjusting its +hyperparameters, and training it with new data. + +References + + +[1] +C. R. UK. "Cancer incidence by age." Cancer +Research UK. http://www.cancerresearchuk.org/health- +professional/cancer-statistics/incidence/age (accessed. + +[2] +Z. Qiao, J. Ge, W. He, X. Xu, and J. He, "Artificial +Intelligence Algorithm-Based Computerized Tomography +Image Features Combined with Serum Tumor Markers for +Diagnosis of Pancreatic Cancer," (in eng), Comput Math +Methods Med, vol. 2022, p. 8979404, 2022, doi: +10.1155/2022/8979404. + +[3] +S. Uddin, A. Khan, M. E. Hossain, and M. A. Moni, +"Comparing different supervised machine learning algorithms +for disease prediction," (in eng), BMC Med Inform Decis +Mak, vol. 19, no. 1, p. 281, Dec 21 2019, doi: 10.1186/s12911- +019-1004-8. + +[4] +Y. Lecun, L. Bottou, Y. Bengio, and P. Haffner, +"Gradient-based learning applied to document recognition," +(in English), P Ieee, vol. 86, no. 11, pp. 2278-2324, Nov 1998, +doi: Doi 10.1109/5.726791. + +[5] +M. Gjoreski et al., "Facial EMG sensing for +monitoring affect using a wearable device," (in eng), Sci Rep, +vol. 12, no. 1, p. 16876, Oct 7 2022. + +[6] +E. A. Smirnov, D. M. Timoshenko, and S. N. +Andrianov, "Comparison of Regularization Methods for +ImageNet Classification with Deep Convolutional Neural +Networks," (in English), Aasri Proc, vol. 6, pp. 89-94, 2014, +doi: 10.1016/j.aasri.2014.05.013. + +[7] +P. Grohs and G. Kutyniok, Eds. Mathematical +Aspects of Deep Learning. Cambridge: Cambridge University +Press, 2022. + +[8] +A. S. El-Baz and J. S. Suri, Artificial intelligence in +cancer diagnosis and prognosis. Volume 3, Brain and prostate +cancer. Bristol [England] (No.2 The Distillery, Glassfields, +Avon Street, Bristol, BS2 0GR, UK): IOP Publishing, 2022. + +[9] +J. Brownlee, XGBoost With Python: Gradient +Boosted Trees with XGBoost and scikit-learn. Machine +Learning Mastery, 2016. + +[10] +A. L. Ross Russell et al., "Spectrum, risk factors and +outcomes of neurological and psychiatric complications of +COVID-19: a UK-wide cross-sectional surveillance study," +Brain Communications, vol. 3, no. 3, 2021, doi: +10.1093/braincomms/fcab168. + +[11] +G. Alfonso Perez and J. Caballero Villarraso, +"Neural Network Aided Detection of Huntington Disease," (in +eng), J Clin Med, vol. 11, no. 8, Apr 10 2022, doi: +10.3390/jcm11082110. + +[12] +J. Zhou, Q. Zhang, and B. Zhang, "Two-phase non- +invasive multi-disease detection via sublingual region," (in +eng), Comput Biol Med, vol. 137, p. 104782, Oct 2021, doi: +10.1016/j.compbiomed.2021.104782. + +[13] +-based diagnosis +International Conference on Pattern Recognition (ICPR), +2018, pp. 1236 1241. + +[14] +M. Ezzat, A. M. Elshahat, K. A. Elsharkawy, M. +Zaky, and M. M. El- +progression of Al +in 2018 10th International Conference on Machine Learning +and Computing (ICMLC), 2018, pp. 277 281. + +[15] +M. P. George, B. Agrawal, and P. J. George, +International +Conference +on +Computer +Science +and +Engineering, 2017, pp. 1 4. + +[16] +S. P. Mavroforakis, G. A. Vouros, and I. +-invasive +ual International Conference of +the IEEE Engineering in Medicine and Biology Society, 2017, +pp. 842 845. +[17] +deep learning approach for predicting the severity of diabetic +national Conference of +the IEEE Engineering in Medicine and Biology Society, 2017, +pp. 4596 4599. + +[18] +M. G. Pechenizkiy, T. P. Kersting, J. F. F. Ziegler, +7th Annual International +Conference of the IEEE Engineering in Medicine and Biology +Society, 2015. + +[19] +likelihood of stroke using a hybrid machine learning +Conference of +the IEEE Engineering in Medicine and Biology Society, 2014, +pp. 5580 5583. + +[20] +2012 34th Annual International Conference of the IEEE +Engineering in Medicine and Biology Society (EMBC), 2012, +pp. 4644 4647. + +[21] + Kotu, N., & Alahakoon, D. (2018). Classification +model performance evaluation metrics. In 2018 IEEE 9th +International Conference on Software Quality, Reliability, and +Security Companion (QRS-C) (pp. 109-115). IEEE. + + + +[22] +Hossain, M. J., & Rahman, M. M. (2015). +Performance evaluation of classification models. International +Journal of Computer Applications, 111(7) + + + diff --git a/mdE0T4oBgHgl3EQfZQAg/content/tmp_files/load_file.txt b/mdE0T4oBgHgl3EQfZQAg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..49a04d97ee8754a8a1ad78e3602accc4fdf111fb --- /dev/null +++ b/mdE0T4oBgHgl3EQfZQAg/content/tmp_files/load_file.txt @@ -0,0 +1,985 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf,len=984 +page_content='Convolutional XGBoost (C XGBOOST) Model for Brain Tumor Detection Abstract Brain tumors are masses or abnormal growths of cells within the brain or the central spinal canal with symptoms such as headaches, seizures, weakness or numbness in the arms or legs, changes in personality or behaviour, nausea, vomiting, vision or hearing problems and dizziness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Conventional diagnosis of brain tumour involves some tests and procedure which may include the consideration of medical history, physical examination, imaging tests (such as CT or MRI scans), and biopsy (removal and examination of a small piece of the tumor tissue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' These procedures, while effective, are mentally strenuous and time demanding due to the manual examination of the brain scans and the thorough evaluation of test results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' It has been established in lots of medical research that brain tumours diagnosed and treated early generally tends to have a better prognosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Deep learning techniques have evolved over the years and have demonstrated impressive and faster outcomes in the classification of brain tumours in medical imaging, with very little to no human interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This study proposes a model for the early detection of brain tumours using a combination of convolutional neural networks (CNNs) and extreme gradient boosting (XGBoost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The proposed model, named C-XGBoost has a lower model complexity compared to purely CNNs, making it easier to train and less prone to overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' It is also better able to handle imbalanced and unstructured data, which are common issues in real-world medical image classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' To evaluate the effectiveness of the proposed model, we employed a dataset of brain MRI images with and without tumours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The dataset used for training and testing was retrieved from figshare public repository, it contains 2 folders with 253 positive and negative brain MRI Images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The data was pre- processed and augmented to ensure a diverse and more representative sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The C-XGBoost model was trained and validated on the dataset, and the results were compared to those of a non-hybrid CNN-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The model extracts data features using the DenseNet-121 transfer learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The proposed model achieved an F1 score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='97 and an accuracy of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='02%, outperforming the CNN-based model, which achieved an accuracy of 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='8%, demonstrating its potential as a more reliable technique for detecting brain tumours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The C-XGBoost model had a lower training and validation loss, indicating better generalization to the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Our experimental results showed that the proposed model achieved high levels of accuracy in detecting brain tumours from medical images, making it a viable approach for early detection of brain tumours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Keywords - Brain tumour;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' MRI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' CNN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' XGBoost;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' C-XGBoost;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' DenseNet121;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Transfer learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' deep learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' convolutional neural network I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' INTRODUCTION According to data from Cancer Research UK, in 2018, there were an estimated 10,800 new cases of brain and central nervous system tumors diagnosed in the United Kingdom [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This number includes both benign (non-cancerous) and malignant (cancerous) tumors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The incidence rate of brain tumors in the UK varies by age, with the highest rates occurring in children under the age of 15 and in adults over the age of 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The most common type of brain tumors in the UK is a glioma, which accounts for around half of all brain tumors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' WHO estimates that brain and other central nervous system tumors accounted for around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='2% of all deaths from cancer in 2018, which was approximately 784,000 deaths worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This represents a relatively high mortality rate, as brain tumors are often difficult to treat and can be aggressive in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Brain tumors can be caused by a variety of factors, including genetics, radiation exposure, and certain viral infections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' However, in many cases, the cause of a brain tumor is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Treatment for brain tumors may include surgery, radiation therapy, chemotherapy, or a combination of these approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=" The specific treatment plan will depend on the type and stage of the tumor, as well as the patient's age and overall health." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=" The prognosis for brain tumor patients varies widely, depending on the type and stage of the tumor, as well as the patient's age and overall health." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Some brain tumors can be cured with surgery or other treatments, while others are more difficult to treat and may be incurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Early detection of brain tumors can potentially improve the chances of successful treatment and increase the survival rate of patients diagnosed with the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Machine learning techniques, such as ConvXGB, can be used to identify tumors at an earlier stage when they are more likely to be successfully treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The CNN component of our proposed model extracts distinctive features from medical images, while the xgboost component uses these features to make predictions about the presence of tumours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The model was trained on a dataset of medical brain scan images and was able to detect different types of brain tumours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' In addition, the model was able to handle a wide variety of image types and conditions, making it highly versatile and applicable to a wide range of medical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The results of this research demonstrate the potential of machine learning algorithms for early and accurate detection of brain tumours, paving the way for improved patient outcomes and reduced healthcare costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' UTHUSORMT OlaeoajuRAT LS Tsonemmoummesiy Dounemoumvesiy Bomemoumommveisyounemoum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='k mmemoum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' CTOCECS II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' LITERATURE There is no doubt that machine learning has become a crucial field in the biomedical sciences because it provides techniques for analysing high-dimensional and multimodal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' However, current approaches have difficulty incorporating structural and functional imagery, as well as genomic, proteomic, and ancillary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' As far as learning machines are concerned, the human brain is by far the most flexible and powerful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' It is for this reason that the machine learning community has become increasingly interested in neuroscience, seeking to understand brain-based learning systems as well as identify new theories and architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Machine learning-aided diagnosis aims to assist doctors in disease diagnosis using artificial intelligence (AI) and deep learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' One of such technique is Convolutional neural networks, which can be used to analyse and evaluate medical images to detect and extract abnormal features or lesions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Qiao Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' et al - examined the use of CT image characteristics in combination with blood tumour markers and a convolutional neural network (CNN) algorithm to diagnose pancreatic cancer [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The model uses machine learning algorithms, tensor flow, and neural networks to analyse various related health parameters which are parsed as input variables through the fully connected layers of the CNN model, to predict the likelihood of the presence of the diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The study also examined the use of CT image characteristics in combination with blood tumour markers and the three- dimensional (3D) convolution neural network (CNN) algorithm to make a clinical diagnosis of pancreatic cancer [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Neural networks are known to excel at prediction problems involving unstructured data such as images [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' However, numerous disease processes in patients present a significant barrier in segmentation, radionics and various detection methods that are relevant for early treatment and prognosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Machine learning has made significant advances in the medical field by providing tools and techniques for analysing complex data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' By applying machine learning algorithms, such as decision trees, random forests, and naive Bayes, to medical records, researchers can assess the performance of these algorithms using various evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The accuracy of these algorithms can sometimes reach up to accuracy levels that medical experts routinely operate at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' In this study [3], researchers compared the performance of various supervised machine learning techniques for the prediction of disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' To ensure a fair comparison of performance analysis, they selected studies that used multiple machine learning algorithms on the same dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' They established a common benchmark for datasets and scope to account for the wide variability in clinical data and research scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This research showed that the algorithms being used were effective in detecting diseases in their early stages with a high degree of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Convolution Neutral Network (CNN) In AI Convolutional neural networks (CNNs) are a state-of-the-art development in artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' CNNs were introduced by Yann LeCun as a new generation of technology for image processing, replacing older methods [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' CNNs are a variant of regular neural networks that process images by taking in a 3D input volume and transforming it into a 3D output volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' CNNs consist of several layers that learn directly from images and transform them to produce output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' In this research, the CNN model was used to perform image analysis tasks such as object recognition, image classification and segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' To predict and detect diseases in patients, the CNN model uses a key concept called shared weights and biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Shared weights and biases in CNNs refer to the process of training the model to detect the same features in different parts of an image, allowing the network to recognize objects in an image [5] [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This reduces the number of parameters the network needs to predict and detect images by defining kernels or filters in a convolutional layer based on various feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' CNNs are a powerful tool for image analysis and can be used to detect and predict diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' In the research paper [7], CNN is represented as a mathematical theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Assume the input equation as: (1) Is the intensity of the pixel in the kernels ), is the convolution which produces the map from the image and is apply to filter .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' are the filters that are slid through the image by .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This is defined as discrete convolution which is show in the equation below: (2) The convolution operation is applied to the input along with the additive bias indexed by so the output is derived from the previous layer: , (3) = rectified linear unit (ReLU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' = Bais Matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' = this is the filter of size This is the output of the layer, , , , position of is: (4) This output with the highest valuable within the neighbourhood is replaced by the output of the previous layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The previous output in the layer shows that activation function with (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=') is the pooling process that stride , and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' There are two types of pooling in this operation which are medium pooling and maximum pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The maximum pooling is the max value in the window that are taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The maximum pooling output process function is made below: (5) If the max function in this operation is appliable to the maximum pooling process in each window dimension, then the operation will become: (6) The final layer of the CNN model is called the Fully Connected (FC) layer, and it takes the output from the p=(x(m,n)[1 ≤m≤W,1 ≤n≤H),y Jm,n m,nU =max U m,n mn previous pooling layer as its input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' In the FC layer, all the neurons in the preceding layer are connected to the following layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' We apply a multilayer perception equation that includes L FC layers, where f_1^((L)) is the numerical value representing the feature size of the maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' (7) This operation above explains is the connection of the weight unit at in the maps inside , layer as in position of at the feature map, and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' CNN Model Architecture In a convolutional neural network (CNN), there are three main types of layers: input, convolutional, and fully connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The input layer receives the input data, which may be an image, text, or numerical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The convolutional layers use filters to identify important features in the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The features extracted by the convolutional layers are then processed by one or more fully connected layers, which utilize them to make predictions, classification or perform other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The accuracy of the CNN model depends on the ability of the convolutional layers to effectively learn and extract meaningful features from the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 1 CNN model C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Feature Learning Data pre-processing: Normalization and standardization can be applied to the input data in a machine learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Normalization refers to the process of scaling the input data to have a range of values between 0 and 1, while standardization refers to the process of scaling the random variable of the input data by subtracting the mean of the data from each data point and then dividing the result by the standard deviation of the data in order to ascertain a mean of zero and a standard deviation of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Input layer: The input layer is the first layer in a convolutional neural network (CNN) and it receives the raw input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' In the context of the proposed model C-XGBoost, the input data may be a dataset of tuples ( ) where is the index in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' is defined as the feature matrix and represent the class label to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The input layer is responsible for providing the raw input data to the rest of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' It is important to note that normalization and standardization are not always necessary or appropriate for all types of input data or models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' In some cases, these pre-processing steps may improve the performance of the model, while in other cases they may have little or no effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=" It is usually a good idea to experiment with different pre-processing techniques and evaluate their effect on the model's performance." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Convolutional layer: The convolutional layer is responsible for applying convolutional filters to the input data to extract features from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' In C-XGBoost, the convolutional layer is used to extract features from the input data to feed into the XGBoost layer, which learns a decision tree model from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The output of the convolutional layer is typically a tensor with dimensions using the tensor in its dimension, where z^((l)) is the number of filters applied in the -th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 2 CNN layer architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=" The number and complexity of the layers in a convolutional neural network (CNN) should be carefully chosen to balance the model's ability to learn complex patterns in the data with the risk of overfitting." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Overfitting occurs when a model is trained on a small or insufficiently diverse dataset, or when the model is too complex for the underlying data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' To avoid overfitting, it is often necessary to use techniques such as regularization or early stopping to reduce the complexity of the model and consequently improve its generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' In the case of a CNN designed to predict and detect diseases, it may often be beneficial to use a deeper network with more layers in order to capture more complex patterns in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' However, it is important to carefully evaluate the performance of the model on a validation set and tune the number and complexity of the layers to achieve the best balance between accuracy and overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Class Label Prediction In this part of a convolutional neural network (CNN), there are typically three types of layers that work together to predict class labels from input data: reshape layers, fully connected layers, and output layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Reshape layer: This layer is used to change the shape or dimensions of the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' A reshape layer may be used to convert the output of the convolutional layers from a tensor with three dimensions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=', width, height, and depth) to a tensor with two dimensions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=', a vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This is often done to prepare the input data for the fully connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Fully connected layer: In this layer, all the neurons in one layer are connected to all the neurons in the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' In a CNN designed to predict class labels, the fully connected layer(s) may be used to learn more complex relationships between the features extracted by the convolutional layers and the target classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The class prediction layer in the statement provided is likely referring to one or more fully connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Output layer: The output layer is the final layer in a CNN, and it is responsible for producing the final prediction or output of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' In a classification task, the output layer may have as many neurons as there are classes in the dataset, and each input DataPreprocessing Convolutional ReshapeClassPrediction Outp Loyer Layer Lavers Layer Laver Layetensorform layer +(N,N,2) N/NXz)loyer tayernduFeature Learning Predict Class LabelsClassprediction layer Convolutional layer Vectorandno layer neuron corresponds to a different class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The output layer may use a variety of activation functions, such as SoftMax, to produce a probability distribution over the classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Extreme Gradient Boosting (Xgboost) In AI Jason et al proposed the classification of data features using extreme gradient boosting, which is a supervised learning algorithm that can be used to accurately predict various diseases by combining several weaker models [8,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This method involves ensembling K classification and regression trees, each with K nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The prediction is the sum of the scores of each tree in the K-node: This operation states that is defined as the members of the training set, is set of the equivalent class title in the dataset, is the total value for the given K tree and is the set that contain the number in K scores for all the K classification and regression tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' To improve this operation, we added regularization process to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The first operation shows that is the missing function that measure the total variance between the key value and the prediction and represents the importance result of the model: The second operation shows that both and are the constants that controls the learning algorithm degree, is defined as the number of leaves in the tree and is the measured weight of each leaf of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Gradient boosting has great effect in both regression and classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' They are used with the missing functions for extending the constant terms reduction to give a simpler objective term: Where = indicate the data set of leaf , and = , = The and are the total image statistics of the missing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The perfect weight of the leaf in the K tree, the condition of the tree, for a given tree can be obtained in below operation: is the perfect weight given in the leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' is the leaf of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' is the condition of a tree structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' is the tree structure that is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' In this operation below, we are showing the evaluation of a splitting candidate score in the total datasets of the right and left nodes then it was occurred to loss reduction in its process: F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' C XGBoost Machine Learning Technique Convolutional Extreme Gradient Boosting (C-XGBoost) is a variant of the Extreme Gradient Boosting (XGBoost) algorithm, which is a popular and efficient implementation of the Gradient Boosting algorithm for machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Like the original XGBoost algorithm, C-XGBoost is designed for efficient training of decision tree models for regression, classification, and ranking tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' However, C-XGBoost extends the capabilities of XGBoost by incorporating convolutional neural network (CNN) layers into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This allows C-XGBoost to handle input data that has a grid- like structure, such as images, and to learn more complex patterns from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' They are utilized in a wide range of machine learning applications like NLP and genomics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The ConvXGB algorithm combines the strengths of both CNNs and XGBoost to reduce model complexity and the number of parameters required for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This is achieved by using CNNs without pooling or fully connected layers, and by using XGBoost as the final layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This reduces the risk of overfitting and makes the model more efficient and easier to train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The C-XGBoost algorithm is a powerful tool for image analysis and disease prediction, and it has the potential to greatly improve the accuracy and efficiency of these tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The individual contributions of CNN and XGBoost in the proposed C-XGBoost model are as follows: Convolutional neural networks have several layers of convolutional, pooling, and fully connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Convolutional layers apply a set of filters to the input data, which are used to extract features from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Pooling layers reduce the dimensionality of the data by applying a function such as max pooling or average pooling to subregions of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Fully connected layers connect all the neurons in one layer to all the neurons in the next layer, allowing the network to learn more complex relationships between the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The XGBoost algorithm is used to learn a decision tree model from the input data, and the convolutional layers are used to identify features from the data to feed into the XGBoost model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' C XGBOOST MODEL APPROACH Given a C-XGBoost model trained on a dataset of sample size M, where each element ( , ) Consists of a vector of tree numbers in and a corresponding vector , The model can be implemented following this approach: Importing Library: We utilize python libraries such as pandas, matplotlib, Kera, NumPy, seaborn, TensorFlow, CNN and cv2 to import, analyse and process our data before the model is trained on the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The proposed brain tumour prediction 1 [(Ziel, gi)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' (ZieIR, gi)2 (Eiel, gi)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' split 2 ZielL, hi +^ ZieIR, hi + ^ Ziel, hi + ^ technique identifies the likelihood of a patient having a brain tumour from the MRI scan of the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The model is trained to classify the presence or absence of brain tumours based on observable imaging features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Loading of Data: The proposed machine learning model is trained on a dataset sourced from public repositories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The dataset used for the model is sourced from figshare public repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Exploratory Data Analysis: A cross-classified analysis is carried out on the dataset using graphical and non-graphical techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This helps to collect, clean, pre-process and visualize data to have insight into the data features and types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This step is crucial for correcting missing data before pre- processing them into comprehensible formats for further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This step involves image normalization and standardisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Another crucial benefit of this step is to remove biases from skewed data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Prepare the input data: It is important to pre-process the input data to ensure that it is in a suitable format for input into the C-XGBoost model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This may involve normalizing or standardizing the data as stated in step three above, converting it to a tensor with appropriate dimensions, and splitting the data into a training set and a test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The training set is a subset of the data that is used to train the model, while the test set is a separate subset of the data that is used to evaluate the performance of the trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Define the CNN architecture: The architecture of the CNN in a C-XGBoost model plays a critical role in its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This involves choosing the number and type of layers to use, as well as the hyperparameters for each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' For example, the number of filters and kernel size in a convolutional layer can be chosen to extract relevant features from the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Initialize the weights and biases: In a CNN, the weights and biases of the layers are typically initialized randomly using a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The mean and standard deviation of the distribution can be chosen based on the characteristics of the data and the desired properties of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Feed the input data through the CNN: Given a set of input data X and a set of corresponding labels Y, the input data can be fed through the CNN using the following equation: Z = f (W X + b) where Z is the output of the CNN, f is an activation function, W is the weight matrix for the layer, and b is the bias vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Calculate the loss function: The loss function is used to measure the discrepancy between the predicted output of the CNN and the true labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This can be done using a variety of loss functions, such as the cross-entropy loss: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Backpropagate the error: After calculating the loss, the error can be backpropagated through the CNN using the following equation: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=" gradient of the loss function with to the biases, and f'(Z) is the derivative of the activation function with respect to the output of the CNN." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Update the weights and biases: After backpropagating the error, the weights and biases can be updated using an optimization algorithm, such as stochastic gradient descent, using the following equation: are the updated weights and biases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Extract features from the CNN: After the CNN has been trained, the output of the convolutional layers can be extracted as features for use in the XGBoost model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' These features can be fed into the XGBoost model along with the target labels to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Train the XGBoost model: The XGBoost model can be trained using the gradient boosting algorithm to minimize the loss function and improve prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This involves building a series of decision trees and adding them together to form a strong prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Evaluate the model: After training the ConvXGB model, it is important to evaluate its performance on a separate test dataset to assess its generalization ability and determine its prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This can be done by comparing the predicted labels to the true labels and calculating evaluation metrics such as accuracy, precision, and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Existing Disease Prediction Models There has been significant research using machine learning techniques to detect and diagnose various diseases, some of which have achieved notable success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' For example, Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' (2020) used machine learning to identify a devastating neurological disorder at various diagnostic stages, for which there is currently no known cure [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Perez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=" proposed a method that uses a neural network to analyse DNA CpG methylation data in order to improve the early detection of Huntington's Disease (HD) [11]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' (2021) utilized artificial intelligence and computer vision to segment the sublingual vein region in order to improve the non-invasive identification of multiple diseases [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='3 summarizes the performance of existing machine learning models that have been used to predict various diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' (4 - ) 60l × (人 - 1) + (4)601 × ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='人!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content="3- = 7aLIaW = (y - Y) * f'(Z) * X aL/ab = (y - Y) * f(Z)Wnew=W-n*aL/aw b_new=b-n*aL/ab Table 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' performance characteristics of existing machine learning models for various diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Reference Study Title Machine Learning Technique Used Accuracy Koo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [13] Deep learning-based diagnosis of breast cancer using mammography images Deep Convolutional Neural Network 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='50% Ezzat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=" [14] Predicting the onset and progression of Alzheimer's disease using machine learning Convolutional Neural Network 95." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='00% George et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [15] Predicting diabetes using logistic regression Logistic Regression 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='70% Mavroforakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=" [16] A machine learning approach for early diagnosis of Parkinson's disease using non-invasive biomarkers Convolutional Neural Network 95." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='60% Imran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [17] A deep learning approach for predicting the severity of diabetic retinopathy Deep Convolutional Neural Network 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='70% Pechenizkiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [18] Predicting heart failure using decision tree and random forest algorithms Decision Tree, Random Forest 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='70% Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [19] Predicting the likelihood of stroke using a hybrid machine learning approach Decision Tree, Artificial Neural Network 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='60% Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [20] Predicting the likelihood of breast cancer using support vector machine Support Vector Machine 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='30% III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' METHODOLOGY The methodology proposed in this research investigates the effectiveness of convolutional extreme gradient boosting (c- XGBoost) method for training a dataset of brain MRI images with and without tumors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' To achieve this, we pre-processed the dataset by applying appropriate image augmentation techniques and normalization to ensure that the input data was in a suitable format for the C-XGBoost model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Next, we extracted features from the pre-processed images using a convolutional neural network (CNN) model, specifically the DenseNet121 architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The extracted features were then fed into the c-XGBoost model for training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' We evaluated the performance of the c-XGBoost model using standard metrics such as accuracy and F1 score and compared its performance to other commonly used machine learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Our results showed that the c-XGBoost model achieved superior performance in predicting the presence of brain tumours in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The proposed model utilizes the CCO (creative common) licensed figshare dataset containing 253 images of human brain MRI images which are classified into 3 classes: glioma - meningioma - and pituitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Here are a few key advantages of using a convolutional extreme gradient boosting (C-XGBoost) approach over a non- hybrid convolutional neural network (CNN) approach: C-XGBoost combines the ability of CNNs to extract features and patterns from images with the ability of XGBoost to effectively model and make predictions using those features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This combination allows for the model to not only effectively identify relevant features in the data, but also to make accurate predictions using those features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Additionally, XGBoost can handle large and complex datasets, as well as handle missing or incomplete data, which can be beneficial in the medical field where data may be limited or unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Overall, the use of C-XGBoost in medical image analysis can lead to improved performance and accuracy compared to the use of a single method, such as a CNN, on its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The lower model complexity of C-XGBoost is achieved using XGBoost, a gradient boosting algorithm, which combines the predictions of multiple weak models, rather than relying on a single complex model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This ensemble approach results in a more robust model that is less likely to overfit to the training data, and thus, is more likely to generalize well to unseen data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Additionally, the use of XGBoost can also help to reduce the risk of overfitting by implementing regularization techniques such as early stopping and tree pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Overall, the lower model complexity of C-XGBoost makes it easier to train and more reliable for use in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' While CNNs can extract feature maps from images and identify patterns, they are not as interpretable as traditional machine learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' On the other hand, XGBoost is a powerful and interpretable decision tree-based algorithm that is widely used in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' By combining these two approaches, C-XGBoost can retain the ability to extract features from images and make accurate predictions, while also providing a level of interpretability and explainability that may be important in a medical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This can be particularly useful when it comes to understanding why a particular diagnosis or treatment was recommended, and how certain features in the images were used to arrive at that decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' C-XGBoost can handle imbalanced datasets better than CNNs because it uses gradient boosting, which focuses on minimizing the misclassification error rather than the overall error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This means that C-XGBoost will give more weight to the minority class and try to classify those samples correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' In medical image classification tasks, where the number of brain tumour samples belonging to the minority class is often much smaller than the number of samples without brain tumours belonging to the majority class, this can lead to improved performance and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' C-XGBoost can handle missing values and noisy data better than CNNs, which can be an issue in real-world medical datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Noisy data can negatively impact the performance of a machine learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' In the context of medical image classification, noisy data can come from various sources, such as errors in data collection or annotation, or variations in image quality or resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' C-XGBoost is less sensitive to such noise compared to CNNs, as it uses a combination of feature maps extracted from CNNs and the XGBoost algorithm to make predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The XGBoost algorithm can handle noise and missing values in the data by using decision trees, which can be robust to such issues and make more accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 3 Block Diagram of Proposed C-XGBoost Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=" FinalModel Evaluation Test DitasetPreorocessirg Detring and PusingEtactod Evluatng theInages ESEENOTSSE TranrestSot F833785 ondataset MSUO4 adAugner achiectune XIGBcostMcce modeData Preprocessing The C-XGBOOST algorithm for detecting brain tumours in python is demonstrated showing results from the code including evaluation metrics for analysing the algorithm's performance." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The implementation of pooling layers and the output of the algorithm are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Importing Required Library The necessary python libraries were imported which include NumPy, pandas, TensorFlow, pathlib, matplotlib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='pyplot, pymatreader, sklearn, keras and xgboost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' All of which are tools and functions essential for implementing and training the proposed C-XGBOOST model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Shown below is the screenshot of the library importation code as implemented in Jupyter notebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Loading dataset A dataset of medical brain scan images that includes both examples of brain tumor presence and healthy brain scans, was collected from the figshare public repository, unzipped and loaded into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The code snippet shown in fig 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='3 downloads a zip file from the specified URL and saves it to the local system with the specified file name "brain_tumor_dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='zip".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The "wget" command is used to download files from the source, and the "-O" flag specifies the name to be used for the saved file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The URL provided in the command points to a file on figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='com, a public online scientific repository which contains the utilized dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The unzipped files are in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='mat format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The already imported python library read_mat reads data in MATLAB format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 4 Dataset loading from figshare repository Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 4 Data unzipping C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Data pre processing Data pre-processing is a crucial step in the model training process, as it helps to ensure that the data is in a suitable format for further analysis and modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Some common steps involved in data pre-processing include but not limited to: Data cleaning: This involves removing or correcting any inaccuracies, inconsistencies, or missing data in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Data integration: This involves combining data from different sources or formats to create a single, unified dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Data transformation: This involves transforming the data into a suitable format for further analysis, such as scaling or standardizing numerical variables or encoding categorical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Data reduction: This involves selecting a subset of the data that is relevant for the analysis and discarding the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Data discretization: This involves converting continuous numerical variables into discrete bins or categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Data visualization: This involves creating charts, graphs, or other visualizations to help understand the data and identify trends or patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Shown in fig 4 are some loaded dataset images snipped from the Jupyter notebook IDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 5 some examples of visualised image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Collecting Feature labels In the proposed model, the collect_features_labels function is responsible for collecting the features and labels of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This is done after the data has been loaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The features are the input data that the model will use for prediction, while the labels are the correct output or target values that the model is trying to predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Collecting the features and labels is an important step in preparing the data for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Shown below in fig 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='5 are feature collected from the dataset when it initiates the collect_features_ labels function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' In [1]:Iwget -Obrain_tumor_dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='zip https://figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='com/ndownloader/articles/15] --2022-12-2717:39:18--.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='https://figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='com/ndownloader/articles/1512427/ver Resolvingfigshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='com(figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='212.' metadata={'source': 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0100200300400500 0100defcollectfeatureslabels(self): images=[] labels[] for single file in self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='Datasetiles: mat obj self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='read mat file(single_file) imgmatobjcjdata lf image] img=img/255 img=np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='float32(img)img=cv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='resize(img,(224,224)) img=cv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='cVtColor(img,cv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='COLOR GRAY2BGR) imgimgnp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='mean(img) img=img/np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='std(img) lab=int(matobj[cjdataJ[label ]) images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='append(img) labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='append(lab) self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='images = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='asarray(images)self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='labels n np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='asarray(labeisbrain 838.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='76M 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='39MB/s in3m4s 2022- 17:42:22 umor .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='zip saved 【879501695/8In[9]: unzipbraintumordataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='zip Archive:brain tumor dataset,zip extracting:brainTumorDataPublic_1-766.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='zip extracting:brainTumorDataPublic_1533-2298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='2ip extracting:brainTumorDatapublic767-1532.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='zip extracting:brainTumorDataPublic2299-3064.' metadata={'source': 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+page_content='mat inflating:brain_tumor_dataset/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='mat inflating:brain_tumor_dataset/1oo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='mat inflating:brain_tumor_dataset/1ol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='mat inflating:brain_tumor_dataset/102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='mat inflating:brain_tumor_dataset/103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='matinflating:brain_tumor_dataset/105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='mat inflating:brain_tumor dataset/io6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='mat inflating:brain_tumordataset/io7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='mat inflating:brain_tumor dataset/io8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='mat inflating:braintumor dataset/io9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='matgioma meningioma glioma 0 100 100 100 200200 200 200 200 300 300 300 300 400 400 400500 500 500 500 100200300 500 400 500 400 s00 100 ghoma pituitary glioma 0 0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 6 Function for the collection feature labels Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 7 Some examples of extracted features E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Data augmentation A convolutional neural network is set to be optimized when its parameter tuning can correctly map a set of input (images, numeric variables, categorical variables etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=') to a label, also known as the output, without loss to its model regardless of the orientation of the image input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' One of the ways to ensure this is to train the model until invariance is ascertained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Regardless of object orientation, size or angle of perspective, the model correctly classifies the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Fig 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='6 shows the change in orientation of the data features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' In a real-life application, the dataset of images used to train a convolutional neural network may exist in limited forms, which may not look exactly like the images that may serve as input at the application end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Since it will be time demanding to manually change the orientation for each image, it is advised to augment the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=" This serves both to skew the parameters and increase the set of data the model is being trained on, thus enhancing the model's performance." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' In our model, we trained new samples from existing ones by applying random transformations to old samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This is often done in order to increase the size of the training dataset and to reduce overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The function shown in fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='2 demonstrates how the data augmentation is carried out in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 8 Code snippet of the python function created for data augmentation The results of the augmented images are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 9 Some examples of augmented images F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Class labels The code snippet shown in fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='3 defines a class called Brain_Tumor that has a constructor method __init__.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The __init__ method is called when an object of the Brain_Tumor class is created, and it initializes the object with the provided arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The arguments of the __init__ method are epochs, batch_size, dataset_folder, optimizer, and loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' These arguments are assigned to instance variables of the same name, which can be accessed and modified through the object as shown in fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='4a,b,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 10 Code snippet for creating class labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 11 Identified class labels of data features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Train test splitting Splitting the dataset into training and testing sets would allow us to evaluate the performance of the model on new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The train_test_split() calls on an object of the Brain_Tumor class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The function is used in the code to split the dataset of brain tissue images into a training set and a test set, with the test set comprising 10% of the total dataset (indicated by the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='1 argument as shown in figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This is a step in the machine learning pipelines used to evaluate the performance of the model on new data after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The training set is used to fit glioma meningoma glioma glioma 0 50 50 30 75 75Brain Tumor obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='augmentated images() plt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='figure(figsize(20,8)) fori inrange(8): plt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='subplot(2, 4, 1+1) plt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='imshow(Brain Tumor_obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='images[i])plt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='show()meningioma pituitary meningicma glioma 150 30200 100 100 300 125 125 25 150 150 150 250 175 125 175 200 200 200 200150 200 100 150 200 150 200 50 150 200 pitaitary glioma gioma ghoma 0 0 25 25 25 25 50 50 5075 75 15 200 100 100 325 125 12:5 150 150 150 150 175 175 sit 175200 200 200 200 150 200 100 150 200 50 100 150 200 50 100 150 200classBrain Tumor: def init (self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' epochs, batch size, dataset folder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' optimizer, loss):self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='epochs=epochs self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='batch size batch size self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='datasetfolderdataset folder self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='optimizer = optimizer self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='loss= loss self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='Datasetiles=list(pathlib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='Path(os-path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='join(self.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='=[] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='labelsafteraug ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='[]augmentation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='objectself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='DataAugmentationObject() for index, imageinenumerate(self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='images): foriinrange(3)：imagesafteraug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='append(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='reshape(img，（22422243）)) labels after aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='append(self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content="labels[index]) the model, while the test set is used to evaluate the model's performance." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 12 Train-test splitting function with an argument of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='1 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' CNN Architecture CNNs are commonly used for image classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' They are well-suited for this task because they can automatically learn hierarchical representations of visual data, which makes them effective at identifying patterns in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The Model() function defines the architecture of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' It starts by creating a DenseNet121 model with no pre-trained weights and an input shape of (224, 224, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Then, it adds a Dropout layer with a rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Next, it adds a GlobalAveragePooling2D layer, which takes the average of all feature maps and flattens them into a single 1D tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This is followed by another Dropout layer with a rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' It adds a Dense layer with 3 output units, using the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' It also includes L2 regularization on the kernel and bias weights to help prevent overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' is set to be the output of the DenseNet121 model, and the o be the output of the Dense layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The model then loads pre-trained weights from the "chexnet- weights/brucechou1983_CheXNet_Keras_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='0_weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='h5" file and sets the first 200 layers to be non-trainable, while allowing the remaining layers to be trainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The model is then stored as an attribute of the Brain_Tumor object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The compile() function then compiles the model using the specified optimizer, loss function and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Finally, the fit_model() function trains the model on the training data (self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='X_train and self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='y_train) for the specified number of epochs and batch size, using the validation data (self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='X_test and self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content="y_test) to evaluate the model's performance at each epoch." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' It also includes two callbacks: a ReduceLROnPlateau callback, which reduces the learning rate when the validation loss has stopped improving, and an EarlyStopping callback, which stops the training when the validation loss has not improved for a specified number of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The training history is then stored as an attribute of the Brain_Tumor object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 13 Non hybrid CNN model Defining and compiling the architecture of the CNN, including the number and size of the convolutional and pooling layers, as well as the activation functions and regularization techniques to be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The Brain_Tumor_obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='Model() function initializes the model architecture and loads the pre- trained weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The Brain_Tumor_obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='compile() function sets the optimizer, loss function, and any other metrics to be used during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The Brain_Tumor_obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='fit_model() function trains the model on the training dataset and evaluates its performance on the validation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 14 Batch of input data passed through a total of 25 Epochs I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' XGBOOST Architecture Convolutional eXtreme Gradient Boosting consists of several stacked convolutional layers to recognize input features and is able to learn the features automatically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' after performing the required training of the convolutional neural network on brain tumour images,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' we will extract the last layer preceding the Dense classification layers (the layer named GlobalAveragePooling2D_ while building the proposed neural network) and from it we will extract the features and train XGBoost in the last layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=" The output of the CNN models is parsed as input and fitted into the xgboost framework to minimize the loss function and improve the model's accuracy." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' We do this by loading the previously trained CNN model from two separate files: the model architecture ("brain_tumor_model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='h5") and the model weights ("brain_tumor_weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='h5").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Then, we created a new model that takes the input and output layers of the loaded model, but only includes the layers up to and including the layer called "GlobalAveragePooling2D_".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This new model is referred to as "new_model".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Finally, the "predict" method of the new model generates features for the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The shape of the resulting feature matrix is then printed to the console.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' XGBoost is a gradient boosting algorithm that is commonly used for classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' It is trained on the feature maps extracted from the CNN model (X_train_features) and the corresponding ground truth labels (Brain_Tumor_obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='y_train).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The model is then used to make predictions on the test set (X_test_features) and the predicted labels (y_pred) are compared to the ground truth labels (y_test) to evaluate the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The learning rate, maximum depth of the tree, and number of estimators are specified as hyperparameters for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=" The model performs a multi-class classification task through use of the 'multiclass: Softmax' function to predict the class probabilities as shown in Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Brain Tumor_obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='train test_split(O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='l)Brain Tumor_obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='to categorical label() Brain Tumorobj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='train test_split(o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='1) Brain Tumor obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='Model() Brain Tumorobj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='compile() Brain Tumor obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='fit model() Epoch1/25 4136/4136[ =1-171s36ms/step-loss0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='9056-accurac 1r:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='0010 Epoch2/25defModel(self): d =densenet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='DenseNet121(weightsNone,includetop =False,input shape (224,224,3)) m=tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='keras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='Dropout(o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='8)(d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='output) m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='(=tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='keras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='GlobalAveragePooling2D(name=/"GlobalAveragePooling2D")(m) m=tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='keras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='Dropout(o,8)(m) m=tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='keras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='Dense(3,kernel_initializer=GlorotNormal(), activationmsoftmaxrkernel regularizertf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='keras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='regularizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='1 bias_regularizer=tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='keras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='regularizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='L2(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='000l))(m)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='loadweights("chexnet-weights/brucechou1983ChexNet_Keras0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='0 weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='h5"by_name=True for layer in m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='layers[:200]: layer,trainable =False for layer in m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='layers[200:l: layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='trainablePrue self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='mmm def compile(self): Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 15 The output of the CNN models is parsed as input and fitted into the xgboost framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' DISCUSSION AND ANALYSIS We created a plot of the training and validation loss over the course of the training epochs as shown in figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The history attribute of the Brain_Tumor object stores the training history of the model, including the loss and validation loss at each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The pyplot function of the matplotlib module is used to create the plot, with the training loss and validation loss being plotted using the training loss and val_loss keys of the history dictionary, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=" The training loss is used to evaluate the model's performance on the training set and to adjust the model's weights and biases during training." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=" The validation loss is used to evaluate the model's generalization ability, or its ability to perform well on new data." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 16 Training and Validation loss and accuracy for the DenseNet20-based non-hybrid CNN model Minimum Loss is achieved for both DenseNet201-CNN and DenseNet201-C-XGboost, whereas validation accuracy for C- XGBoost as shown in Figure 17b is slightly higher than that of purely DenseNet201 based CNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' A validation accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='99021739 is achieved for the proposed model while a slightly lower accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='9880434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' If the training loss is much lower than the validation loss, it could indicate that the model is overfitting, meaning it is performing well on the training data but not generalizing well to new data, in which case, the model may benefit from regularization techniques such as weight decay or dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' If the training loss is much higher than the validation loss, it could indicate that the model is underfitting, meaning it is not able to capture the underlying pattern in the training data well, in which case, the model may benefit from increasing the complexity of the model, such as adding more layers or increasing the number of units in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' For our hybrid model, the training loss and validation loss are both low and directionally close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This indicates that the model is performing well and is not overfitting or underfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 17 Training and Validation loss and accuracy for the DenseNet201 based C-Xgboost model A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Evaluation Metrics We observe and evaluate the performance of the model on a test dataset to assess its accuracy and identify any potential improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Evaluation metrics are used to evaluate the efficiency of an AI model, usually based on accuracy, sensitivity, specificity, F1-score, and area under the curve (AUC) of ROC charts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Accuracy: This is a simple metric that measures correct predictions made by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' It is computed by dividing the number of correct predictions by the total number of predictions made by the model [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' However, accuracy can be misleading when the classes in the dataset are imbalanced, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='95 0600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='70 accuracyvalidationaccuracy F5 10 15 20 2510 loss val_loss0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='0 T T T 0 20 40 09 8010 accuracyT05S validationloss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='8val accuracy 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='60 20 40 09 80fromxgboostimportxGBclassifier xgb =xGBclassifier(objective=multiclass:softmax,learning rate=o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='l, maxdepth=15,n_estimators=500) xgb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='fit(X_train_features,np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='argmax(Brain_Tumorobj-ytrain,axis =I))xGBclassifier(maxdepth=15,n_estimators=500,objective=multi:softprob) X_testfeatures=newmodel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='predict(Brain_Tumor obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='xtest) y_pred=xgb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='predict(x_test_features)y_test np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='argmax(Brain Tumorobj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='ytest, axisl)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='0 T T T T 0 5 10 15 20 251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='00 since the model can achieve high accuracy by always predicting the majority class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Hence, we computed our outcomes with other metrics such as F1 score and AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Sensitivity, also known as true positive rate, measures the percentage of positive examples that were correctly predicted by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' It is calculated by dividing the number of true positive predictions by the total number of positive examples in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=" Sensitivity is useful for evaluating the model's ability to correctly identify positive examples." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Specificity, also known as true negative rate, measures the percentage of negative examples that were correctly predicted by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' It is calculated by dividing the number of true negative predictions by the total number of negative examples in the dataset [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=" Specificity is useful for evaluating the model's ability to correctly identify negative examples." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Where TP is the number of true positive examples, FN is the number of false negative examples, FP is the number of false positive examples, and TN is the number of true negative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' TO is the number of total observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' F1-score is a metric that combines precision and recall into a single score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' It is calculated by taking the harmonic mean of precision and recall [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' F1-score is useful for evaluating the overall performance of a model, since it considers both the number of correct predictions and the number of false positives and false negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' performance characteristics of proposed C-XGBoost model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Implementation Of Evaluation Metrics for CNN Model and Hybrid CNN-XGBoost Model in Python 1) Accuracy The lines of code shown in Figure 18 perform prediction tasks on the test set using the trained model and calculate the accuracy of the model on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The y_pred uses the predict method of the trained model to make predictions on the test set and converts the predicted probabilities for each class into class labels by taking the argmax along the axis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The y_test also converts the true labels for the test set into class labels in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Finally, the accuracy_score function from the sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='metrics module is imported and used to calculate the accuracy of the model on the test set, by comparing the predicted class labels with the true labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The accuracy is returned as a float value between 0 and 1 which in the case of our CNN model is an impressive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='9880434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 18 Accuracy outcome for non-hybrid CNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 19 Accuracy outcome for CNN-XGBOOST model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This study shows the difference between the accuracy that was achieved using standalone CNN model, and that of when combined with XGBoost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The result displayed in Figure 19 shows that the integration of CNN with XGBoost leads to a higher accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='99021739 than that achieved using CNN only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 2) Confusion Matrix Confusion matrix is a table that is often used to describe the performance of a classification model on a dataset for which the true values are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' It is called a "confusion" matrix because it allows us to visualize the predicted and actual classifications of a model, and "confuse" these predictions with the true values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' A confusion matrix typically has four entries: true positive (TP), false positive (FP), true negative (TN), and false negative (FN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' These entries are calculated by comparing the predicted values of the model with the true values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' For example, a true positive is an example where the model correctly predicts the positive class, while a false positive is an example where the model incorrectly predicts the positive class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Accuracy F1 score Specificity Sensitivity CNN MODEL 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='97 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='2 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='4 C XGBOOST MODEL 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='98 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='4 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='5 Accuracy = (TP + TN) / TOfrom sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='metrics import accuracy scoreas acac(ypred,y test) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='9902173913043478Sensitivity=TP/(TP+FN)Specificity=TN/(TN+FP)F1-score = 2 * (P * R) / (P + R) P=precision R=RecallBrain Tumor obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='evaluate(Brain Tumor obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='X test, Brain Tumor obj-ytest,batch size2) 460/460t j7s15ms/step-loss:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='0381-accuracy:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='9880 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='03809912130236626,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='988043487071991] or_obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='xtest)y_pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='=np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='argmax(y_pred,axis1) y_test=np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='argmax(Brain_Tumor_obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='y_test,axis=1) from sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='metrics import accuracy_score as ao0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content="9880434782608696xGBclassifier(max depth-l5,nestimators=5oo,objective='multi:softprob')X test featuresnew model." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='predict(Brain Tumor obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='X test) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 19 Accuracy outcome for non-hybrid CNN (left) and CNN-XGBOOST model (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' We used the seaborn library to generate a heatmap visualization of a confusion matrix for the evaluation of the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' It compares the predicted class labels (y_pred) with the true class labels (y_test) and shows the number of correct and incorrect predictions for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The heatmap is a visual representation of the confusion matrix, with the colour intensity indicating the number of predictions in each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='confusion_matrix function is used to compute the confusion matrix from the true labels (y_test) and predicted labels (y_pred).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The annot parameter specifies that the values of the confusion matrix should be displayed in each cell of the heatmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This visualization identifies which classes are being misclassified and how well the model is performing overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The three classes represented in the matrix represents glioma - meningioma - and pituitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The precise number of images classified by the model can be determined using the diagonal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' From the result above, we observed that the total frequency of misclassifications is fewer in C-Xgboost than purely CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Random samples of 15 test images, with their predicted labels and ground truth is shown below using the proposed convolutional xgboost model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Figure 20: Random samples of 15 test images showing their predicted and ground truth labels C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Comparative Performance Analysis Performance results obtained from existing deep learning models on the brain tumor MRI images are shown in comparison with that of the proposed models in Table 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The table shows the machine learning technique employed used by the authors and their outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' performance characteristics of different prediction models trained using brain MRI images V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' CONCLUSION This paper analysed the performance of C-XGBoost models for predicting brain tumors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The C-XGBoost model which is based on DenseNet201 outperformed the non-hybrid CNN model in various metrics as discussed in the result analysis section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The dataset was acquired from the figshare public repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The proposed hybrid C-XGBoost model leveraged the strengths of both CNNs and XGBoost, resulting in an improved ability to accurately classify brain MRIs with tumour and without tumours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' The model achieved an accuracy of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='0% and F1 score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' To assess the relative effectiveness of the proposed C-XGBoost model, its performance was also compared to that of other existing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' One of the benefits of using the XGBoost component Reference Study Title Machine Learning Technique Used Accuracy Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [23] Automatic Detection of Brain Tumors in magnetic resonance imaging (MRI) Using Machine Learning Algorithms A combination of support vector machines, decision trees, and neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='50% Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [24] Brain Tumor Detection and Classification Using Multimodal MRI Scans and Deep Convolutional Neural Networks Convolutional Neural Network Sensitivity: 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='3% Specificity: 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='7% Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [25] Brain Tumor Detection and Segmentation in MRI Images Using Deep Learning Deep learnig technique Dice co-efficient: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='82 Kamnitsas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [26] A Deep Learning Framework for Brain Tumor Detection and Segmentation in 3D Magnetic Resonance Images 3D Convolutional Neural Network Dice co-efficient: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='81 El-Shenawee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [27] Convolutional Neural Networks (CNN) and adaptive thresholding Deep Convolutional Neural Network 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='44% Al-Shara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [28] Detection of Brain Tumor in MRI Images Using Deep Learning and Fast R-CNN Deep learning and Fast R-CNN 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='31% Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [29] Deep Learning-Based Detection of Brain Tumors in MRI Images Using U-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Deep learning and U- Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='90% Krishna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [30] Detection of Brain Tumors in MRI Images Using a Hybrid Approach Based on Deep Learning and Transfer Learning Deep learning and transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='40% CNN model 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='80% C-XGBOOST model 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='02% Brain tumor detection using convolutional XGBoost Proposed Methodology import seaborn as snsimport seaborn as snssns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='heatmap(tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='confusion matrix( y_test, ypred,)，annot一True)4002e+02 0 6 350 300250 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='2e+02 I 200150100 0 29e+02 2 500 1 NPrediction Yes (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='o0) Prediction:No (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='03) Prediction:No (o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='o0) Prediction Yes (1 oo) Ground Truth:Yes Ground Truth:No Ground Truth:No Ground Truth YesY_test, y_pred, otmaPredictionYes:(1D0) Prediction:Yes (o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='66) Prediction:Yest1oo1 Prediction:No to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='01)Ground Thuth:Yes Ground Truth:Yes Ground Truth:Yes Groand Truth:NoPredictianYes (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='00) Prediction:Yes (1 o0) Predictipon,No (o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='03) PredictionYes (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='00) GroiandTruthYes Ground Truth:Yes Gtound Truth:No Ground TruthYes400 2e+02 一350300 2504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content="2e+02 1 200 150-100 29e+02 2 -50-2 0 in the proposed C-XGBoost model is its ability to provide interpretability and explainability for the model's decisions." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' This can be a useful diagnostic tool for medical practitioners when applied in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Further analysis and validation on larger and more diverse datasets are necessary to fully assess the capabilities of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=" To further improve the model's ability to detect brain tumors at very early stages, it may be necessary to address challenges related to adjusting image capture, improving image quality, combining diverse data formats, and correcting weight misalignment." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' These challenges can be addressed by continually fine-tuning the model, adjusting its hyperparameters, and training it with new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' References [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' "Cancer incidence by age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='" Cancer Research UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='cancerresearchuk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='org/health- professional/cancer-statistics/incidence/age (accessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [2] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Qiao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Ge, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Xu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' He, "Artificial Intelligence Algorithm-Based Computerized Tomography Image Features Combined with Serum Tumor Markers for Diagnosis of Pancreatic Cancer," (in eng), Comput Math Methods Med, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 2022, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 8979404, 2022, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='1155/2022/8979404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Uddin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Khan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Hossain, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Moni, "Comparing different supervised machine learning algorithms for disease prediction," (in eng), BMC Med Inform Decis Mak, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 19, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 281, Dec 21 2019, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='1186/s12911- 019-1004-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [4] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Lecun, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Bottou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Bengio, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Haffner, "Gradient-based learning applied to document recognition," (in English), P Ieee, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 86, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 2278-2324, Nov 1998, doi: Doi 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='1109/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='726791.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Gjoreski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=', "Facial EMG sensing for monitoring affect using a wearable device," (in eng), Sci Rep, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 16876, Oct 7 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [6] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Smirnov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Timoshenko, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Andrianov, "Comparison of Regularization Methods for ImageNet Classification with Deep Convolutional Neural Networks," (in English), Aasri Proc, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 89-94, 2014, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='aasri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [7] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Grohs and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Kutyniok, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Mathematical Aspects of Deep Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Cambridge: Cambridge University Press, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' El-Baz and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Suri, Artificial intelligence in cancer diagnosis and prognosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Volume 3, Brain and prostate cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Bristol [England] (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='2 The Distillery, Glassfields, Avon Street, Bristol, BS2 0GR, UK): IOP Publishing, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Brownlee, XGBoost With Python: Gradient Boosted Trees with XGBoost and scikit-learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Machine Learning Mastery, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Ross Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=', "Spectrum, risk factors and outcomes of neurological and psychiatric complications of COVID-19: a UK-wide cross-sectional surveillance study," Brain Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 3, 2021, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='1093/braincomms/fcab168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [11] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Alfonso Perez and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Caballero Villarraso, "Neural Network Aided Detection of Huntington Disease," (in eng), J Clin Med, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 8, Apr 10 2022, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='3390/jcm11082110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Zhou, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Zhang, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Zhang, "Two-phase non- invasive multi-disease detection via sublingual region," (in eng), Comput Biol Med, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 137, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 104782, Oct 2021, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='compbiomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content='104782.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [13] -based diagnosis International Conference on Pattern Recognition (ICPR), 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 1236 1241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Ezzat, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Elshahat, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Elsharkawy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Zaky, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' El- progression of Al in 2018 10th International Conference on Machine Learning and Computing (ICMLC), 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 277 281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [15] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' George, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Agrawal, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' George, International Conference on Computer Science and Engineering, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Mavroforakis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Vouros, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' -invasive ual International Conference of the IEEE Engineering in Medicine and Biology Society, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 842 845.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [17] deep learning approach for predicting the severity of diabetic national Conference of the IEEE Engineering in Medicine and Biology Society, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 4596 4599.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Pechenizkiy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Kersting, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Ziegler, 7th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [19] likelihood of stroke using a hybrid machine learning Conference of the IEEE Engineering in Medicine and Biology Society, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 5580 5583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [20] 2012 34th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 4644 4647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [21] Kotu, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=', & Alahakoon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Classification model performance evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' In 2018 IEEE 9th International Conference on Software Quality, Reliability, and Security Companion (QRS-C) (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' 109-115).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' [22] Hossain, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=', & Rahman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' Performance evaluation of classification models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} +page_content=' International Journal of Computer Applications, 111(7)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQfZQAg/content/2301.02317v1.pdf'} diff --git a/mtAyT4oBgHgl3EQfyvmi/content/2301.00690v1.pdf b/mtAyT4oBgHgl3EQfyvmi/content/2301.00690v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..85ebcc496910c87f6e8ad113f33e99473783e4a8 --- /dev/null +++ b/mtAyT4oBgHgl3EQfyvmi/content/2301.00690v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:70dd5b7d012d96b448763d88b948aefd2127dfb566e30cb3e3bf2661215539a2 +size 565371 diff --git a/mtAyT4oBgHgl3EQfyvmi/vector_store/index.faiss b/mtAyT4oBgHgl3EQfyvmi/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..06124666b499440feb14402b4dcff563acbf4b2c --- 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+3Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA +4Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK +5Kavli Institute for Cosmology, Madingley Road, Cambridge CB3 0HA, UK +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +The reported detection of the global 21-cm signal by the EDGES collaboration is significantly stronger than standard astrophysical +predictions. One possible explanation is an early radio excess above the cosmic microwave background. Such a radio background +could have been produced by high redshift galaxies, if they were especially efficient in producing low-frequency synchrotron +radiation. We have previously studied the effects of such an inhomogeneous radio background on the 21-cm signal; however, +we made a simplifying assumption of isotropy of the background seen by each hydrogen cloud. Here we perform a complete +calculation that accounts for the fact that the 21-cm absorption occurs along the line of sight, and is therefore sensitive to radio +sources lying behind each absorbing cloud. We find that the complete calculation strongly enhances the 21-cm power spectrum +during cosmic dawn, by up to two orders of magnitude; on the other hand, the effect on the global 21-cm signal is only at +the 5% level. In addition to making the high-redshift 21-cm fluctuations potentially more easily observable, the line of sight +radio effect induces a new anisotropy in the 21-cm power spectrum. While these effects are particularly large for the case of an +extremely-enhanced radio efficiency, they make it more feasible to detect even a moderately-enhanced radio efficiency in early +galaxies. This is especially relevant since the EDGES signal has been contested by the SARAS experiment. +Key words: +methods: numerical – methods: statistical – dark ages, reionization, first stars – cosmology: observations – +cosmology: theory +1 INTRODUCTION +The redshifted 21-cm signal which originates due to the hyperfine +splitting of the neutral hydrogen in the intergalactic medium (IGM) is +the most promising probe of the early universe, most importantly, the +epoch of the first stars and the Epoch of Reionization (EoR). The rest- +frame frequency of 1420 MHz is redshifted due to the expansion of +the universe and can be detected using ground-based radio telescopes +at frequencies below 200 MHz against the background radiation, +which is usually assumed to be the cosmic microwave background +(CMB). +The first claimed detection of the all-sky averaged global 21-cm +signal from 𝑧 ∼ 13−17 was the EDGES low band observation in the +frequency range of 50 − 100 MHz (Bowman et al. 2018). The signal +was centered at 𝑧 ∼ 17 (which corresponds to 𝜈 ∼ 78 MHz) with +a strong absorption feature of 𝑇21 = −500+200 +−500 mK. While disputed +at 95% significance by the SARAS experiment (Singh et al. 2021), +with further measurements expected to resolve this tension, the ten- +tative EDGES signal has inspired various theories. Specifically, this +anomalously strong trough has two main categories of explanations. +One category is that an additional cooling mechanism can cool the +★ E-mail: sudiptas@mail.tau.ac.il +gas faster than only adiabatic cooling due to the cosmic expansion. +An additional cooling mechanism has been suggested (Barkana 2018; +Berlin et al. 2018; Barkana et al. 2018; Muñoz & Loeb 2018; Liu +et al. 2019; Barkana et al. 2022) that involves a non-gravitational +interaction between the baryons and dark matter particles (e.g., via +Rutherford-like scattering) that drives down the temperature of the +gas leading to the strong observed absorption. The other category of +explanation is the presence of an excess radio background at high red- +shifts, well over the CMB level (Bowman et al. 2018; Feng & Holder +2018; Ewall-Wice et al. 2018; Fialkov & Barkana 2019; Mirocha +& Furlanetto 2019; Ewall-Wice et al. 2020). Specifically, Fialkov & +Barkana (2019) showed that the EDGES signal could be explained +by a homogeneous external radio background with a synchrotron +spectrum. However, this external radio background is not directly re- +lated to astrophysical sources. Exotic processes such as dark matter +annihilation or superconducting cosmic strings (Fraser et al. 2018; +Pospelov et al. 2018; Brandenberger et al. 2019) could give rise to this +kind of homogeneous external radio excess. A more astrophysically- +grounded approach is to assume that radio-loud sources such as active +galactic nuclei (AGN, Urry & Padovani 1995; Biermann et al. 2014; +Bolgar et al. 2018; Ewall-Wice et al. 2018, 2020) or star-forming +galaxies (Condon 1992; Jana et al. 2019) at high redshift could pro- +duce an excess radio background, which in this case would be inho- +© 2022 The Authors +arXiv:2301.04585v1 [astro-ph.CO] 11 Jan 2023 + +2 +S. Sikder et al. +mogeneous. Reis et al. (2020) first incorporated the inhomogeneous +excess galactic radio background into semi-numerical simulations of +the early Universe, and explored the effect on the global 21-cm signal +and on the 21-cm power spectrum. Interestingly, at low frequencies, +ARCADE2 (Fixsen et al. 2011; Seiffert et al. 2011) detected an ex- +cess radio background over that CMB that was confirmed by LWA1 +(Dowell & Taylor 2018) in the frequency range 40 − 80 MHz. This +observed excess radio could be explained by extragalactic sources, +but it is unclear what fraction of the observed excess originates from +Galactic compared to extragalactic sources (e.g., Subrahmanyan & +Cowsik 2013). +In our previous work (Reis et al. 2020) we made a simplifying ap- +proximation and assumed that the effect of the radio background on +a given hydrogen cloud can be determined from the isotropically- +averaged radio intensity at that position. However, since 21-cm +absorption occurs along the line-of-sight (hereafter LoS), the 21- +cm effect effectively involves two different radio intensities. The +isotropically-averaged radio intensity is appropriate for effects such +as the physical heating of the gas, while the calculation of radiative +transfer along the line of sight depends on the radio intensity coming +only from radio sources lying behind the hydrogen cloud, along our +line of sight. This can potentially enhance the 21-cm power spectrum +due to the LoS radio fluctuations, especially early in cosmic dawn +when the number of contributing radio sources is small. We perform +this complete calculation in this work, and also quantify the resulting +LoS anisotropy in the 21-cm power spectrum using the anisotropy +ratio (following Fialkov et al. 2015). +This paper is organized as follows: we briefly describe our semi- +numerical simulation in section 2. In section 3, we review the theo- +retical framework of the 21-cm signal in the presence of an excess +radio background and show how we include the line of sight effect of +radio fluctuations in the simulation. In section 4, we explore the effect +of the line of sight radio fluctuations on the 21-cm signal including +the anisotropic power spectrum due to this line of sight effect. We +conclude the paper with a summary in section 5. +2 BASIC METHOD +We use our semi-numerical 21-cm simulation code (e.g., Visbal +et al. 2012; Fialkov & Barkana 2014; Cohen et al. 2017; Fialkov +& Barkana 2019) to calculate the 21-cm signal over a wide range of +redshifts. This simulation code was originally inspired by 21cmFAST +(Mesinger et al. 2011), but it is entirely an independent implemen- +tation. The code simulates the realization of the universe in a 3843 +Mpc3 comoving cosmological volume with a resolution of 3 comov- +ing Mpc. The simulation is based on the following algorithm: we +create a random realization of the large-scale linear density field, i.e., +the three dimensional cubes of density fluctuations and the relative +velocity between the dark matter and the baryons (Tseliakhovich & +Hirata 2010) given the power spectra of initial Gaussian random den- +sity fields and velocity fields (calculated using the publicly available +code CAMB, Lewis et al. 2000). Given the large scale density fields +and the relative velocity, we obtain the population of the collapsed +dark matter halos inside each cell of 33 Mpc3 comoving volume, +using the modified Press-Schechter model (Press & Schechter 1974; +Sheth & Tormen 1999; Barkana & Loeb 2004). The baryon fraction +contained in each halo is assumed to be the cosmic mean, except that +it is reduced due to the streaming velocity (Tseliakhovich & Hirata +2010; Fialkov et al. 2012). Star formation takes place where the dark +matter halos are massive enough to radiatively cool the infalling gas. +This sets the minimum mass of star forming halos (given by a min- +imum circular velocity 𝑉𝑐), except that feedback can also affect this +free parameter. Another important parameter is the star formation +efficiency, 𝑓∗, and we also take into account the suppression of star +formation due to the above-mentioned relative velocity between dark +matter and baryons, Lyman-Werner feedback on molecular-hydrogen +cooling halos (Haiman et al. 1997; Fialkov et al. 2013), and photo- +heating feedback (Rees 1986; Sobacchi & Mesinger 2013; Cohen +et al. 2016). +Once we have a population of galaxies, we calculate the radiation +fields emitted by those galaxies. The most relevant radiation fields +that affect the 21-cm signal are ionizing, Ly-𝛼 and X-ray radiation. To +calculate the intensity of the Ly-𝛼 radiation field, i.e., 𝐽𝛼, we assume +that galaxies contain population II stars. The X-ray luminosity (𝐿X) +of the galaxies is assumed to scale with the star formation rate (SFR), +which is based on X-ray observations of low-redshift galaxies (e.g., +Grimm et al. 2003; Gilfanov et al. 2004; Mineo et al. 2012; Fragos +et al. 2013; Fialkov et al. 2014; Pacucci et al. 2014): +𝐿X +SFR = 3 × 1040 𝑓X erg s−1M−1 +⊙ yr , +(1) +where the standard normalization factor 𝑓X is the X-ray efficiency +of the sources, a free parameter in our simulation. Here 𝑓X = 1 +corresponds to the typical observed value for low metallicity galaxies. +In addition to the luminosity, the shape of the X-ray spectral energy +density (SED) affects the 21-cm signal. We assume that the shape of +the X-ray SED is determined by a power law slope (𝛼, which we set +equal to 1.5) and a low energy cutoff (𝐸min). The hard X-ray SED +sourced by a population of high redshift X-ray binaries (XRBs) peaks +at energy ∼ 3 keV. Soft X-ray sources (with typical energy ∼ 0.5 kev) +produce strong fluctuations on small scales (up to a few tens of Mpc), +whereas the typical mean free path of hard X-ray photons is so large +that the fluctuations are reduced and also they lose some of their +energy due to the redshift effect. +After the heating transition due to X-ray photons, the universe +starts to reionize. The phase transition known as the epoch of reion- +ization is expected to have occured inside out, meaning that the +high-density regions containing most of the sources proceeded to +reionize first (Barkana & Loeb 2004; Furlanetto et al. 2004). As the +21-cm brightness temperature is proportional to the fraction (𝑥HI) of +the neutral hydrogen atoms in the IGM, the amplitude of the global +signal decreases as reionization proceeds. Another free parameter +in our simulation is the ionization efficiency, 𝜁. The late stages of +reionization also depend on the maximum mean free path of the +ionizing photons, 𝑅mfp (Greig & Mesinger 2015). Dense regions of +neutral hydrogen (specifically, Lyman-limit systems) that appear at +high redshifts due to structure formation, effectively absorb the ioniz- +ing radiation and set an upper limit on the effective ionization bubble +size. Since here we focus on significantly higher redshifts, we simply +set 𝑅mfp = 30 Mpc (comoving) and 𝜁 = 30 for all the cases used +in this work; this gives an optical depth to the CMB that is consis- +tent with Planck measurements (Planck Collaboration et al. 2018). +Finally, an excess radio background above the cosmic microwave +background can be included as we discuss in the next section. +3 21-CM SIGNAL +The 21-cm brightness temperature, 𝑇21, depends on the contrast +between the spin temperature, 𝑇S, of the neutral hydrogen and the +background radiation temperature, 𝑇rad, and can be written as +𝑇21 = 𝑇S − 𝑇rad +1 + 𝑧 +(1 − 𝑒−𝜏21) . +(2) +MNRAS 000, 1–10 (2022) + +LoS effect on the 21-cm signal from radio galaxies +3 +Usually the background radiation is assumed to be the CMB (at +redshift 𝑧), in which case 𝑇rad = 𝑇CMB = 2.725(1+𝑧) K, unless there +is an excess radio background above the CMB (discussed below). +Here 𝜏21 is the optical depth of the 21-cm signal and is given by +𝜏21 = +3ℎpl𝐴10𝑐𝜆2 +21𝑛H +32𝜋𝑘𝐵𝑇S(1 + 𝑧)𝑑𝑣/𝑑𝑟 , +(3) +where ℎpl is the Planck constant, 𝐴10 is the spontaneous decay rate +of the hyperfine transition of the neutral hydrogen, 𝑐 is the speed +of light, 𝜆21 = 21.1 cm is the rest frame wavelength of the 21-cm +line, 𝑛𝐻 is the number density of the neutral hydrogen, 𝑘𝐵 is the +Boltzmann constant, 𝑑𝑣/𝑑𝑟 = 𝐻(𝑧)/(1+ 𝑧) is the gradient of the line +of sight component of the comoving velocity field and 𝐻(𝑧) is the +Hubble constant (Our code also includes fluctuations with respect to +this expression for the mean velocity gradient). +The spin temperature 𝑇𝑆 can be written as (Madau et al. 1997) +𝑇𝑆 = +𝑥rad + 𝑥tot +𝑥rad𝑇−1 +rad + 𝑥tot𝑇−1 +K +, +(4) +where +𝑥rad = 1 − 𝑒−𝜏21 +𝜏21 +(5) +is the radiative coupling (Venumadhav et al. 2018), and the coupling +coefficient 𝑥tot is the sum of the contributions of Ly-𝛼 coupling (𝑥𝛼) +and the collisional coupling (𝑥𝑐), i.e., 𝑥tot = 𝑥𝛼 + 𝑥𝑐, with +𝑥𝛼 = +1 +𝐴10𝑇rad +16𝜋2𝑇∗𝑒2 𝑓𝛼 +27𝑚𝑒𝑐 +𝐽𝛼 , +(6) +and +𝑥𝑐 = +1 +𝐴10𝑇rad +𝜅1−0(𝑇K)𝑛H𝑇★ . +(7) +Here 𝑓𝛼 = 0.4162 is the oscillator strength of the Ly-𝛼 transition, 𝐽𝛼 +is the intensity of the Ly-𝛼 radiation, 𝑇∗ = 0.0682K and 𝜅1−0(𝑇𝐾 ) is +the known atomic coefficient (Allison & Dalgarno 1969; Zygelman +2005). +When the optical depth 𝜏21 << 1, the 21-cm brightness tempera- +ture is given by +𝑇21 ≈ 26.8 +� Ωbℎ +0.0327 +� � Ωm +0.307 +�−1/2 � 1 + 𝑧 +10 +�1/2 +(1 + 𝛿)𝑥HI +𝑥tot +1 + 𝑥tot +� +1 − 𝑇rad +𝑇K +� +mK , +(8) +where we have included the effect of the neutral hydrogen fraction +𝑥HI, and of the density contrast 𝛿. +When calculating the kinetic gas temperature we include the usual +effects of adiabatic evolution, Compton heating, and X-ray heating. +The effect of the radio background on the kinetic gas temperature +based on the CMB heating mechanism introduced by Venumadhav +et al. (2018) is also included here (but see objections by Meiksin +2021). The heating rate due to the radio background is +𝜖rad = 𝑥HI𝐴10 +2𝐻(𝑧) 𝑥rad +�𝑇rad +𝑇S +− 1 +� 𝑇21 +𝑇K +. +(9) +In practice we use eq. 3, including the effect of the inhomogeneous +density and velocity gradient, and do not assume the linearized form +as in eq. 8; we note, though, that the linearized expression is in most +cases rather accurate. +3.1 The excess radio background: previous work +In the presence of an excess radio background, we can rewrite the +background radiation temperature 𝑇rad, as +𝑇rad = 𝑇Radio + 𝑇CMB , +(10) +where 𝑇Radio is the brightness temperature of the excess radio +background. In our previous work we calculated the isotropically- +averaged radio intensity at each pixel, and used the resulting 𝑇Radio +in all of the above equations. +One type of excess radio background that has been considered is +a homogeneous external radio model that is not directly related to +astrophysical sources. This excess radio background could possibly +be generated by exotic processes, e.g., annihilating dark matter or +super-conducting cosmic strings (Fraser et al. 2018; Pospelov et al. +2018; Brandenberger et al. 2019). A simple formulation of such a +model (Fialkov & Barkana 2019) sets the brightness temperature of +the excess radio background at the 21-cm rest frame frequency at +redshift 𝑧 as +𝑇Radio = 2.725(1 + 𝑧) 𝐴r × +� +1420 +78(1 + 𝑧) +� 𝛽 +K , +(11) +where 2.725 K is the CMB temperature today, the spectral index +of the synchrotron spectrum is 𝛽 = −2.6 (set to match the slope of +the observed extragalactic radio background, so that the exotic excess +background is consistent with observational limits), and 𝐴𝑟 measures +the amplitude of the radio background (relative to the CMB at the +central redshift of the EDGES claimed absorption feature). +An excess radio background over the CMB can also be produced +by high redshift galaxies if they emit strongly in the radio (Reis et al. +2020). Based on the empirical relation of Gürkan et al. (2018), we +can write the galaxy radio luminosity per unit frequency, which is +proportional to the star formation rate (SFR), as +𝐿Radio(𝜈, 𝑧) = 𝑓Radio1022 � +𝜈 +150 MHz +�−𝛼Radio +SFR +𝑀⊙ yr−1 , +(12) +in units of W Hz−1. In eq. 12, the spectral index in the radio band +𝛼Radio is set to 0.7 as in Mirocha & Furlanetto (2019) and Gürkan +et al. (2018); see also Condon et al. (2002) and Heesen et al. (2014). +Here 𝑓Radio is the normalization of the radio emissivity, where +𝑓Radio = 1 for present-day star-forming galaxies. In our work, we +assume for simplicity a uniform value of 𝑓Radio, though we note that +there is significant scatter in 𝑓Radio from observations. +In our previous work (Reis et al. 2020), the brightness temperature +of the radio background at redshift 𝑧 at the 21-cm frequency was +calculated by summing the contribution from all the galaxies within +the past light-cone (following Ewall-Wice et al. 2020): +𝑇Radio(𝜈21, 𝑧) = +𝜆2 +21 +2𝑘B +𝑐(1 + 𝑧)3 +4𝜋 +∫ +𝜖Radio +� +𝜈21 +1 + 𝑧em +1 + 𝑧 , 𝑧em +� +(1 + 𝑧em)−1𝐻(𝑧em)−1𝑑𝑧em , +(13) +where 𝑧em > 𝑧 is the redshift at which a photon was emitted, and +𝜖Radio is the comoving radio emissivity, i.e., the luminosity per unit +frequency per unit comoving volume, averaged over radial shells +within this spherical integral. The radius of each spherical shell is +given by the light travel distance between 𝑧em and 𝑧. This calculation +is thus similar to that for finding the Ly-𝛼 and X-ray radiation fields +in our semi-numerical simulation, except that for the Ly-𝛼 radiation +field, modified window functions are used in order to include the +effect of multiple scattering of the Ly-𝛼 photons (Reis et al. 2022). +MNRAS 000, 1–10 (2022) + +4 +S. Sikder et al. +3.2 LoS effect of the radio background from galaxies +As outlined in the previous subsection, in our previous work we +calculated the isotropically-averaged radio intensity at each pixel, and +used the resulting 𝑇Radio in eq. 10. This is accurate for all the direct +physical effects of the radiation, i.e., in eqs. 6, 7, 4, and 9. However, +it is only approximately true in the radiative transfer equation 2. +As the 21-cm absorption occurs along the line of sight, in this +work we accurately consider the line of sight contribution of the +excess radio background originating from high redshift galaxies, and +examine the effect on the 21-cm signal. Our previous approximation +becomes accurate in the limit of many radio sources reaching each +pixel, since in that case the radio background does become nearly +isotropic. However, at the highest redshifts the radio background at +a point is still dominated by a small number of nearby sources, and +it then matters whether a source is behind our line of sight to the +absorbing pixel, or not. Indeed, we expect the LoS effect to amplify +the 21-cm fluctuations at early times. +Here we will continue to refer to the isotropically-averaged radio +intensity at a given pixel as 𝑇Radio, but we also use a different window +function and calculate the brightness temperature of the radio back- +ground from sources lying behind the pixel along our LoS; we refer +to the latter quantity as 𝑇R,los. For clarity, we first find the observed +21-cm brightness temperature relative to the CMB: +𝑇CMB +21 += +�𝑇𝑅,los + 𝑇CMB +� 𝑒−𝜏21 + 𝑇𝑆 (1 − 𝑒−𝜏21) − 𝑇CMB +1 + 𝑧 +, +(14) +where here 𝑇𝑆 and 𝜏21 depend on 𝑇Radio. Now, if there is indeed a +strong excess radio background, it too is observed and is subtracted +out in any method of foreground removal (since it is assumed here +to have a smooth power-law synchrotron spectrum). Another way of +expressing this is that we must subtract out the 𝜏21 = 0 case in order +to arrive at the final expression: +𝑇21 = 𝑇𝑆 − �𝑇𝑅,los + 𝑇CMB +� +1 + 𝑧 +�1 − 𝑒−𝜏21� . +(15) +This replaces eq. 2 (which also was written after subtracting out +the 𝜏21 = 0 case under the previous approximation). Since the 21- +cm optical depth is usually quite small, we also note the linearized +form of this full expression including the line-of-sight effect, which +replaces eq. 8: +𝑇21 ≈ 26.8 +� Ωbℎ +0.0327 +� � Ωm +0.307 +�−1/2 � 1 + 𝑧 +10 +�1/2 +(1 + 𝛿)𝑥HI +1 +1 + 𝑥tot +� +𝑥tot +� +1 − 𝑇𝑅,los + 𝑇CMB +𝑇K +� ++ +� +1 − 𝑇𝑅,los + 𝑇CMB +𝑇Radio + 𝑇CMB +�� +mK . +(16) +As we noted above, the last two equations also contain implicit depen- +dencies on 𝑇Radio (the isotropically-averaged radio intensity) through +eq. 10, which affects eqs. 6, 7, 4, and 9. +The last factor in eq. 16 shows how the spin temperature varies +between𝑇Radio+𝑇CMB (in the uncoupled case) and𝑇K (after saturated +coupling), while the observations always probe the contrast between +𝑇𝑆 and 𝑇𝑅,los +𝑇CMB. The uncoupled case (where the previous result +gave no signal) shows explicitly how the line-of-sight effect provides +a new source of 21-cm fluctuations. Meanwhile, the last factor in +eq. 16 together with eqs. 6 and 7 shows that the Ly-𝛼 coupling +transition (when 𝑥tot ∼ 1) is substantially delayed by an intense radio +background. Also, the sign transition (when the mean 21-cm signal +goes from absorption to emission) is significantly delayed, as it no +longer occurs around the heating transition (normally defined as the +average gas temperature𝑇K reaching𝑇CMB), but must wait for the gas +to heat to the higher temperature given by the average of𝑇𝑅,los+𝑇CMB +(assuming 𝑥tot is large at that time). Even if CMB heating is effective +(given the radio-background boost in eq. 9), the sign transition is still +substantially delayed when the excess radio background is strong. +3.3 LoS anisotropies in the 21-cm signal +Among the various sources of fluctuations that contribute to the +21-cm fluctuations, many are statistically isotropic. This is true, for +example, for the gas density fluctuations that arise from the initial +conditions and are a potential probe of the cosmological parameters. +Once the first stars and galaxies form, various forms of radiation in- +cluding Ly-𝛼 photons, ionizing photons, and X-ray photons, become +additional sources of 21-cm fluctuations. These radiation fields are +produced by the processes of star and galaxy formation that are com- +plex and non-linear, but have no overall preferred directions. Thus, +the 21-cm signal due to these radiation fluctuations is isotropic. How- +ever, since the 21-cm signal is redshifted and is determined by LoS +absorption, there are a number of effects that make it anisotropic. It +is important to quantify the LoS anisotropy in the redshifted 21-cm +signal for a better understanding of the 21-cm signal itself and also +since the anisotropy is potentially directly observable with upcoming +telescope arrays such as the Square Kilometre Array (SKA). +The coherent inflow of matter into over-dense regions and outflow +of matter from under-dense regions, namely the peculiar velocity +of the baryonic matter, makes the 21-cm signal anisotropic along +the LoS. As a result, the 21-cm power spectrum is expected to be +anisotropic due to the radial component of the peculiar velocity +gradient (Bharadwaj & Ali 2004; Barkana & Loeb 2005). The light- +cone effect, whereby only the LoS direction corresponds to a varying +redshift, also produces a LoS anisotropy in the 21-cm fluctuations +(Barkana & Loeb 2006; Datta et al. 2012). Due to the uncertainty +in the values of the cosmological parameters, another potentially +observable source of anisotropy in the 21-cm power spectrum is the +Alcock-Paczyński effect (Alcock & Paczyński 1979; Nusser 2005; +Ali et al. 2005; Barkana 2006). In our calculations we include the +main expected source of 21-cm anisotropy out of these, i.e., the +LoS anisotropy due to peculiar velocities. Now, since we consider +in this work the line of sight dependence of the radio fluctuations, +this naturally introduces a new potential sources of anisotropy in the +21-cm signal. +In order to understand the anisotropy, we first note that in the +presence of the anisotropy due to peculiar velocities, the 21-cm power +spectrum (in linear theory) can be written as a polynomial 𝑃(𝑘, 𝜇), +where 𝜇 is the cosine of the angle between k and the line of sight +(Barkana & Loeb 2005): +𝑃(𝑘, 𝜇) = 𝑃𝜇0 (𝑘) + 𝜇2𝑃𝜇2 (𝑘) + 𝜇4𝑃𝜇4 (𝑘) . +(17) +Here 𝑃𝜇0 (𝑘) results from the fluctuations from all the isotropic +sources, 𝑃𝜇4 (𝑘) is proportional to the primordial density power +spectrum, and 𝑃𝜇2 (𝑘) is proportional to the correlation between the +density and the radiation from the isotropic sources. This decompo- +sition of the 21-cm power spectrum can potentially provide valuable +information beyond just the spherically-averaged power spectrum +(Barkana & Loeb 2005; Fialkov et al. 2015). However, in the pres- +ence of non-linearities and more generally, Fialkov et al. (2015) +proposed a simpler method to measure the anisotropy in the 21-cm +power spectrum by defining the anisotropy ratio: +𝑟𝜇(𝑘, 𝑧) ≡ +⟨𝑃(k, 𝑧)|𝜇𝑘 |>0.5⟩ +⟨𝑃(k, 𝑧)|𝜇𝑘 |<0.5⟩ − 1 . +(18) +Here the angular brackets denote an angular averaging over a range +of 𝜇 values. The value of 𝑟𝜇(𝑘, 𝑧) captures in one number (at each +MNRAS 000, 1–10 (2022) + +LoS effect on the 21-cm signal from radio galaxies +5 +-100 +0 +100 +X [Mpc] +-192 +-92 +0 +92 +192 +Y [Mpc] +z = 20 +Previous work +-100 +0 +100 +Z [Mpc] +Previous work (LoS) +-100 +0 +100 +X [Mpc] +This work +-100 +0 +100 +Z [Mpc] +This work (LoS) +−8000 +−6000 +−4000 +−2000 +T21 [mK] +Figure 1. Line-of-sight effect of fluctuations in the excess radio background on the cosmic dawn 21-cm signal, illustrated at 𝑧 = 20. We compare the case of +a fluctuating radio background from our previous work (left two panels) to the full calculation including the LoS effect of the fluctuating radio background +emitted by galaxies (right two panels). From a single cubic simulation box, we show both the 21-cm slice perpendicular to the LoS (𝑋𝑌 plane) and one that +includes the LoS (𝑍) direction. Both models have the same radio production efficiency, with astrophysical model parameters: 𝑉𝑐 = 16.5 km s−1, 𝑓∗ = 0.1, and +𝑓Radio = 3000; note that the circular velocity corresponds to the minimum halo mass for star formation being set by atomic cooling, and at 𝑧 = 20 the mass is +3 × 107𝑀⊙. In this example we show the 21-cm slices from 𝑧 = 20, when X-ray heating and reionization are rather insignificant, but for completeness we note +that the parameters are hard X-rays (𝐸min = 1 keV with 𝑓X = 1) and reionization parameters as noted in section 2. We also note that we show the 50’th slice +from the simulation box along each axis, i.e., centered at 𝑍 = −43.5 Mpc (for the 𝑋𝑌 plane) and 𝑋 = −43.5 Mpc (for the 𝑍𝑌 plane). +𝑘 and 𝑧) the overall angular dependence of the power spectrum. If +𝑟𝜇 is close to zero (i.e., much smaller than unity in absolute value), +the power spectrum shows little angular dependence, when it is large +and positive the fluctuations are stronger along the LoS, and when it +is large and negative the fluctuations are stronger in directions on the +sky (i.e., perpendicular to the LoS). In this work, we use equation 18 +in order to quantitatively explore the anisotropy in the 21-cm power +spectrum. A more detailed analysis that quantifies the anisotropy +using other methods is left for future work. +4 RESULTS +4.1 Implications of the LoS effect of the radio background for +the 21-cm signal +In order to study the impact of an excess radio background from early +galaxies on the 21-cm signal, we compare several simulated cases. +We start with the cases considered in Reis et al. (2020), i.e., the CMB- +only case (without any radio excess) and the radio excess case in the +isotropically-averaged approximation of eq. 2, and compare them to +the fully accurate calculation including the LoS effect as in eq. 15. +We find that the LoS effect on radio fluctuations can significantly +affect the 21-cm signal in the redshift range relevant for current +and upcoming radio telescopes. However, this effect varies between +astrophysical models, scales and different epochs. Here we mainly +focus on the dependence on the radio efficiency parameter 𝑓Radio +from eq. 12. In Reis et al. (2020) we found that models that can explain +the EDGES low-band absorption require 𝑓Radio × 𝑓∗ ∼ 140 or higher +(depending on the other astrophysical parameters). As our main case +we consider 𝑓Radio = 3000 (with 𝑓∗ = 0.1), which lies well within +the range compatible with EDGES. We also, though, consider much +lower 𝑓Radio values, that are still well above unity (i.e., moderately +enhanced compared to low-redshift galaxies) but do not depend on +the veracity of the EDGES measurement. We emphasize that there +are few observational constraints on the radio efficiency of galaxies +at very high redshifts, and given the very different astrophysical +conditions at that epoch compared to those at low redshift, it is +important to keep an open mind on the possible radio efficiency, +until new observational constraints can be established. +Fig. 1 shows a comparison of 21-cm slices at 𝑧 = 20 for 𝑓Radio = +3000, simulated using two different models of radio fluctuations: as +in our previous work (Reis et al. 2020, two left-most panels), or as in +this work including the line-of-sight effect (two right-most panels). +In this case in which early galaxies were unusually bright in low- +frequency radio emission relative to star formation, the resulting radio +background strongly enhances the 21-cm signal map, and brings out +the regions surrounding early radio galaxies as strong peaks of 21- +cm absorption. The full inclusion of the LoS effect further brightens +these regions by a factor of 2 or 3 in 21-cm brightness temperature. +Especially interesting is the clear induced structure along the line +of sight, which is potentially a clear observational signature of the +presence of a strong background from high-redshift radio galaxies. +However, these slices represent pure theoretical predictions of the 21- +cm signal, and do not include observational effects that are expected +to make this structure significantly more difficult to discern, as we +explore further below. +At high redshifts during the epoch of the first stars, the 21-cm signal +is normally dominated by Ly-𝛼 fluctuations along with some contri- +butions from the density and temperature fluctuations. As shown in +the previous section, in the presence of a strong radio background, +the LoS effect produces 21-cm fluctuations even before significant +Ly-𝛼 coupling, and this then mixes in with Ly-𝛼 fluctuations (in an +interplay of the two terms in the last factor in eq. 16). At this stage +we find that the 21-cm signal is enhanced by up to a factor of a +few by the radio background; once the delayed Ly-𝛼 coupling does +occur (i.e., 𝑥tot ∼ 1 is reached) in some region, the enhancement can +become much larger (in proportion to the radio intensity), but this +is tempered by the stronger CMB-radio heating. We note that the +radio and Ly-𝛼 intensity fluctuations are positively correlated (also +with the underlying density fluctuations), since both fields originate +MNRAS 000, 1–10 (2022) + +6 +S. Sikder et al. +from the same high redshift galaxies. Thus, these fluctuations en- +hance each other. However, the temperature fluctuations (whether +from CMB-radio heating or, later on, from X-rays) work mostly in +the opposite direction due to the inverse gas temperature dependence +of the 21-cm signal (as long as it is an absorption signal, relative +to the radio background). When we consider the line-of-sight effect +of the radio background (which is unique and does not occur in the +other relevant radiation backgrounds), it is positively correlated with +the radio intensity (since a nearby source along the line of sight con- +tributes also to the overall radio intensity), but it adds a strong source +of random fluctuations due to the directional dependence. +The statistics of the fluctuations are well captured in the 21-cm +power spectrum, shown in Fig. 2 for two different values of wavenum- +ber, as a function of redshift. In addition to showing the enhanced +radio background from this work (with the LoS effect) compared to +our previous work, here we also show two other comparison cases. +One is the case without an excess radio background, i.e., the CMB +only case with 𝑓Radio = 0 (dotted black line in both panels). The other +comparison case is the uniform radio case, i.e., when we take the cos- +mic mean excess radio background (which is the same for this work +and our previous work) and spread it out evenly over the entire box. +This allows us to separate the effect of the overall enhanced radiative +background from the effect of the fluctuations in this background. +We find that the radio fluctuations boost the 21-cm power spectrum +at high redshifts, from the beginning of cosmic dawn on small scales +(𝑘 = 1 Mpc−1) and from 𝑧 ∼ 25 even on large scales (𝑘 = 0.1 +Mpc−1). At 𝑘 = 0.1 Mpc−1, the enhancement during cosmic dawn +would be by 2 orders of magnitude even with a spatially uniform +excess radio background, but the radio fluctuations add more than an +additional order of magnitude, mostly due to the line of sight effect +of the radio fluctuations. At 𝑘 = 1 Mpc−1 the enhancement due to +radio fluctuations starts earlier, but the behaviour is quite similar at +𝑧 = 10 − 20. Regardless of scale, the radio fluctuations eventually +die down as the number of sources becomes large, and the power +spectrum becomes the same as it would be with a uniform excess +radio background. +In order to explore some of the dependence on the various unknown +astrophysical parameters, we show the case of a soft X-ray SED in the +top panel of Fig. 3, for 𝑘 = 0.1 Mpc−1. As noted above, the heating +fluctuations are anti-correlated with the other 21-cm fluctuations, so +once the first generation of X-ray sources heat up the IGM, this heat- +ing mechanism reduces the 21-cm fluctuations, and then produces +a 21-cm fluctuation peak when the heating fluctuations dominate. +X-ray photons with lower energies are absorbed locally, while hard +X-ray photons (≳ 1 keV) lose their energy due to redshifting as they +have a much longer mean free path. As a result, heating is delayed +and the resulting fluctuations are smaller for a hard X-ray SED com- +pared to a soft X-ray SED. Thus, the CMB-only case does show three +peaks (Ly-𝛼 coupling, heating, and reionization), but the strong heat- +ing peak in the soft X-ray case is barely present in the case of hard +X-rays. +In the presence of a strong radio background, the coupling tran- +sition is delayed due to the inverse dependence of the coupling co- +efficients on the radiation background, and the heating transition is +delayed as well since the kinetic temperature now needs to reach +the higher value 𝑇CMB + 𝑇radio (where 𝑇radio is replaced by 𝑇R,los +in the full calculation with the LoS effect). At the same time, the +fluctuations in the radio background compete with, and sometimes +dominate over, the Ly-𝛼 and heating fluctuations. For hard X-rays, +the two normal peaks are washed out and a single overall peak ap- +pears, close to the redshift of the CMB-only heating peak. When we +include the LoS radio fluctuations, this overall peak gets a significant +6 +10 +15 +20 +25 +30 +35 +z +10−4 +10−2 +100 +102 +104 +106 +∆2 [mK2] +k = 0.1 Mpc−1 +This work +Previous work +Uniform radio +CMB only +6 +10 +15 +20 +25 +30 +35 +z +10−4 +10−2 +100 +102 +104 +106 +∆2 [mK2] +k = 1.0 Mpc−1 +This work +Previous work +Uniform radio +CMB only +Figure 2. The 21-cm power spectrum at two wavenumbers, 𝑘 = 0.1 Mpc−1 +and 𝑘 = 1.0 Mpc−1, as a function of redshift, for various simulation runs with +a fixed excess radio background of 𝑓Radio = 3000. We compare the fluctuating +radio background with the LoS effect (solid blue line) to the fluctuating radio +background from our previous work (solid cyan line). For added comparison, +the 21-cm power spectrum due to a uniform excess radio background is +shown (dashed blue line), along with the standard case with no excess radio +background, i.e., the CMB only case (dotted black line). The uniform radio +background case has the same mean intensity of the excess radio at each +redshift as in the cases of the fluctuating radio background. Shown here is the +model with a hard X-ray SED with 𝐸min = 1 keV. +boost (and a slight delay). In the case of a soft X-ray SED, without +the LoS effect (i.e., in the previous work case) the strong heating +fluctuations dominate and maintain a clear heating peak in the 21- +cm power spectrum, which is boosted and delayed compared to the +case without excess radio (CMB only). However, when we take into +account the LoS radio fluctuations, these fluctuations become strong +enough compared to the heating fluctuations to again wash out the +heating peak from the power spectrum and produce a single overall +peak. +Besides the X-ray SED parameters, the X-ray radiation efficiency +( 𝑓X) is another free parameter in our simulation, defined in equation +1. In the bottom panel of Fig. 3 we consider a low efficiency case +( 𝑓X = 0.01, compared to our fiducial 𝑓X = 1). In this case the +heating peak disappears from the 21-cm power spectrum even in +the CMB only case. In the presence of a strong radio background, +the dominance of the radio fluctuations (in this case with hard X- +rays) means that lowering the X-ray efficiency has little effect at high +redshifts, but it does boost the low-redshift signal since in this case +the 21-cm signal is maintained in absorption down to the reionization +epoch. +Since the (isotropically averaged) 21-cm power spectrum is a func- +tion of two variables (𝑘 and 𝑧), in Fig. 4 we show the other cut, i.e., +the function of 𝑘 at a given redshift. This shows the effect of the LoS +radio fluctuations on the shape of the 21-cm power spectrum. Here +MNRAS 000, 1–10 (2022) + +LoS effect on the 21-cm signal from radio galaxies +7 +6 +10 +15 +20 +25 +30 +35 +z +10−4 +10−2 +100 +102 +104 +106 +∆2 [mK2] +Soft X-rays +k = 0.1 Mpc−1 +This work +Previous work +Uniform radio +CMB only +6 +10 +15 +20 +25 +30 +35 +z +10−4 +10−2 +100 +102 +104 +106 +∆2 [mK2] +Low fX +k = 0.1 Mpc−1 +This work +Previous work +Uniform radio +CMB only +Figure 3. The 21-cm power spectrum at 𝑘 = 0.1 Mpc−1 as a function of +redshift for various X-ray parameters. Top panel: a soft X-ray SED (𝐸min = +0.1 keV) with the fiducial X-ray efficiency 𝑓X = 1. Bottom panel: a hard +X-ray SED (𝐸min = 1.0 keV) with a low 𝑓X = 0.01. +we use the same astrophysical model parameters as in Fig. 2. At high +redshifts (right panel of Fig. 4), when the first stars and galaxies begin +to form, the Ly-𝛼 fluctuations are the dominant source of the 21-cm +fluctuations in the standard case. The Ly-𝛼 photons typically travel a +significant distance (Reis et al. 2022), which washes out small-scale +fluctuations, while the strong radio intensity near sources increases +small-scale fluctuations, even more when the LoS effect is included. +At lower redshift (left panel), the LoS effect has less of an effect on +the power spectrum shape, and just gives an overall boost. At later +times, 𝑧 ∼ 10, the LoS radio fluctuations have almost no effect on the +shape of the 21-cm power spectrum compared to the previous radio +fluctuation work (Reis et al. 2020) due to the disappearance of the +excess radio fluctuations at low redshift. +Up to now in this section we have illustrated the consequences of +the LoS effect on the radio background using a particularly strong +radio background. Next, we examine the effect of the LoS radio fluc- +tuations on the 21-cm power spectrum while varying the parameter +𝑓Radio that regulates the strength of the excess radio background. We +show the 21-cm power spectrum as a function of redshift at 𝑘 = 0.1 +Mpc−1 for various values of 𝑓Radio in Fig. 5. The LoS radio fluctua- +tions for moderate values of 𝑓Radio can have a significant effect (∼ an +order of magnitude) after the onset of star formation till the end of +the heating transition, as can be seen in the left panel of Fig.5. The +right panel of Fig. 5 shows the shape of the 21-cm power spectrum +for various values of 𝑓Radio at 𝑧 = 25. Among the values in the plot, +at high redshift only 𝑓Radio of at least 300 has a large effect, but later +in cosmic dawn even 𝑓Radio = 30 has quite a significant effect; since +we set 𝑓∗ = 0.1, the latter value corresponds to a value of 𝑓Radio × 𝑓∗ +that is lower by two orders of magnitude than the value required to +match the EDGES measurement. +The effect of the radio fluctuations on the global 21-cm signal is +shown in Fig. 6. The LoS effect of the radio fluctuations has only +a small effect on the global signal compared to the radio fluctua- +tion model considered in our previous work (Reis et al. 2020). The +LoS radio background (solid blue line) results in a slightly shallower +minimum (a difference of ∼ 50 mK). While the mean radio back- +ground is unchanged by the LoS effect, the non-linearity of the 21-cm +fluctuations causes a slight change in the mean global signal. +4.2 Quantifying the LoS anisotropy in the 21-cm power +spectrum +In the previous subsection, we illustrated the effect of the LoS radio +fluctuations on the 21-cm signal in detail. In this subsection, we ana- +lyze and quantify the anisotropy present in the 21-cm power spectrum +using the anisotropy ratio of eq. 18. This anisotropy includes a) the +normal LoS anisotropy due to the radial component of the peculiar +velocity gradient (Barkana & Loeb 2005), and b) the anisotropy due +to the LoS radio fluctuations. +We show in Fig. 7 the angular dependence of the 21-cm power +spectrum as a function of redshift for two wavenumbers: 𝑘 = +0.1 Mpc−1 (top panel) and 𝑘 = 1.0 Mpc−1 (bottom panel). The dot- +ted black line represents the case with no excess radio background +(CMB only case), in wh ich the anisotropy showing is only due to the +LoS peculiar velocity gradient. In the top panel of Fig. 7, we see two +clear peaks (in the dotted black line) at 𝑧 ∼ 20 − 21 (when the Ly-𝛼 +fluctuations dominate the 21-cm signal) and 𝑧 ∼ 12 − 13 (during +the heating transition). The colored lines show the cases when we +include the excess radio background from previous work (solid teal +line), the radio background with the LoS effect from this work (solid +blue line), and the comparison case of a uniform excess radio back- +ground (dashed blue line). The case of a uniform radio background +shows a single peak of 𝑟𝜇 at 𝑧 ∼ 15−16, while the case of the fluctu- +ating radio background from the previous work also shows a single +peak in 𝑟𝜇, at a slightly lower redshift (𝑧 ∼ 14). In the bottom panel +of Fig. 7, these three cases again show a single significant positive +peak of 𝑟𝜇, at a redshift between 17 and 22. +In contrast, the LoS effect on the radio fluctuations (solid blue +lines in the two panels of Fig. 7) completely changes the anisotropy. +It makes the anisotropy ratio 𝑟𝜇 negative (even approaching its lowest +possible value of −1) throughout cosmic dawn, until late in the +epoch of reionization. The reason is that each radio source lights +up a pencil beam between us and the source (see the right-most +panel of Fig. 1). This smooths out the fluctuations along the line of +sight, making them small compared to those that are perpendicular +to the LoS. Fig. 8 shows the much smaller value of 𝑓Radio = 30, +corresponding to moderate enhancement of the radio background. In +this case, the anisotropy ratio would be only slightly changed by the +radio background without the LoS effect, but it is still substantially +lowered by the LoS effect; it goes down to negative values at 𝑧 below +18, and the peak at 𝑧 ∼ 12 − 13 is erased for 𝑘 = 0.1 Mpc−1, while a +negative peak is created for 𝑘 = 1 Mpc−1. +These results show that the anisotropy yields a potentially clear ob- +servational signature of the LoS effect of bright early radio sources. +However, we must add a note of caution. These are theoretical, simu- +lated results, while in practice observations face additional obstacles. +Specifically, in any 21-cm measurement, the strong radio synchrotron +foreground (including the emission coming from the Milky Way) has +a rather smooth spectrum, so foreground removal will likely elimi- +nate much of the pencil beam signature seen in Fig. 1. To illustrate +this, we show in Fig. 9 a realistic observational version (correspond- +ing to expectations for the SKA) of Fig. 1; following Reis et al. (2022) +MNRAS 000, 1–10 (2022) + +8 +S. Sikder et al. +10−1 +100 +k [Mpc−1] +101 +102 +103 +104 +105 +∆2 [mK2] +z = 20 +This work +Previous work +Uniform radio +CMB only +10−1 +100 +k [Mpc−1] +102 +103 +104 +∆2 [mK2] +z = 25 +This work +Previous work +Uniform radio +CMB only +Figure 4. The shape of the 21-cm power spectrum at a given redshift during the epoch of cosmic dawn, comparing previous work with our full calculation that +includes the line of sight effect on the excess radio background. The panels show the power spectrum at two different redshifts. Shown here is the case with +𝑓R = 3000 and a hard X-ray SED (𝐸min = 1 keV) as in Fig. 2. +6 +10 +15 +20 +25 +30 +35 +z +10−4 +10−2 +100 +102 +104 +106 +∆2 [mK2] +k = 0.1 Mpc−1 +fRadio = 3000 +fRadio = 300 +fRadio = 30 +fRadio = 5 +CMB only +10−1 +100 +k [Mpc−1] +102 +103 +104 +∆2 [mK2] +z = 25 +fRadio = 3000 +fRadio = 300 +fRadio = 30 +fRadio = 5 +CMB only +Figure 5. Left panel: The 21-cm power spectrum as a function of redshift at 𝑘 = 0.1 Mpc−1 for various values of 𝑓Radio. Right panel: The shape of the 21-cm +power spectrum at 𝑧 = 25 for various values of 𝑓Radio. We show the full model with the LoS effect on the radio background (solid) compared to the fluctuating +radio background (dashed) considered in our previous work (Reis et al. 2020). We also show the case without any excess radio background (i.e., the CMB-only +case, black dotted line). Shown here is the model with a hard X-ray SED (𝐸min = 1 keV). +6 +10 +15 +20 +25 +30 +35 +z +−1200 +−1000 +−800 +−600 +−400 +−200 +0 +T21[mK] +This work +Previous work +CMB only +Figure 6. The effect of the radio fluctuations and of the line-of-sight compo- +nent on the 21-cm global signal, for the astrophysical model of a high radio +production efficiency, 𝑓𝑅 = 3000, and halos with 𝑉𝑐 = 16.5 km/s, 𝑓𝑋 = 1, +𝑓∗ = 0.1, and a hard X-ray SED. +(see that reference for details), we include angular smoothing cor- +responding to the angular resolution, thermal noise, and foreground +avoidance corresponding to the removal of a foreground-dominated +wedge in k space. Comparing the 𝑋𝑌 slices, we see that the LoS ef- +fect on the excess radio background produces a strong enhancement +of the bright regions in the SKA images (corresponding to galaxy +concentrations). Comparing the 𝑍𝑌 slices, we again see the enhance- +ment in the rightmost panel with the LoS effect, but while the angular +smoothing is seen in the 𝑌 direction, the pencil beam features in the +𝑍 direction are not apparent. We leave for future work a quantitative +assessment of methods for detecting the anisotropy that account for +the need for foreground avoidance. +5 SUMMARY +In this paper, we showed that individual radio sources acting as back- +ground 21-cm sources along the line of sight significantly change +the expected 21-cm signal, if early galaxies were particularly bright +in low-frequency radio emission. In particular, we found that the +LoS effect on the radio fluctuations boosts the 21-cm power spec- +trum throughout cosmic dawn, by an additional order of magnitude +beyond just the enhanced radio background and its angle-averaged +fluctuations. The radio fluctuations, enhanced by the LoS effect, wash +out the Ly-𝛼 and heating peaks and produce a single broad peak at +cosmic dawn, when the power spectrum is considered as a function +of redshift; this is true at 𝑘 = 0.1 Mpc−1 as well as 1 Mpc−1, and +whether the SED of the X-ray heating sources is hard or soft. The +LoS radio fluctuations have almost no effect at the end of the epoch +of reionization due to the fact that the radio sources become numer- +ous and the radio fluctuations disappear at lower redshifts. The radio +fluctuations also change the shape of the 21-cm power spectrum, +increasing small-scale fluctuations at the higher redshifts (particular +around 𝑧 = 25), even more when the LoS effect is included. +MNRAS 000, 1–10 (2022) + +LoS effect on the 21-cm signal from radio galaxies +9 +6 +10 +15 +20 +25 +30 +z +−1.0 +−0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +rµ +k = 0.1 Mpc−1 +This work +Previous work +Uniform radio +CMB only +6 +10 +15 +20 +25 +30 +z +−1.0 +−0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +rµ +k = 1.0 Mpc−1 +This work +Previous work +Uniform radio +CMB only +Figure 7. The anisotropy ratio (𝑟𝜇) of the 21-cm power spectrum as a +function of redshift at two wavenumbers: 𝑘 = 0.1 Mpc−1 (top panel) and +𝑘 = 1.0 Mpc−1 (bottom panel) for various simulation runs: no excess ra- +dio or CMB only case (dotted black line), uniform excess radio background +(dashed blue line), fluctuating radio background without the LoS effect (solid +teal line) and full fluctuating radio background including the LoS effect (solid +blue line). The dotted dark red horizontal line indicates 𝑟𝜇 = 0. +We also explored the LoS effect on the radio fluctuations for mod- +erate values of radio enhancement, that are well below the values +that are required to explain the EDGES feature. We found that even +models with 𝑓Radio = 30 can significantly enhance the 21-cm power +spectrum (by a half to one order of magnitude) during most of cosmic +dawn. When modelling and interpreting the 21-cm signal, it is there- +fore important to consider a possible enhanced radio background, +and to include the LoS effect. We note that the LoS radio fluctuations +also slightly affect the global signal due to the non-linearity of the +21-cm fluctuations. +Finally, we quantified the anisotropy in the 21-cm power spectrum +and showed that the LoS effect on the radio background introduces +a new anisotropy in the 21-cm power spectrum. The LoS effect +specifically produces a negative anisotropy ratio almost over a wide +redshift range, at least in the case of a very high (EDGES motivated) +radio production efficiency ( 𝑓𝑅 = 3000). Even a moderate radio +efficiency ( 𝑓Radio = 30) still produces a unique signature in the +anisotropy ratio, with the LoS effect producing a negative anisotropy +ratio for part of cosmic dawn. However, we caution that further +investigation is needed to see whether the anisotropy ratio can be +measured in the case of a realistic power spectrum, such as that +expected from SKA observations. +ACKNOWLEDGEMENTS +SS and RB acknowledge the support of the Israel Science Foundation +(grant No. 2359/20). RB also thanks the Ambrose Monell Founda- +6 +10 +15 +20 +25 +30 +z +−1.0 +−0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +rµ +k = 0.1 Mpc−1 +This work +Previous work +Uniform radio +CMB only +6 +10 +15 +20 +25 +30 +z +−1.0 +−0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +rµ +k = 1.0 Mpc−1 +This work +Previous work +Uniform radio +CMB only +Figure 8. The anisotropy ratio (𝑟𝜇) of the 21-cm power spectrum. Same as +Fig. 7 but for 𝑓Radio = 30. +tion and the Institute for Advanced Study as well as the Vera Rubin +Presidential Chair in Astronomy at UCSC and the Packard Founda- +tion. AF was supported by the Royal Society University Research +Fellowship. This research made use of: Numpy (Harris et al. 2020), +Scipy (Virtanen et al. 2020), matplotlib (Hunter 2007) and the +NASA Astrophysics Data System Bibliographic Services. +DATA AVAILABILITY +The data underlying this article will be shared on reasonable request +to the corresponding author. +REFERENCES +Alcock C., Paczyński B., 1979, Nature, 281, 358 +Ali S. S., Bharadwaj S., Pandey B., 2005, MNRAS, 363, 251 +Allison A. C., Dalgarno A., 1969, ApJ, 158, 423 +Barkana R., 2006, MNRAS, 372, 259 +Barkana R., 2018, Nature, 555, 71 +Barkana R., Loeb A., 2004, ApJ, 609, 474 +Barkana R., Loeb A., 2005, ApJ, 624, L65 +Barkana R., Loeb A., 2006, MNRAS, 372, L43 +Barkana R., Outmezguine N. J., Redigol D., Volansky T., 2018, Phys. Rev. D, +98, 103005 +Barkana R., Fialkov A., Liu H., Outmezguine N. J., 2022, arXiv e-prints, p. +arXiv:2212.08082 +Berlin A., Hooper D., Krnjaic G., McDermott S. D., 2018, Phys. Rev. Lett., +121, 011102 +Bharadwaj S., Ali S. S., 2004, MNRAS, 352, 142 +Biermann P. L., Nath B. B., Caramete L. I., Harms B. C., Stanev T., Becker +Tjus J., 2014, MNRAS, 441, 1147 +Bolgar F., Eames E., Hottier C., Semelin B., 2018, MNRAS, 478, 5564 +MNRAS 000, 1–10 (2022) + +10 +S. Sikder et al. +-100 +0 +100 +X [Mpc] +-192 +-92 +0 +92 +192 +Y [Mpc] +z = 20 +Previous work +-100 +0 +100 +Z [Mpc] +Previous work (LoS) +-100 +0 +100 +X [Mpc] +This work +-100 +0 +100 +Z [Mpc] +This work (LoS) +−800 +−600 +−400 +−200 +0 +200 +T21 [mK] +Figure 9. Mock SKA images of the cosmic dawn 21-cm signal from 𝑧 = 20. The panels show the same simulated slices as in Fig. 1, but with several observational +effects, simulated as expected for the SKA: angular resolution, thermal noise, and foreground avoidance. +Bowman J. D., Rogers A. E. E., Monsalve R. A., Mozdzen T. J., Mahesh N., +2018, Nature, 555, 67 +Brandenberger R., Cyr B., Shi R., 2019, J. Cosmology Astropart. Phys., 2019, +009 +Cohen A., Fialkov A., Barkana R., 2016, MNRAS, 459, L90 +Cohen A., Fialkov A., Barkana R., Lotem M., 2017, MNRAS, 472, 1915 +Condon J. J., 1992, ARA&A, 30, 575 +Condon J. J., Cotton W. D., Broderick J. J., 2002, AJ, 124, 675 +Datta K. K., Mellema G., Mao Y., Iliev I. T., Shapiro P. R., Ahn K., 2012, +MNRAS, 424, 1877 +Dowell J., Taylor G. B., 2018, ApJ, 858, L9 +Ewall-Wice A., Chang T. C., Lazio J., Doré O., Seiffert M., Monsalve R. A., +2018, ApJ, 868, 63 +Ewall-Wice A., Chang T.-C., Lazio T. J. W., 2020, MNRAS, 492, 6086 +Feng C., Holder G., 2018, ApJ, 858, L17 +Fialkov A., Barkana R., 2014, MNRAS, 445, 213 +Fialkov A., Barkana R., 2019, MNRAS, 486, 1763 +Fialkov A., Barkana R., Tseliakhovich D., Hirata C. M., 2012, MNRAS, 424, +1335 +Fialkov A., Barkana R., Visbal E., Tseliakhovich D., Hirata C. M., 2013, +MNRAS, 432, 2909 +Fialkov A., Barkana R., Visbal E., 2014, Nature, 506, 197 +Fialkov A., Barkana R., Cohen A., 2015, Phys. Rev. Lett., 114, 101303 +Fixsen D. J., et al., 2011, ApJ, 734, 5 +Fragos T., et al., 2013, ApJ, 764, 41 +Fraser S., et al., 2018, Physics Letters B, 785, 159 +Furlanetto S. R., Zaldarriaga M., Hernquist L., 2004, ApJ, 613, 16 +Gilfanov M., Grimm H. J., Sunyaev R., 2004, MNRAS, 347, L57 +Greig B., Mesinger A., 2015, Monthly Notices of the Royal Astronomical +Society, 449, 4246 +Grimm H. J., Gilfanov M., Sunyaev R., 2003, MNRAS, 339, 793 +Gürkan G., et al., 2018, MNRAS, 475, 3010 +Haiman Z., Rees M. J., Loeb A., 1997, ApJ, 476, 458 +Harris C. R., et al., 2020, Nature, 585, 357 +Heesen V., Brinks E., Leroy A. K., Heald G., Braun R., Bigiel F., Beck R., +2014, AJ, 147, 103 +Hunter J. D., 2007, Computing in Science & Engineering, 9, 90 +Jana R., Nath B. B., Biermann P. L., 2019, MNRAS, 483, 5329 +Lewis A., Challinor A., Lasenby A., 2000, ApJ, 538, 473 +Liu H., Outmezguine N. J., Redigolo D., Volansky T., 2019, Phys. Rev. D, +100, 123011 +Madau P., Meiksin A., Rees M. J., 1997, ApJ, 475, 429 +Meiksin A., 2021, Research Notes of the American Astronomical Society, 5, +126 +Mesinger A., Furlanetto S., Cen R., 2011, Monthly Notices of the Royal +Astronomical Society, 411, 955 +Mineo S., Gilfanov M., Sunyaev R., 2012, MNRAS, 419, 2095 +Mirocha J., Furlanetto S. R., 2019, MNRAS, 483, 1980 +Muñoz J. B., Loeb A., 2018, arXiv e-prints, p. arXiv:1802.10094 +Nusser A., 2005, MNRAS, 364, 743 +Pacucci F., Mesinger A., Mineo S., Ferrara A., 2014, MNRAS, 443, 678 +Planck Collaboration et al., 2018, arXiv e-prints, p. arXiv:1807.06209 +Pospelov M., Pradler J., Ruderman J. T., Urbano A., 2018, Phys. Rev. Lett., +121, 031103 +Press W. H., Schechter P., 1974, ApJ, 187, 425 +Rees M. J., 1986, MNRAS, 222, 27P +Reis I., Fialkov A., Barkana R., 2020, MNRAS, 499, 5993 +Reis I., Barkana R., Fialkov A., 2022, ApJ, 933, 51 +Seiffert M., et al., 2011, ApJ, 734, 6 +Sheth R. K., Tormen G., 1999, MNRAS, 308, 119 +Singh S., et al., 2021, arXiv e-prints, p. arXiv:2112.06778 +Sobacchi E., Mesinger A., 2013, MNRAS, 432, 3340 +Subrahmanyan R., Cowsik R., 2013, The Astrophysical Journal, 776, 42 +Tseliakhovich D., Hirata C., 2010, Phys. Rev. D, 82, 083520 +Urry C. M., Padovani P., 1995, PASP, 107, 803 +Venumadhav T., Dai L., Kaurov A., Zaldarriaga M., 2018, Phys. Rev. D, 98, +103513 +Virtanen P., et al., 2020, Nature Methods, 17, 261 +Visbal E., Barkana R., Fialkov A., Tseliakhovich D., Hirata C. M., 2012, +Nature, 487, 70 +Zygelman B., 2005, ApJ, 622, 1356 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–10 (2022) + diff --git a/n9E3T4oBgHgl3EQfjAqo/content/tmp_files/load_file.txt b/n9E3T4oBgHgl3EQfjAqo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..27eafbf02166e45223f8957efdf34f80928f29f2 --- /dev/null +++ b/n9E3T4oBgHgl3EQfjAqo/content/tmp_files/load_file.txt @@ -0,0 +1,889 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf,len=888 +page_content='MNRAS 000, 1–10 (2022) Preprint 12 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 Strong 21-cm fluctuations and anisotropy due to the line-of-sight effect of radio galaxies at cosmic dawn Sudipta Sikder1★,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Rennan Barkana1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Anastasia Fialkov4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' and Itamar Reis1 1School of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Tel-Aviv University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Tel-Aviv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 69978,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Israel 2Institute for Advanced Study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 1 Einstein Drive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Princeton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' New Jersey 08540,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Madingley Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' CB3 0HA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' UK 5Kavli Institute for Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Madingley Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Cambridge CB3 0HA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' UK Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' in original form ZZZ ABSTRACT The reported detection of the global 21-cm signal by the EDGES collaboration is significantly stronger than standard astrophysical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' One possible explanation is an early radio excess above the cosmic microwave background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Such a radio background could have been produced by high redshift galaxies, if they were especially efficient in producing low-frequency synchrotron radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' We have previously studied the effects of such an inhomogeneous radio background on the 21-cm signal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' however, we made a simplifying assumption of isotropy of the background seen by each hydrogen cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Here we perform a complete calculation that accounts for the fact that the 21-cm absorption occurs along the line of sight, and is therefore sensitive to radio sources lying behind each absorbing cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' We find that the complete calculation strongly enhances the 21-cm power spectrum during cosmic dawn, by up to two orders of magnitude;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' on the other hand, the effect on the global 21-cm signal is only at the 5% level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In addition to making the high-redshift 21-cm fluctuations potentially more easily observable, the line of sight radio effect induces a new anisotropy in the 21-cm power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' While these effects are particularly large for the case of an extremely-enhanced radio efficiency, they make it more feasible to detect even a moderately-enhanced radio efficiency in early galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' This is especially relevant since the EDGES signal has been contested by the SARAS experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Key words: methods: numerical – methods: statistical – dark ages, reionization, first stars – cosmology: observations – cosmology: theory 1 INTRODUCTION The redshifted 21-cm signal which originates due to the hyperfine splitting of the neutral hydrogen in the intergalactic medium (IGM) is the most promising probe of the early universe, most importantly, the epoch of the first stars and the Epoch of Reionization (EoR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The rest- frame frequency of 1420 MHz is redshifted due to the expansion of the universe and can be detected using ground-based radio telescopes at frequencies below 200 MHz against the background radiation, which is usually assumed to be the cosmic microwave background (CMB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The first claimed detection of the all-sky averaged global 21-cm signal from 𝑧 ∼ 13−17 was the EDGES low band observation in the frequency range of 50 − 100 MHz (Bowman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The signal was centered at 𝑧 ∼ 17 (which corresponds to 𝜈 ∼ 78 MHz) with a strong absorption feature of 𝑇21 = −500+200 −500 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' While disputed at 95% significance by the SARAS experiment (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2021), with further measurements expected to resolve this tension, the ten- tative EDGES signal has inspired various theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Specifically, this anomalously strong trough has two main categories of explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' One category is that an additional cooling mechanism can cool the ★ E-mail: sudiptas@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='il gas faster than only adiabatic cooling due to the cosmic expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' An additional cooling mechanism has been suggested (Barkana 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Berlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Barkana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Muñoz & Loeb 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Barkana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2022) that involves a non-gravitational interaction between the baryons and dark matter particles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', via Rutherford-like scattering) that drives down the temperature of the gas leading to the strong observed absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The other category of explanation is the presence of an excess radio background at high red- shifts, well over the CMB level (Bowman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Feng & Holder 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Ewall-Wice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Fialkov & Barkana 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Mirocha & Furlanetto 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Ewall-Wice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Specifically, Fialkov & Barkana (2019) showed that the EDGES signal could be explained by a homogeneous external radio background with a synchrotron spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' However, this external radio background is not directly re- lated to astrophysical sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Exotic processes such as dark matter annihilation or superconducting cosmic strings (Fraser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Pospelov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Brandenberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2019) could give rise to this kind of homogeneous external radio excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' A more astrophysically- grounded approach is to assume that radio-loud sources such as active galactic nuclei (AGN, Urry & Padovani 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Biermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Bolgar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Ewall-Wice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2018, 2020) or star-forming galaxies (Condon 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Jana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2019) at high redshift could pro- duce an excess radio background, which in this case would be inho- © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='04585v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='CO] 11 Jan 2023 2 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Sikder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' mogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Reis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' (2020) first incorporated the inhomogeneous excess galactic radio background into semi-numerical simulations of the early Universe, and explored the effect on the global 21-cm signal and on the 21-cm power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Interestingly, at low frequencies, ARCADE2 (Fixsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Seiffert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2011) detected an ex- cess radio background over that CMB that was confirmed by LWA1 (Dowell & Taylor 2018) in the frequency range 40 − 80 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' This observed excess radio could be explained by extragalactic sources, but it is unclear what fraction of the observed excess originates from Galactic compared to extragalactic sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Subrahmanyan & Cowsik 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In our previous work (Reis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2020) we made a simplifying ap- proximation and assumed that the effect of the radio background on a given hydrogen cloud can be determined from the isotropically- averaged radio intensity at that position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' However, since 21-cm absorption occurs along the line-of-sight (hereafter LoS), the 21- cm effect effectively involves two different radio intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The isotropically-averaged radio intensity is appropriate for effects such as the physical heating of the gas, while the calculation of radiative transfer along the line of sight depends on the radio intensity coming only from radio sources lying behind the hydrogen cloud, along our line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' This can potentially enhance the 21-cm power spectrum due to the LoS radio fluctuations, especially early in cosmic dawn when the number of contributing radio sources is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' We perform this complete calculation in this work, and also quantify the resulting LoS anisotropy in the 21-cm power spectrum using the anisotropy ratio (following Fialkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' This paper is organized as follows: we briefly describe our semi- numerical simulation in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In section 3, we review the theo- retical framework of the 21-cm signal in the presence of an excess radio background and show how we include the line of sight effect of radio fluctuations in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In section 4, we explore the effect of the line of sight radio fluctuations on the 21-cm signal including the anisotropic power spectrum due to this line of sight effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' We conclude the paper with a summary in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2 BASIC METHOD We use our semi-numerical 21-cm simulation code (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Visbal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Fialkov & Barkana 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Fialkov & Barkana 2019) to calculate the 21-cm signal over a wide range of redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' This simulation code was originally inspired by 21cmFAST (Mesinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2011), but it is entirely an independent implemen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The code simulates the realization of the universe in a 3843 Mpc3 comoving cosmological volume with a resolution of 3 comov- ing Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The simulation is based on the following algorithm: we create a random realization of the large-scale linear density field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', the three dimensional cubes of density fluctuations and the relative velocity between the dark matter and the baryons (Tseliakhovich & Hirata 2010) given the power spectra of initial Gaussian random den- sity fields and velocity fields (calculated using the publicly available code CAMB, Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Given the large scale density fields and the relative velocity, we obtain the population of the collapsed dark matter halos inside each cell of 33 Mpc3 comoving volume, using the modified Press-Schechter model (Press & Schechter 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Sheth & Tormen 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Barkana & Loeb 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The baryon fraction contained in each halo is assumed to be the cosmic mean, except that it is reduced due to the streaming velocity (Tseliakhovich & Hirata 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Fialkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Star formation takes place where the dark matter halos are massive enough to radiatively cool the infalling gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' This sets the minimum mass of star forming halos (given by a min- imum circular velocity 𝑉𝑐), except that feedback can also affect this free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Another important parameter is the star formation efficiency, 𝑓∗, and we also take into account the suppression of star formation due to the above-mentioned relative velocity between dark matter and baryons, Lyman-Werner feedback on molecular-hydrogen cooling halos (Haiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Fialkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2013), and photo- heating feedback (Rees 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Sobacchi & Mesinger 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Once we have a population of galaxies, we calculate the radiation fields emitted by those galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The most relevant radiation fields that affect the 21-cm signal are ionizing, Ly-𝛼 and X-ray radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' To calculate the intensity of the Ly-𝛼 radiation field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 𝐽𝛼, we assume that galaxies contain population II stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The X-ray luminosity (𝐿X) of the galaxies is assumed to scale with the star formation rate (SFR), which is based on X-ray observations of low-redshift galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Grimm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Gilfanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Mineo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Fragos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Fialkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Pacucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2014): 𝐿X SFR = 3 × 1040 𝑓X erg s−1M−1 ⊙ yr , (1) where the standard normalization factor 𝑓X is the X-ray efficiency of the sources, a free parameter in our simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Here 𝑓X = 1 corresponds to the typical observed value for low metallicity galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In addition to the luminosity, the shape of the X-ray spectral energy density (SED) affects the 21-cm signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' We assume that the shape of the X-ray SED is determined by a power law slope (𝛼, which we set equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5) and a low energy cutoff (𝐸min).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The hard X-ray SED sourced by a population of high redshift X-ray binaries (XRBs) peaks at energy ∼ 3 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Soft X-ray sources (with typical energy ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5 kev) produce strong fluctuations on small scales (up to a few tens of Mpc), whereas the typical mean free path of hard X-ray photons is so large that the fluctuations are reduced and also they lose some of their energy due to the redshift effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' After the heating transition due to X-ray photons, the universe starts to reionize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The phase transition known as the epoch of reion- ization is expected to have occured inside out, meaning that the high-density regions containing most of the sources proceeded to reionize first (Barkana & Loeb 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Furlanetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' As the 21-cm brightness temperature is proportional to the fraction (𝑥HI) of the neutral hydrogen atoms in the IGM, the amplitude of the global signal decreases as reionization proceeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Another free parameter in our simulation is the ionization efficiency, 𝜁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The late stages of reionization also depend on the maximum mean free path of the ionizing photons, 𝑅mfp (Greig & Mesinger 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Dense regions of neutral hydrogen (specifically, Lyman-limit systems) that appear at high redshifts due to structure formation, effectively absorb the ioniz- ing radiation and set an upper limit on the effective ionization bubble size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Since here we focus on significantly higher redshifts, we simply set 𝑅mfp = 30 Mpc (comoving) and 𝜁 = 30 for all the cases used in this work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' this gives an optical depth to the CMB that is consis- tent with Planck measurements (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Finally, an excess radio background above the cosmic microwave background can be included as we discuss in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 3 21-CM SIGNAL The 21-cm brightness temperature, 𝑇21, depends on the contrast between the spin temperature, 𝑇S, of the neutral hydrogen and the background radiation temperature, 𝑇rad, and can be written as 𝑇21 = 𝑇S − 𝑇rad 1 + 𝑧 (1 − 𝑒−𝜏21) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' (2) MNRAS 000, 1–10 (2022) LoS effect on the 21-cm signal from radio galaxies 3 Usually the background radiation is assumed to be the CMB (at redshift 𝑧), in which case 𝑇rad = 𝑇CMB = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='725(1+𝑧) K, unless there is an excess radio background above the CMB (discussed below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Here 𝜏21 is the optical depth of the 21-cm signal and is given by 𝜏21 = 3ℎpl𝐴10𝑐𝜆2 21𝑛H 32𝜋𝑘𝐵𝑇S(1 + 𝑧)𝑑𝑣/𝑑𝑟 , (3) where ℎpl is the Planck constant, 𝐴10 is the spontaneous decay rate of the hyperfine transition of the neutral hydrogen, 𝑐 is the speed of light, 𝜆21 = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1 cm is the rest frame wavelength of the 21-cm line, 𝑛𝐻 is the number density of the neutral hydrogen, 𝑘𝐵 is the Boltzmann constant, 𝑑𝑣/𝑑𝑟 = 𝐻(𝑧)/(1+ 𝑧) is the gradient of the line of sight component of the comoving velocity field and 𝐻(𝑧) is the Hubble constant (Our code also includes fluctuations with respect to this expression for the mean velocity gradient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The spin temperature 𝑇𝑆 can be written as (Madau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 1997) 𝑇𝑆 = 𝑥rad + 𝑥tot 𝑥rad𝑇−1 rad + 𝑥tot𝑇−1 K , (4) where 𝑥rad = 1 − 𝑒−𝜏21 𝜏21 (5) is the radiative coupling (Venumadhav et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2018), and the coupling coefficient 𝑥tot is the sum of the contributions of Ly-𝛼 coupling (𝑥𝛼) and the collisional coupling (𝑥𝑐), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 𝑥tot = 𝑥𝛼 + 𝑥𝑐, with 𝑥𝛼 = 1 𝐴10𝑇rad 16𝜋2𝑇∗𝑒2 𝑓𝛼 27𝑚𝑒𝑐 𝐽𝛼 , (6) and 𝑥𝑐 = 1 𝐴10𝑇rad 𝜅1−0(𝑇K)𝑛H𝑇★ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' (7) Here 𝑓𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='4162 is the oscillator strength of the Ly-𝛼 transition, 𝐽𝛼 is the intensity of the Ly-𝛼 radiation, 𝑇∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0682K and 𝜅1−0(𝑇𝐾 ) is the known atomic coefficient (Allison & Dalgarno 1969;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Zygelman 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' When the optical depth 𝜏21 << 1, the 21-cm brightness tempera- ture is given by 𝑇21 ≈ 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='8 � Ωbℎ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0327 � � Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='307 �−1/2 � 1 + 𝑧 10 �1/2 (1 + 𝛿)𝑥HI 𝑥tot 1 + 𝑥tot � 1 − 𝑇rad 𝑇K � mK , (8) where we have included the effect of the neutral hydrogen fraction 𝑥HI, and of the density contrast 𝛿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' When calculating the kinetic gas temperature we include the usual effects of adiabatic evolution, Compton heating, and X-ray heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The effect of the radio background on the kinetic gas temperature based on the CMB heating mechanism introduced by Venumadhav et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' (2018) is also included here (but see objections by Meiksin 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The heating rate due to the radio background is 𝜖rad = 𝑥HI𝐴10 2𝐻(𝑧) 𝑥rad �𝑇rad 𝑇S − 1 � 𝑇21 𝑇K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' (9) In practice we use eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 3, including the effect of the inhomogeneous density and velocity gradient, and do not assume the linearized form as in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' we note, though, that the linearized expression is in most cases rather accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1 The excess radio background: previous work In the presence of an excess radio background, we can rewrite the background radiation temperature 𝑇rad, as 𝑇rad = 𝑇Radio + 𝑇CMB , (10) where 𝑇Radio is the brightness temperature of the excess radio background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In our previous work we calculated the isotropically- averaged radio intensity at each pixel, and used the resulting 𝑇Radio in all of the above equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' One type of excess radio background that has been considered is a homogeneous external radio model that is not directly related to astrophysical sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' This excess radio background could possibly be generated by exotic processes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', annihilating dark matter or super-conducting cosmic strings (Fraser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Pospelov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Brandenberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' A simple formulation of such a model (Fialkov & Barkana 2019) sets the brightness temperature of the excess radio background at the 21-cm rest frame frequency at redshift 𝑧 as 𝑇Radio = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='725(1 + 𝑧) 𝐴r × � 1420 78(1 + 𝑧) � 𝛽 K , (11) where 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='725 K is the CMB temperature today, the spectral index of the synchrotron spectrum is 𝛽 = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='6 (set to match the slope of the observed extragalactic radio background, so that the exotic excess background is consistent with observational limits), and 𝐴𝑟 measures the amplitude of the radio background (relative to the CMB at the central redshift of the EDGES claimed absorption feature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' An excess radio background over the CMB can also be produced by high redshift galaxies if they emit strongly in the radio (Reis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Based on the empirical relation of Gürkan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' (2018), we can write the galaxy radio luminosity per unit frequency, which is proportional to the star formation rate (SFR), as 𝐿Radio(𝜈, 𝑧) = 𝑓Radio1022 � 𝜈 150 MHz �−𝛼Radio SFR 𝑀⊙ yr−1 , (12) in units of W Hz−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 12, the spectral index in the radio band 𝛼Radio is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='7 as in Mirocha & Furlanetto (2019) and Gürkan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' see also Condon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' (2002) and Heesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Here 𝑓Radio is the normalization of the radio emissivity, where 𝑓Radio = 1 for present-day star-forming galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In our work, we assume for simplicity a uniform value of 𝑓Radio, though we note that there is significant scatter in 𝑓Radio from observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In our previous work (Reis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2020), the brightness temperature of the radio background at redshift 𝑧 at the 21-cm frequency was calculated by summing the contribution from all the galaxies within the past light-cone (following Ewall-Wice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2020): 𝑇Radio(𝜈21, 𝑧) = 𝜆2 21 2𝑘B 𝑐(1 + 𝑧)3 4𝜋 ∫ 𝜖Radio � 𝜈21 1 + 𝑧em 1 + 𝑧 , 𝑧em � (1 + 𝑧em)−1𝐻(𝑧em)−1𝑑𝑧em , (13) where 𝑧em > 𝑧 is the redshift at which a photon was emitted, and 𝜖Radio is the comoving radio emissivity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', the luminosity per unit frequency per unit comoving volume, averaged over radial shells within this spherical integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The radius of each spherical shell is given by the light travel distance between 𝑧em and 𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' This calculation is thus similar to that for finding the Ly-𝛼 and X-ray radiation fields in our semi-numerical simulation, except that for the Ly-𝛼 radiation field, modified window functions are used in order to include the effect of multiple scattering of the Ly-𝛼 photons (Reis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' MNRAS 000, 1–10 (2022) 4 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Sikder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='2 LoS effect of the radio background from galaxies As outlined in the previous subsection, in our previous work we calculated the isotropically-averaged radio intensity at each pixel, and used the resulting 𝑇Radio in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' This is accurate for all the direct physical effects of the radiation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 6, 7, 4, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' However, it is only approximately true in the radiative transfer equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' As the 21-cm absorption occurs along the line of sight, in this work we accurately consider the line of sight contribution of the excess radio background originating from high redshift galaxies, and examine the effect on the 21-cm signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Our previous approximation becomes accurate in the limit of many radio sources reaching each pixel, since in that case the radio background does become nearly isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' However, at the highest redshifts the radio background at a point is still dominated by a small number of nearby sources, and it then matters whether a source is behind our line of sight to the absorbing pixel, or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Indeed, we expect the LoS effect to amplify the 21-cm fluctuations at early times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Here we will continue to refer to the isotropically-averaged radio intensity at a given pixel as 𝑇Radio, but we also use a different window function and calculate the brightness temperature of the radio back- ground from sources lying behind the pixel along our LoS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' we refer to the latter quantity as 𝑇R,los.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' For clarity, we first find the observed 21-cm brightness temperature relative to the CMB: 𝑇CMB 21 = �𝑇𝑅,los + 𝑇CMB � 𝑒−𝜏21 + 𝑇𝑆 (1 − 𝑒−𝜏21) − 𝑇CMB 1 + 𝑧 , (14) where here 𝑇𝑆 and 𝜏21 depend on 𝑇Radio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Now, if there is indeed a strong excess radio background, it too is observed and is subtracted out in any method of foreground removal (since it is assumed here to have a smooth power-law synchrotron spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Another way of expressing this is that we must subtract out the 𝜏21 = 0 case in order to arrive at the final expression: 𝑇21 = 𝑇𝑆 − �𝑇𝑅,los + 𝑇CMB � 1 + 𝑧 �1 − 𝑒−𝜏21� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' (15) This replaces eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2 (which also was written after subtracting out the 𝜏21 = 0 case under the previous approximation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Since the 21- cm optical depth is usually quite small, we also note the linearized form of this full expression including the line-of-sight effect, which replaces eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 8: 𝑇21 ≈ 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='8 � Ωbℎ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0327 � � Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='307 �−1/2 � 1 + 𝑧 10 �1/2 (1 + 𝛿)𝑥HI 1 1 + 𝑥tot � 𝑥tot � 1 − 𝑇𝑅,los + 𝑇CMB 𝑇K � + � 1 − 𝑇𝑅,los + 𝑇CMB 𝑇Radio + 𝑇CMB �� mK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' (16) As we noted above, the last two equations also contain implicit depen- dencies on 𝑇Radio (the isotropically-averaged radio intensity) through eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 10, which affects eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 6, 7, 4, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The last factor in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 16 shows how the spin temperature varies between𝑇Radio+𝑇CMB (in the uncoupled case) and𝑇K (after saturated coupling), while the observations always probe the contrast between 𝑇𝑆 and 𝑇𝑅,los +𝑇CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The uncoupled case (where the previous result gave no signal) shows explicitly how the line-of-sight effect provides a new source of 21-cm fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Meanwhile, the last factor in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 16 together with eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 6 and 7 shows that the Ly-𝛼 coupling transition (when 𝑥tot ∼ 1) is substantially delayed by an intense radio background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Also, the sign transition (when the mean 21-cm signal goes from absorption to emission) is significantly delayed, as it no longer occurs around the heating transition (normally defined as the average gas temperature𝑇K reaching𝑇CMB), but must wait for the gas to heat to the higher temperature given by the average of𝑇𝑅,los+𝑇CMB (assuming 𝑥tot is large at that time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Even if CMB heating is effective (given the radio-background boost in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 9), the sign transition is still substantially delayed when the excess radio background is strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='3 LoS anisotropies in the 21-cm signal Among the various sources of fluctuations that contribute to the 21-cm fluctuations, many are statistically isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' This is true, for example, for the gas density fluctuations that arise from the initial conditions and are a potential probe of the cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Once the first stars and galaxies form, various forms of radiation in- cluding Ly-𝛼 photons, ionizing photons, and X-ray photons, become additional sources of 21-cm fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' These radiation fields are produced by the processes of star and galaxy formation that are com- plex and non-linear, but have no overall preferred directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Thus, the 21-cm signal due to these radiation fluctuations is isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' How- ever, since the 21-cm signal is redshifted and is determined by LoS absorption, there are a number of effects that make it anisotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' It is important to quantify the LoS anisotropy in the redshifted 21-cm signal for a better understanding of the 21-cm signal itself and also since the anisotropy is potentially directly observable with upcoming telescope arrays such as the Square Kilometre Array (SKA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The coherent inflow of matter into over-dense regions and outflow of matter from under-dense regions, namely the peculiar velocity of the baryonic matter, makes the 21-cm signal anisotropic along the LoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' As a result, the 21-cm power spectrum is expected to be anisotropic due to the radial component of the peculiar velocity gradient (Bharadwaj & Ali 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Barkana & Loeb 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The light- cone effect, whereby only the LoS direction corresponds to a varying redshift, also produces a LoS anisotropy in the 21-cm fluctuations (Barkana & Loeb 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Datta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Due to the uncertainty in the values of the cosmological parameters, another potentially observable source of anisotropy in the 21-cm power spectrum is the Alcock-Paczyński effect (Alcock & Paczyński 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Nusser 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Ali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Barkana 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In our calculations we include the main expected source of 21-cm anisotropy out of these, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', the LoS anisotropy due to peculiar velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Now, since we consider in this work the line of sight dependence of the radio fluctuations, this naturally introduces a new potential sources of anisotropy in the 21-cm signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In order to understand the anisotropy, we first note that in the presence of the anisotropy due to peculiar velocities, the 21-cm power spectrum (in linear theory) can be written as a polynomial 𝑃(𝑘, 𝜇), where 𝜇 is the cosine of the angle between k and the line of sight (Barkana & Loeb 2005): 𝑃(𝑘, 𝜇) = 𝑃𝜇0 (𝑘) + 𝜇2𝑃𝜇2 (𝑘) + 𝜇4𝑃𝜇4 (𝑘) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' (17) Here 𝑃𝜇0 (𝑘) results from the fluctuations from all the isotropic sources, 𝑃𝜇4 (𝑘) is proportional to the primordial density power spectrum, and 𝑃𝜇2 (𝑘) is proportional to the correlation between the density and the radiation from the isotropic sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' This decompo- sition of the 21-cm power spectrum can potentially provide valuable information beyond just the spherically-averaged power spectrum (Barkana & Loeb 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Fialkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' However, in the pres- ence of non-linearities and more generally, Fialkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' (2015) proposed a simpler method to measure the anisotropy in the 21-cm power spectrum by defining the anisotropy ratio: 𝑟𝜇(𝑘, 𝑧) ≡ ⟨𝑃(k, 𝑧)|𝜇𝑘 |>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5⟩ ⟨𝑃(k, 𝑧)|𝜇𝑘 |<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5⟩ − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' (18) Here the angular brackets denote an angular averaging over a range of 𝜇 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The value of 𝑟𝜇(𝑘, 𝑧) captures in one number (at each MNRAS 000, 1–10 (2022) LoS effect on the 21-cm signal from radio galaxies 5 100 0 100 X [Mpc] 192 92 0 92 192 Y [Mpc] z = 20 Previous work 100 0 100 Z [Mpc] Previous work (LoS) 100 0 100 X [Mpc] This work 100 0 100 Z [Mpc] This work (LoS) −8000 −6000 −4000 −2000 T21 [mK] Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Line-of-sight effect of fluctuations in the excess radio background on the cosmic dawn 21-cm signal, illustrated at 𝑧 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' We compare the case of a fluctuating radio background from our previous work (left two panels) to the full calculation including the LoS effect of the fluctuating radio background emitted by galaxies (right two panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' From a single cubic simulation box, we show both the 21-cm slice perpendicular to the LoS (𝑋𝑌 plane) and one that includes the LoS (𝑍) direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Both models have the same radio production efficiency, with astrophysical model parameters: 𝑉𝑐 = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5 km s−1, 𝑓∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1, and 𝑓Radio = 3000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' note that the circular velocity corresponds to the minimum halo mass for star formation being set by atomic cooling, and at 𝑧 = 20 the mass is 3 × 107𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In this example we show the 21-cm slices from 𝑧 = 20, when X-ray heating and reionization are rather insignificant, but for completeness we note that the parameters are hard X-rays (𝐸min = 1 keV with 𝑓X = 1) and reionization parameters as noted in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' We also note that we show the 50’th slice from the simulation box along each axis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', centered at 𝑍 = −43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5 Mpc (for the 𝑋𝑌 plane) and 𝑋 = −43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5 Mpc (for the 𝑍𝑌 plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 𝑘 and 𝑧) the overall angular dependence of the power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' If 𝑟𝜇 is close to zero (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', much smaller than unity in absolute value), the power spectrum shows little angular dependence, when it is large and positive the fluctuations are stronger along the LoS, and when it is large and negative the fluctuations are stronger in directions on the sky (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', perpendicular to the LoS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In this work, we use equation 18 in order to quantitatively explore the anisotropy in the 21-cm power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' A more detailed analysis that quantifies the anisotropy using other methods is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 4 RESULTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1 Implications of the LoS effect of the radio background for the 21-cm signal In order to study the impact of an excess radio background from early galaxies on the 21-cm signal, we compare several simulated cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' We start with the cases considered in Reis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' (2020), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', the CMB- only case (without any radio excess) and the radio excess case in the isotropically-averaged approximation of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2, and compare them to the fully accurate calculation including the LoS effect as in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' We find that the LoS effect on radio fluctuations can significantly affect the 21-cm signal in the redshift range relevant for current and upcoming radio telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' However, this effect varies between astrophysical models, scales and different epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Here we mainly focus on the dependence on the radio efficiency parameter 𝑓Radio from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In Reis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' (2020) we found that models that can explain the EDGES low-band absorption require 𝑓Radio × 𝑓∗ ∼ 140 or higher (depending on the other astrophysical parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' As our main case we consider 𝑓Radio = 3000 (with 𝑓∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1), which lies well within the range compatible with EDGES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' We also, though, consider much lower 𝑓Radio values, that are still well above unity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', moderately enhanced compared to low-redshift galaxies) but do not depend on the veracity of the EDGES measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' We emphasize that there are few observational constraints on the radio efficiency of galaxies at very high redshifts, and given the very different astrophysical conditions at that epoch compared to those at low redshift, it is important to keep an open mind on the possible radio efficiency, until new observational constraints can be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 1 shows a comparison of 21-cm slices at 𝑧 = 20 for 𝑓Radio = 3000, simulated using two different models of radio fluctuations: as in our previous work (Reis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2020, two left-most panels), or as in this work including the line-of-sight effect (two right-most panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In this case in which early galaxies were unusually bright in low- frequency radio emission relative to star formation, the resulting radio background strongly enhances the 21-cm signal map, and brings out the regions surrounding early radio galaxies as strong peaks of 21- cm absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The full inclusion of the LoS effect further brightens these regions by a factor of 2 or 3 in 21-cm brightness temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Especially interesting is the clear induced structure along the line of sight, which is potentially a clear observational signature of the presence of a strong background from high-redshift radio galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' However, these slices represent pure theoretical predictions of the 21- cm signal, and do not include observational effects that are expected to make this structure significantly more difficult to discern, as we explore further below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' At high redshifts during the epoch of the first stars, the 21-cm signal is normally dominated by Ly-𝛼 fluctuations along with some contri- butions from the density and temperature fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' As shown in the previous section, in the presence of a strong radio background, the LoS effect produces 21-cm fluctuations even before significant Ly-𝛼 coupling, and this then mixes in with Ly-𝛼 fluctuations (in an interplay of the two terms in the last factor in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' At this stage we find that the 21-cm signal is enhanced by up to a factor of a few by the radio background;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' once the delayed Ly-𝛼 coupling does occur (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 𝑥tot ∼ 1 is reached) in some region, the enhancement can become much larger (in proportion to the radio intensity), but this is tempered by the stronger CMB-radio heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' We note that the radio and Ly-𝛼 intensity fluctuations are positively correlated (also with the underlying density fluctuations), since both fields originate MNRAS 000, 1–10 (2022) 6 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Sikder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' from the same high redshift galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Thus, these fluctuations en- hance each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' However, the temperature fluctuations (whether from CMB-radio heating or, later on, from X-rays) work mostly in the opposite direction due to the inverse gas temperature dependence of the 21-cm signal (as long as it is an absorption signal, relative to the radio background).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' When we consider the line-of-sight effect of the radio background (which is unique and does not occur in the other relevant radiation backgrounds), it is positively correlated with the radio intensity (since a nearby source along the line of sight con- tributes also to the overall radio intensity), but it adds a strong source of random fluctuations due to the directional dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The statistics of the fluctuations are well captured in the 21-cm power spectrum, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2 for two different values of wavenum- ber, as a function of redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In addition to showing the enhanced radio background from this work (with the LoS effect) compared to our previous work, here we also show two other comparison cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' One is the case without an excess radio background, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', the CMB only case with 𝑓Radio = 0 (dotted black line in both panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The other comparison case is the uniform radio case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', when we take the cos- mic mean excess radio background (which is the same for this work and our previous work) and spread it out evenly over the entire box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' This allows us to separate the effect of the overall enhanced radiative background from the effect of the fluctuations in this background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' We find that the radio fluctuations boost the 21-cm power spectrum at high redshifts, from the beginning of cosmic dawn on small scales (𝑘 = 1 Mpc−1) and from 𝑧 ∼ 25 even on large scales (𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1 Mpc−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' At 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1 Mpc−1, the enhancement during cosmic dawn would be by 2 orders of magnitude even with a spatially uniform excess radio background, but the radio fluctuations add more than an additional order of magnitude, mostly due to the line of sight effect of the radio fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' At 𝑘 = 1 Mpc−1 the enhancement due to radio fluctuations starts earlier, but the behaviour is quite similar at 𝑧 = 10 − 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Regardless of scale, the radio fluctuations eventually die down as the number of sources becomes large, and the power spectrum becomes the same as it would be with a uniform excess radio background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In order to explore some of the dependence on the various unknown astrophysical parameters, we show the case of a soft X-ray SED in the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 3, for 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' As noted above, the heating fluctuations are anti-correlated with the other 21-cm fluctuations, so once the first generation of X-ray sources heat up the IGM, this heat- ing mechanism reduces the 21-cm fluctuations, and then produces a 21-cm fluctuation peak when the heating fluctuations dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' X-ray photons with lower energies are absorbed locally, while hard X-ray photons (≳ 1 keV) lose their energy due to redshifting as they have a much longer mean free path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' As a result, heating is delayed and the resulting fluctuations are smaller for a hard X-ray SED com- pared to a soft X-ray SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Thus, the CMB-only case does show three peaks (Ly-𝛼 coupling, heating, and reionization), but the strong heat- ing peak in the soft X-ray case is barely present in the case of hard X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In the presence of a strong radio background, the coupling tran- sition is delayed due to the inverse dependence of the coupling co- efficients on the radiation background, and the heating transition is delayed as well since the kinetic temperature now needs to reach the higher value 𝑇CMB + 𝑇radio (where 𝑇radio is replaced by 𝑇R,los in the full calculation with the LoS effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' At the same time, the fluctuations in the radio background compete with, and sometimes dominate over, the Ly-𝛼 and heating fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' For hard X-rays, the two normal peaks are washed out and a single overall peak ap- pears, close to the redshift of the CMB-only heating peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' When we include the LoS radio fluctuations, this overall peak gets a significant 6 10 15 20 25 30 35 z 10−4 10−2 100 102 104 106 ∆2 [mK2] k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1 Mpc−1 This work Previous work Uniform radio CMB only 6 10 15 20 25 30 35 z 10−4 10−2 100 102 104 106 ∆2 [mK2] k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 Mpc−1 This work Previous work Uniform radio CMB only Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The 21-cm power spectrum at two wavenumbers, 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1 Mpc−1 and 𝑘 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 Mpc−1, as a function of redshift, for various simulation runs with a fixed excess radio background of 𝑓Radio = 3000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' We compare the fluctuating radio background with the LoS effect (solid blue line) to the fluctuating radio background from our previous work (solid cyan line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' For added comparison, the 21-cm power spectrum due to a uniform excess radio background is shown (dashed blue line), along with the standard case with no excess radio background, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', the CMB only case (dotted black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The uniform radio background case has the same mean intensity of the excess radio at each redshift as in the cases of the fluctuating radio background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Shown here is the model with a hard X-ray SED with 𝐸min = 1 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' boost (and a slight delay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In the case of a soft X-ray SED, without the LoS effect (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', in the previous work case) the strong heating fluctuations dominate and maintain a clear heating peak in the 21- cm power spectrum, which is boosted and delayed compared to the case without excess radio (CMB only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' However, when we take into account the LoS radio fluctuations, these fluctuations become strong enough compared to the heating fluctuations to again wash out the heating peak from the power spectrum and produce a single overall peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Besides the X-ray SED parameters, the X-ray radiation efficiency ( 𝑓X) is another free parameter in our simulation, defined in equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 3 we consider a low efficiency case ( 𝑓X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='01, compared to our fiducial 𝑓X = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In this case the heating peak disappears from the 21-cm power spectrum even in the CMB only case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In the presence of a strong radio background, the dominance of the radio fluctuations (in this case with hard X- rays) means that lowering the X-ray efficiency has little effect at high redshifts, but it does boost the low-redshift signal since in this case the 21-cm signal is maintained in absorption down to the reionization epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Since the (isotropically averaged) 21-cm power spectrum is a func- tion of two variables (𝑘 and 𝑧), in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 4 we show the other cut, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', the function of 𝑘 at a given redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' This shows the effect of the LoS radio fluctuations on the shape of the 21-cm power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Here MNRAS 000, 1–10 (2022) LoS effect on the 21-cm signal from radio galaxies 7 6 10 15 20 25 30 35 z 10−4 10−2 100 102 104 106 ∆2 [mK2] Soft X-rays k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1 Mpc−1 This work Previous work Uniform radio CMB only 6 10 15 20 25 30 35 z 10−4 10−2 100 102 104 106 ∆2 [mK2] Low fX k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1 Mpc−1 This work Previous work Uniform radio CMB only Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The 21-cm power spectrum at 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1 Mpc−1 as a function of redshift for various X-ray parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Top panel: a soft X-ray SED (𝐸min = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1 keV) with the fiducial X-ray efficiency 𝑓X = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Bottom panel: a hard X-ray SED (𝐸min = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 keV) with a low 𝑓X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' we use the same astrophysical model parameters as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' At high redshifts (right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 4), when the first stars and galaxies begin to form, the Ly-𝛼 fluctuations are the dominant source of the 21-cm fluctuations in the standard case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The Ly-𝛼 photons typically travel a significant distance (Reis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2022), which washes out small-scale fluctuations, while the strong radio intensity near sources increases small-scale fluctuations, even more when the LoS effect is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' At lower redshift (left panel), the LoS effect has less of an effect on the power spectrum shape, and just gives an overall boost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' At later times, 𝑧 ∼ 10, the LoS radio fluctuations have almost no effect on the shape of the 21-cm power spectrum compared to the previous radio fluctuation work (Reis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2020) due to the disappearance of the excess radio fluctuations at low redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Up to now in this section we have illustrated the consequences of the LoS effect on the radio background using a particularly strong radio background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Next, we examine the effect of the LoS radio fluc- tuations on the 21-cm power spectrum while varying the parameter 𝑓Radio that regulates the strength of the excess radio background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' We show the 21-cm power spectrum as a function of redshift at 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1 Mpc−1 for various values of 𝑓Radio in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The LoS radio fluctua- tions for moderate values of 𝑓Radio can have a significant effect (∼ an order of magnitude) after the onset of star formation till the end of the heating transition, as can be seen in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 5 shows the shape of the 21-cm power spectrum for various values of 𝑓Radio at 𝑧 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Among the values in the plot, at high redshift only 𝑓Radio of at least 300 has a large effect, but later in cosmic dawn even 𝑓Radio = 30 has quite a significant effect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' since we set 𝑓∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1, the latter value corresponds to a value of 𝑓Radio × 𝑓∗ that is lower by two orders of magnitude than the value required to match the EDGES measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The effect of the radio fluctuations on the global 21-cm signal is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The LoS effect of the radio fluctuations has only a small effect on the global signal compared to the radio fluctua- tion model considered in our previous work (Reis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The LoS radio background (solid blue line) results in a slightly shallower minimum (a difference of ∼ 50 mK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' While the mean radio back- ground is unchanged by the LoS effect, the non-linearity of the 21-cm fluctuations causes a slight change in the mean global signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='2 Quantifying the LoS anisotropy in the 21-cm power spectrum In the previous subsection, we illustrated the effect of the LoS radio fluctuations on the 21-cm signal in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In this subsection, we ana- lyze and quantify the anisotropy present in the 21-cm power spectrum using the anisotropy ratio of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' This anisotropy includes a) the normal LoS anisotropy due to the radial component of the peculiar velocity gradient (Barkana & Loeb 2005), and b) the anisotropy due to the LoS radio fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 7 the angular dependence of the 21-cm power spectrum as a function of redshift for two wavenumbers: 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1 Mpc−1 (top panel) and 𝑘 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 Mpc−1 (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The dot- ted black line represents the case with no excess radio background (CMB only case), in wh ich the anisotropy showing is only due to the LoS peculiar velocity gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 7, we see two clear peaks (in the dotted black line) at 𝑧 ∼ 20 − 21 (when the Ly-𝛼 fluctuations dominate the 21-cm signal) and 𝑧 ∼ 12 − 13 (during the heating transition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The colored lines show the cases when we include the excess radio background from previous work (solid teal line), the radio background with the LoS effect from this work (solid blue line), and the comparison case of a uniform excess radio back- ground (dashed blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The case of a uniform radio background shows a single peak of 𝑟𝜇 at 𝑧 ∼ 15−16, while the case of the fluctu- ating radio background from the previous work also shows a single peak in 𝑟𝜇, at a slightly lower redshift (𝑧 ∼ 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 7, these three cases again show a single significant positive peak of 𝑟𝜇, at a redshift between 17 and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In contrast, the LoS effect on the radio fluctuations (solid blue lines in the two panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 7) completely changes the anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' It makes the anisotropy ratio 𝑟𝜇 negative (even approaching its lowest possible value of −1) throughout cosmic dawn, until late in the epoch of reionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The reason is that each radio source lights up a pencil beam between us and the source (see the right-most panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' This smooths out the fluctuations along the line of sight, making them small compared to those that are perpendicular to the LoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 8 shows the much smaller value of 𝑓Radio = 30, corresponding to moderate enhancement of the radio background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In this case, the anisotropy ratio would be only slightly changed by the radio background without the LoS effect, but it is still substantially lowered by the LoS effect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' it goes down to negative values at 𝑧 below 18, and the peak at 𝑧 ∼ 12 − 13 is erased for 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1 Mpc−1, while a negative peak is created for 𝑘 = 1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' These results show that the anisotropy yields a potentially clear ob- servational signature of the LoS effect of bright early radio sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' However, we must add a note of caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' These are theoretical, simu- lated results, while in practice observations face additional obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Specifically, in any 21-cm measurement, the strong radio synchrotron foreground (including the emission coming from the Milky Way) has a rather smooth spectrum, so foreground removal will likely elimi- nate much of the pencil beam signature seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' To illustrate this, we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 9 a realistic observational version (correspond- ing to expectations for the SKA) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' following Reis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' (2022) MNRAS 000, 1–10 (2022) 8 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Sikder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 10−1 100 k [Mpc−1] 101 102 103 104 105 ∆2 [mK2] z = 20 This work Previous work Uniform radio CMB only 10−1 100 k [Mpc−1] 102 103 104 ∆2 [mK2] z = 25 This work Previous work Uniform radio CMB only Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The shape of the 21-cm power spectrum at a given redshift during the epoch of cosmic dawn, comparing previous work with our full calculation that includes the line of sight effect on the excess radio background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The panels show the power spectrum at two different redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Shown here is the case with 𝑓R = 3000 and a hard X-ray SED (𝐸min = 1 keV) as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 6 10 15 20 25 30 35 z 10−4 10−2 100 102 104 106 ∆2 [mK2] k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1 Mpc−1 fRadio = 3000 fRadio = 300 fRadio = 30 fRadio = 5 CMB only 10−1 100 k [Mpc−1] 102 103 104 ∆2 [mK2] z = 25 fRadio = 3000 fRadio = 300 fRadio = 30 fRadio = 5 CMB only Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Left panel: The 21-cm power spectrum as a function of redshift at 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1 Mpc−1 for various values of 𝑓Radio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Right panel: The shape of the 21-cm power spectrum at 𝑧 = 25 for various values of 𝑓Radio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' We show the full model with the LoS effect on the radio background (solid) compared to the fluctuating radio background (dashed) considered in our previous work (Reis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' We also show the case without any excess radio background (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', the CMB-only case, black dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Shown here is the model with a hard X-ray SED (𝐸min = 1 keV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 6 10 15 20 25 30 35 z −1200 −1000 −800 −600 −400 −200 0 T21[mK] This work Previous work CMB only Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The effect of the radio fluctuations and of the line-of-sight compo- nent on the 21-cm global signal, for the astrophysical model of a high radio production efficiency, 𝑓𝑅 = 3000, and halos with 𝑉𝑐 = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5 km/s, 𝑓𝑋 = 1, 𝑓∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1, and a hard X-ray SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' (see that reference for details), we include angular smoothing cor- responding to the angular resolution, thermal noise, and foreground avoidance corresponding to the removal of a foreground-dominated wedge in k space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Comparing the 𝑋𝑌 slices, we see that the LoS ef- fect on the excess radio background produces a strong enhancement of the bright regions in the SKA images (corresponding to galaxy concentrations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Comparing the 𝑍𝑌 slices, we again see the enhance- ment in the rightmost panel with the LoS effect, but while the angular smoothing is seen in the 𝑌 direction, the pencil beam features in the 𝑍 direction are not apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' We leave for future work a quantitative assessment of methods for detecting the anisotropy that account for the need for foreground avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 5 SUMMARY In this paper, we showed that individual radio sources acting as back- ground 21-cm sources along the line of sight significantly change the expected 21-cm signal, if early galaxies were particularly bright in low-frequency radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' In particular, we found that the LoS effect on the radio fluctuations boosts the 21-cm power spec- trum throughout cosmic dawn, by an additional order of magnitude beyond just the enhanced radio background and its angle-averaged fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The radio fluctuations, enhanced by the LoS effect, wash out the Ly-𝛼 and heating peaks and produce a single broad peak at cosmic dawn, when the power spectrum is considered as a function of redshift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' this is true at 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1 Mpc−1 as well as 1 Mpc−1, and whether the SED of the X-ray heating sources is hard or soft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The LoS radio fluctuations have almost no effect at the end of the epoch of reionization due to the fact that the radio sources become numer- ous and the radio fluctuations disappear at lower redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The radio fluctuations also change the shape of the 21-cm power spectrum, increasing small-scale fluctuations at the higher redshifts (particular around 𝑧 = 25), even more when the LoS effect is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' MNRAS 000, 1–10 (2022) LoS effect on the 21-cm signal from radio galaxies 9 6 10 15 20 25 30 z −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 rµ k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1 Mpc−1 This work Previous work Uniform radio CMB only 6 10 15 20 25 30 z −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 −0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 rµ k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 Mpc−1 This work Previous work Uniform radio CMB only Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The anisotropy ratio (𝑟𝜇) of the 21-cm power spectrum as a function of redshift at two wavenumbers: 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1 Mpc−1 (top panel) and 𝑘 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 Mpc−1 (bottom panel) for various simulation runs: no excess ra- dio or CMB only case (dotted black line), uniform excess radio background (dashed blue line), fluctuating radio background without the LoS effect (solid teal line) and full fluctuating radio background including the LoS effect (solid blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The dotted dark red horizontal line indicates 𝑟𝜇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' We also explored the LoS effect on the radio fluctuations for mod- erate values of radio enhancement, that are well below the values that are required to explain the EDGES feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' We found that even models with 𝑓Radio = 30 can significantly enhance the 21-cm power spectrum (by a half to one order of magnitude) during most of cosmic dawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' When modelling and interpreting the 21-cm signal, it is there- fore important to consider a possible enhanced radio background, and to include the LoS effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' We note that the LoS radio fluctuations also slightly affect the global signal due to the non-linearity of the 21-cm fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Finally, we quantified the anisotropy in the 21-cm power spectrum and showed that the LoS effect on the radio background introduces a new anisotropy in the 21-cm power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The LoS effect specifically produces a negative anisotropy ratio almost over a wide redshift range, at least in the case of a very high (EDGES motivated) radio production efficiency ( 𝑓𝑅 = 3000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Even a moderate radio efficiency ( 𝑓Radio = 30) still produces a unique signature in the anisotropy ratio, with the LoS effect producing a negative anisotropy ratio for part of cosmic dawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' However, we caution that further investigation is needed to see whether the anisotropy ratio can be measured in the case of a realistic power spectrum, such as that expected from SKA observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' ACKNOWLEDGEMENTS SS and RB acknowledge the support of the Israel Science Foundation (grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2359/20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' RB also thanks the Ambrose Monell Founda- 6 10 15 20 25 30 z −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5 rµ k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='1 Mpc−1 This work Previous work Uniform radio CMB only 6 10 15 20 25 30 z −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='5 rµ k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='0 Mpc−1 This work Previous work Uniform radio CMB only Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The anisotropy ratio (𝑟𝜇) of the 21-cm power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 7 but for 𝑓Radio = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' tion and the Institute for Advanced Study as well as the Vera Rubin Presidential Chair in Astronomy at UCSC and the Packard Founda- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' AF was supported by the Royal Society University Research Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' This research made use of: Numpy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2020), Scipy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 2020), matplotlib (Hunter 2007) and the NASA Astrophysics Data System Bibliographic Services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' DATA AVAILABILITY The data underlying this article will be shared on reasonable request to the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' REFERENCES Alcock C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Paczyński B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 1979, Nature, 281, 358 Ali S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Bharadwaj S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Pandey B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2005, MNRAS, 363, 251 Allison A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Dalgarno A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 1969, ApJ, 158, 423 Barkana R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2006, MNRAS, 372, 259 Barkana R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2018, Nature, 555, 71 Barkana R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Loeb A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2004, ApJ, 609, 474 Barkana R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Loeb A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2005, ApJ, 624, L65 Barkana R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Loeb A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2006, MNRAS, 372, L43 Barkana R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Outmezguine N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Redigol D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Volansky T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2018, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' D, 98, 103005 Barkana R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Fialkov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Liu H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Outmezguine N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='08082 Berlin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Hooper D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Krnjaic G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', McDermott S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2018, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 121, 011102 Bharadwaj S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Ali S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2004, MNRAS, 352, 142 Biermann P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Nath B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Caramete L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Harms B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Stanev T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Becker Tjus J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2014, MNRAS, 441, 1147 Bolgar F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Eames E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Hottier C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Semelin B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2018, MNRAS, 478, 5564 MNRAS 000, 1–10 (2022) 10 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Sikder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 100 0 100 X [Mpc] 192 92 0 92 192 Y [Mpc] z = 20 Previous work 100 0 100 Z [Mpc] Previous work (LoS) 100 0 100 X [Mpc] This work 100 0 100 Z [Mpc] This work (LoS) −800 −600 −400 −200 0 200 T21 [mK] Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Mock SKA images of the cosmic dawn 21-cm signal from 𝑧 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' The panels show the same simulated slices as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' 1, but with several observational effects, simulated as expected for the SKA: angular resolution, thermal noise, and foreground avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Bowman J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Rogers A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Monsalve R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Mozdzen T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Mahesh N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2018, Nature, 555, 67 Brandenberger R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Cyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Shi R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2019, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Cosmology Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2019, 009 Cohen A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Fialkov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Barkana R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2016, MNRAS, 459, L90 Cohen A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Fialkov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Barkana R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Lotem M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2017, MNRAS, 472, 1915 Condon J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 1992, ARA&A, 30, 575 Condon J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Cotton W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Broderick J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2002, AJ, 124, 675 Datta K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Mellema G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Mao Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Iliev I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Shapiro P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Ahn K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2012, MNRAS, 424, 1877 Dowell J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Taylor G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2018, ApJ, 858, L9 Ewall-Wice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Chang T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Lazio J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Doré O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Seiffert M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Monsalve R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2018, ApJ, 868, 63 Ewall-Wice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Chang T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Lazio T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2020, MNRAS, 492, 6086 Feng C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Holder G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2018, ApJ, 858, L17 Fialkov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Barkana R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2014, MNRAS, 445, 213 Fialkov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Barkana R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2019, MNRAS, 486, 1763 Fialkov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Barkana R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Tseliakhovich D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Hirata C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2012, MNRAS, 424, 1335 Fialkov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Barkana R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Visbal E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Tseliakhovich D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Hirata C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2013, MNRAS, 432, 2909 Fialkov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Barkana R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Visbal E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2014, Nature, 506, 197 Fialkov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Barkana R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Cohen A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2015, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 114, 101303 Fixsen D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2011, ApJ, 734, 5 Fragos T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2013, ApJ, 764, 41 Fraser S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2018, Physics Letters B, 785, 159 Furlanetto S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Zaldarriaga M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Hernquist L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2004, ApJ, 613, 16 Gilfanov M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Grimm H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Sunyaev R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2004, MNRAS, 347, L57 Greig B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Mesinger A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2015, Monthly Notices of the Royal Astronomical Society, 449, 4246 Grimm H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Gilfanov M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Sunyaev R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2003, MNRAS, 339, 793 Gürkan G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2018, MNRAS, 475, 3010 Haiman Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Rees M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Loeb A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 1997, ApJ, 476, 458 Harris C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2020, Nature, 585, 357 Heesen V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Brinks E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Leroy A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Heald G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Braun R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Bigiel F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Beck R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2014, AJ, 147, 103 Hunter J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2007, Computing in Science & Engineering, 9, 90 Jana R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Nath B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Biermann P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2019, MNRAS, 483, 5329 Lewis A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Challinor A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Lasenby A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2000, ApJ, 538, 473 Liu H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Outmezguine N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Redigolo D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Volansky T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2019, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' D, 100, 123011 Madau P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Meiksin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Rees M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 1997, ApJ, 475, 429 Meiksin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2021, Research Notes of the American Astronomical Society, 5, 126 Mesinger A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Furlanetto S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Cen R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2011, Monthly Notices of the Royal Astronomical Society, 411, 955 Mineo S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Gilfanov M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Sunyaev R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2012, MNRAS, 419, 2095 Mirocha J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Furlanetto S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2019, MNRAS, 483, 1980 Muñoz J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Loeb A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2018, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='10094 Nusser A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2005, MNRAS, 364, 743 Pacucci F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Mesinger A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Mineo S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Ferrara A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2014, MNRAS, 443, 678 Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2018, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' arXiv:1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='06209 Pospelov M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Pradler J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Ruderman J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Urbano A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2018, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 121, 031103 Press W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Schechter P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 1974, ApJ, 187, 425 Rees M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 1986, MNRAS, 222, 27P Reis I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Fialkov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Barkana R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2020, MNRAS, 499, 5993 Reis I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Barkana R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Fialkov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2022, ApJ, 933, 51 Seiffert M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2011, ApJ, 734, 6 Sheth R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Tormen G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 1999, MNRAS, 308, 119 Singh S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2021, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content='06778 Sobacchi E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Mesinger A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2013, MNRAS, 432, 3340 Subrahmanyan R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Cowsik R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2013, The Astrophysical Journal, 776, 42 Tseliakhovich D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Hirata C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2010, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' D, 82, 083520 Urry C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Padovani P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 1995, PASP, 107, 803 Venumadhav T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Dai L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Kaurov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Zaldarriaga M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2018, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' D, 98, 103513 Virtanen P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2020, Nature Methods, 17, 261 Visbal E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Barkana R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Fialkov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Tseliakhovich D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', Hirata C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2012, Nature, 487, 70 Zygelman B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=', 2005, ApJ, 622, 1356 This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} +page_content=' MNRAS 000, 1–10 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfjAqo/content/2301.04585v1.pdf'} diff --git a/nNE5T4oBgHgl3EQfjQ_b/content/tmp_files/2301.05655v1.pdf.txt b/nNE5T4oBgHgl3EQfjQ_b/content/tmp_files/2301.05655v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f9926c6bb54901f0211da4c5617d70eb05bd799c --- /dev/null +++ b/nNE5T4oBgHgl3EQfjQ_b/content/tmp_files/2301.05655v1.pdf.txt @@ -0,0 +1,1153 @@ +arXiv:2301.05655v1 [math.ST] 13 Jan 2023 +CONSTRICTION FOR SETS OF PROBABILITIES +MICHELE CAPRIO AND TEDDY SEIDENFELD +Abstract. Given a set of probability measures P representing an agent’s knowledge on the +elements of a sigma-algebra F, we can compute upper and lower bounds for the probability +of any event A ∈ F of interest. A procedure generating a new assessment of beliefs is said +to constrict A if the bounds on the probability of A after the procedure are contained in +those before the procedure. It is well documented that (generalized) Bayes’ updating does +not allow for constriction, for all A ∈ F [32]. In this work, we show that constriction can +take place with and without evidence being observed, and we characterize these possibilities. +1. Introduction +Call ∆(Ω, F) the space of countably additive probability measures on a measurable space +(Ω, F) of interest and let P ⊂ ∆(Ω, F) be a set of probability measures. Then P(A) = +infP ∈P P(A) is called the lower probability of A, and its conjugate P(A) = 1−infP ∈P P(Ac) = +supP ∈P P(A) is called the upper probability of A. They are two of the main building blocks +of the literature known as imprecise probability theory [35]. Like in measure theory, where +if outer and inner measures of a set coincide, then we say that the set has a measure, if +upper and lower probabilities coincide, then they are an ordinary probability measure, and +P is a singleton. The parallel we just drew between imprecise probability theory and measure +theory is not merely heuristic in nature: [35, Section 3.1.5] shows that if F is a sigma-algebra, +then the lower probability P(A) of any set A ∈ F is its inner measure, while P(A) is its +outer measure. +The reasons for studying imprecise probabilities are discussed at length in [2, 35] and +references therein; in this work we focus especially on the motivations expressed in [15, 28]. +There, the authors point out how specifying sets of probabilities – and thus their “boundary +elements”, namely lower and upper probabilities – accounts for the ambiguity faced by the +agent carrying out the analysis. This means that since the agent does not know the true data +generating process governing the experiment of interest, they may want to take advantage +of the flexibility of IP theory and specify a set of probability measures to represent their +ignorance. The set will be “wider”, that is, the difference between P(A) and P(A) will be +larger for all A ∈ F, the higher the uncertainty faced by the agent. +The aim of this paper is to study the constriction phenomenon that takes place after a +given procedure. +Definition 1. Consider an event of interest A ∈ F, a generic set of probability measures +P ⊂ ∆(Ω, F), and denote by P and P the lower and upper probabilities associated with P, +2010 Mathematics Subject Classification. Primary: 62A01; Secondary: 60A10, 60A99. +Key words and phrases. Constriction; dilation; sets of probabilities; evidence; conditioning; forgetting. + +2 +Michele Caprio and Teddy Seidenfeld +respectively. Call �ֹנ a generic procedure that produces a new assessment of beliefs, and denote +by P �ֹנ and P +�ֹנ the lower and upper probabilities resulting from such procedure, respectively.1 +Then, we say that procedure �ֹנ(strictly) constricts A, in symbols �ֹנ↫ A, if P �ֹנ(A) > P(A) +and P +�ֹנ(A) < P(A). We say that �ֹנ weakly constricts A if one of the two inequalities is weak. +Trivially, if �ֹנ strictly constricts A, then �ֹנ weakly constricts A. Constriction is sometimes +referred to as contraction [8, 17, 21]. We prefer constriction – as denoted in [22] – because +contraction is used in the belief revision literature to denote an instance of corrigibility for full +beliefs, which happens when an agent gives up some current evidence by moving to a logically +weaker body of evidence [27]. In addition when we say that �ֹנ produces a new assessment of +beliefs, we mean that procedure �ֹנ outputs (a set of) probabilities that represent the belief +of the agent around the elements of F. This should not be confused with AGM theory [21] +where procedures generate a new set of full beliefs (sets of sentences). We keep the same +terminology as, given the context, no confusion arises. +Our interest for constriction stems from surprising results involving the opposite phe- +nomenon, called dilation, which was first observed in the context of (generalized) Bayes’ +updating of P [32]. We remark that P need not be closed or convex. Pick any A ∈ F, +and call P(A) := {P(A) : P ∈ P}. Let X : Ω → R be a P-measurable random variable, +that is, let it be P-measurable for all P ∈ P. +Call then I a generic index set, and let +X := {X = xi}i∈I be the sample space of measurable events associated with X. Denote by +P(A | xi) := {P(A | X = xi) : P ∈ P} the set of conditional probabilities of event A, given +X = xi. In order to avoid issues with conditional probability given a P-null event, P ∈ P, we +assume that the elements of P are mutually absolutely continuous with respect to random +variable X. Let us denote by E = X−1(x) ⊂ Ω the evidence collected after experiment X. +The (generalized) Bayes’ updating procedure, denoted by �ֹנ=(B, E),2 strictly constricts A +if, for each xi in a set of P-probability 1 (that is, a set of P-probability 1, for all P ∈ P), +there exist P1i, P2i ∈ P such that, for all P ∈ P(A | xi), +P1i(A) < P(A | xi) < P2i(A). +(1) +Notice that by putting P (B,E)(A) ≡ P B(A | xi) = infP ∈P P(A | xi), xi ∈ X, equation +(1) implies that the condition in Definition 1 for strict constriction is satisfied. �ֹנ=(B, E) +weakly constricts A if one of the two inequalities in (1) is weak. Define now, for all P ∈ P, +XA+ +P +:= {x ∈ X : P(A | X = x) > P(A)} and XA− +P +:= {x ∈ X : P(A | X = x) < P(A)}. +The following lemma comes immediately from the law of conditional expectations. +Lemma 2. Pick any A ∈ F. Then, for all P ∈ P, P(XA+ +P ) > 0 if and only if P(XA− +P ) > 0. +We then have two important propositions which give two instances in which collecting +evidence in the form of experiment X makes us less certain around the probability of A. +1We use Hebrew letter �ֹנ to denote the procedure because the Hebrew word for procedure, �לַהֹנ(pronounced +nohal), begins with �ֹנ. In addition, Latin, Greek, and Cyrillic letters p, P, π, п, П – that could be associated +with the word “procedure” – are usually associated with probabilities and partitions, while Greek letter ̟ +can be easily confused with ω, which we will use to denote an element of the state space Ω of interest. +2In �ֹנ=(B, E), letter B denotes generalized Bayes’ updating, and E = X−1(x) is the conditioning set. +Throughout the paper, we refer to generalized Bayes’ updating simply as “contitioning”, while other techniques +are referred to as “updating rules”. + +Constriction for sets of probabilities +3 +Proposition 3. Pick any A ∈ F. If P(A) is closed in the Euclidean topology, then no +experiment X is such that (B, E) weakly constricts A. +Proposition 4. Pick any A ∈ F. +No simple experiment X is such that (B, E) weakly +constricts A. That is, if X is a simple random variable (i.e. if the index set I for the sample +space X is finite with P-probability 1, for all P ∈ P), then (B, E) does not weakly constrict +A. +Let us give an example, borrowed from [32], which illustrates that no matter the outcome +of an experiment, we are going to be less certain about an event A. +Example 5. Suppose we flip a fair coin twice so that the flips may be dependent. Denote +by Hi and Ti outcome “heads” and “tails”, respectively, in tosses i ∈ {1, 2}. Let +P := +� +P : P(H1) = P(H2) = 1 +2, P(H1 ∩ H2) = p +� +p∈[0, 1 +2] +. +Now, suppose we flip the coin; we have P(H2) = 1/2, but +0 = P B(H2 | H1) < P(H2) = 1 +2 = P(H2) < P +B(H2 | H1) = 1 +and +0 = P B(H2 | T1) < P(H2) = 1 +2 = P(H2) < P +B(H2 | T1) = 1. +As we can see, we start with a precise belief about the second toss and, no matter what the +outcome of the first toss is, we end up having a vacuous beliefs about the second toss. +The fact that Bayes’ rule of conditioning – arguably the most popular beliefs updating +procedure – can give rise to dilation is one motivation for exploring updating techniques that +instead admit constriction. That is the focus of our work. +The paper is divided as follows. Section 2 studies procedures that allow constriction to +take place when no new evidence is collected. Theorems 8 and 9 are the main results and +give very general conditions for procedures to give the opportunity for constriction in the +absence of new collected evidence. In section 3, each individual in a group applies a (convex) +personal pooling rule with “precise” inputs from the others in order to form their revised +opinion. The process iterates until the individual opinions merge to a fixed point. Because +the pooling rules are convex, the fixed point is a constriction of the original set of opinions. +Section 4 studies constriction when evidence is collected and non-Bayesian updating rules +are used to revise the agent’s beliefs. For a countably additive probability, given a generic +partition E of Ω, conditioning does not allow constriction for all E ∈ E. So the only way of +obtaining constriction for all E ∈ E is to intentionally forget the experiment associated with +E. But if we are able to make assumptions about the nature of P, we can give conditions +for constriction to take place for all E ∈ E. Section 5 concludes our work, and we prove our +results in Appendix A. +2. Constricting without evidence +In this section, we study procedures that give the opportunity for constriction when no +data are collected. + +4 +Michele Caprio and Teddy Seidenfeld +2.1. Coherent extension of a precise probability. Recall that, for de Finetti, a proba- +bility measure P is coherent if for any finite collection {Ai}n +i=1 of nonempty subsets of a state +space Ω of interest, we have that +sup +ω∈Ω +n +� +i=1 +ci[IAi(ω) − P(Ai)] ≥ 0, +for all c1, . . . , cn ∈ R, where IAi denotes the indicator function for set Ai. +De Finetti’s +Fundamental Theorem of Probability [11, Section 3.10] is the following. +Theorem 6. Call Ω the state space of interest. Given the probabilities P(Ai) of a finite +number of events A1, . . . , An ⊂ Ω, the probability P(An+1) of a further event An+1 +(1) either turns out to be determined if An+1 is linearly dependent on the Ai’s; +(2) or can be assigned, coherently, any value in a closed interval [p′, p′′]. +More precisely, p′ is the greatest lower bound (GLB) sup P(X) of the evaluations from below +of the P(X) given by the random quantities X linearly dependent on the Ai’s for which we +certainly have X ≤ An+1.3 The same can be said for p′′ (replacing sup by inf, maximum by +minimum, A′ +n+1 by A′′ +n+1, and changing the direction of the inequalities, etc. It is the least +upper bound of evaluations from above). +Notice that [p′, p′′] can be an illusory restriction, for example if p′ = 0 and p′′ = 1. The +interpretation to this result is the following. Suppose we express our subjective beliefs around +events A1, . . . , An via a precise probability distribution P. The fact that P is precise is a +crucial tenet of de Finetti’s subjective probability theory. Then, if we want to coherently +extend our beliefs to a new event An+1 of interest, either we can do that “for free” if An+1 is +a linear combination of the other events, or we have an interval [p′, p′′] within which to select +the value to assign to P(An+1). De Finetti himself does not say specifically how to choose +a value within [p′, p′′]. The takeaway seems to be along the lines of “you should be able to +think hard enough to come up with a precise number p ∈ [p′, p′′] to attach to P(An+1)”. +Denote by �ֹנ= deFin the procedure of choosing any value in [p′, p′′] to assign to the +probability of event An+1. Then, the following holds. +Theorem 7. Suppose – in the notation of Theorem 6 – that p′ ̸= p′′. Then, deFin ↫ An+1 +if P deFin(An+1) ∈ (p′, p′′); the constriction is weak if P deFin(An+1) ∈ {p′, p′′}. +Recall that, for Walley, a lower probability measure P is coherent if for any finite collection +{Ai}n +i=0 of nonempty subsets of a state space Ω of interest, we have that +sup +ω∈Ω +n +� +i=1 +[IAi(ω) − P(Ai) − s(IA0(ω) − P(A0))] ≥ 0, +for all s, n ∈ Z+. +In [35, Section 3.1] the author gives the imprecise probabilities (IP) +counterpart of Theorem 6. That is, Walley presents a method to extend coherently lower +and upper probabilities P(Ai), P(Ai) from a finite collection of sets {A1, . . . , An} ⊂ 2Ω to +any other An+1 ⊂ Ω. This result does not give opportunity for constriction, though, because +it does not prescribe to select a precise value in [P(An+1), P(An+1)]. The whole point of the +3This inequality has to be interpreted as X(ω) ≤ IAn+1(ω), for all ω ∈ Ω. + +Constriction for sets of probabilities +5 +IP literature is to be “comfortable” with working with sets of probabilities, and not being +forced to select a precise value inside the set. +Let # denote the cardinality operator, Conv(H) the convex hull of a generic set H, and +ex[K] the extreme points of a generic convex set K. We can generalize Theorem 7 to the +following. +Theorem 8. Suppose a generic procedure �ֹנ generates a set P ⊂ ∆(Ω, F) of probabilities on +(Ω, F) such that #P ≥ 2, and then prescribes a way of selecting one element P ⋆ = P �ֹנ= P +�ֹנ +from Conv(P). We have that +• if P ⋆ ∈ ex[Conv(P)], then there may exist a collection { ˜A} ⊂ F for which �ֹנ weakly +constricts ˜A. In addition, �ֹנ↫ A, for all A ∈ F \ { ˜A}; +• if instead P ⋆ ∈ Conv(P) \ ex[Conv(P)], then �ֹנ↫ A, for all A ∈ F. +We can also give a topological version of Theorem 8; call ∂XH and intXH the boundary +and the interior of a generic set H in X, respectively. +Theorem 9. Endow [0, 1] with the Euclidean topology, and call B([0, 1]) the Borel sigma- +algebra on [0, 1]. Fix a generic A ∈ F, and assume that P(A) := {P(A) : P ∈ P} ⊂ B([0, 1]) +and that #P(A) ≥ 2. Then, +• if P(A) is closed in the Euclidean topology and P ⋆(A) ∈ ∂B([0,1])P(A), then �ֹנ weakly +constricts A; +• if instead P ⋆(A) ∈ intB([0,1])P(A), then �ֹנ↫ A. +Remark 10. Notice that if P has countably many extreme points, then the assumption that +P(A) ⊂ B([0, 1]) is verified. It is also verified in the case that P is convex. +It is immediate to see how Theorem 7 is a special case of Theorems 8 and 9. Another +procedure that fits the requirement of Theorems 8 and 9 is Halmos’ extension [19, Exercise +48.4], [6, Section 4.13]. Consider two generic measurable spaces (X, X ) and (Y, Y). Let +µ ∈ ∆(X, X ) and, for all x ∈ X, νx ∈ ∆(Y, Y). Suppose further that +∀B ∈ Y, +ν•(B) : X → [0, 1] +is X -measurable. Then, +(i) map E �→ νx({y ∈ Y : (x, y) ∈ E}) is X -measurable, for all E ∈ X × Y; +(ii) map +π : X × Y → [0, 1], +E �→ π(E) := +� +X +νx ({y ∈ Y : (x, y) ∈ E})µ(dx) +is a probability measure on X × Y. +Suppose now that there exists a set A ⊂ Y such that its inner and outer measures do not +coincide, that is, for all x ∈ X, ν⋆x(A) ̸= ν⋆ +x(A). Then, consider A′ = X × A. We have that +πA′ ≡ π(A′) = +� +X +νx ({y ∈ Y : (x, y) ∈ A′})µ(dx) , +and πA′ ∈ [ν⋆x(A), ν⋆ +x(A)]. So Halmos’ extension prescribes a way to extend a countable +additive probability measure on Y to another countably additive probability measure on + +6 +Michele Caprio and Teddy Seidenfeld +X × Y that gives a well defined measure to a Y-non-measurable set A. This value belongs +to the interval whose endpoints are the inner and outer measures of A, respectively. It is +immediate to see, then, how Halmos’ extension satisfies the conditions of Theorems 8 and 9. +Remark 11. Before going on, we need to mention a noteworthy difference between �ֹנ= deFin +and extension theorems from measure theory (à la Halmos). The relevant contrast is that +for the Fundamental Theorem (applied to probability), de Finetti uses as his domains linear +spans of, e.g. indicator functions. And for the measure theorists, the extension of probabilities +to a larger ring of sets uses (countable) sums of indicators defined: in the finite case from an +algebra, and in the infinite case from a sigma-algebra. Two additional meaningful differences +are the following +1) for finite structures, de Finetti does not require that probabilities are defined over an +algebra, whereas, the others do; +2) for infinite structures, as de Finetti does not require countable additivity, his inner +and outer approximations are by finite sums of indicators. By contrast, the measure +theorists require countably additive probabilities, and so they use countable sums for +constructing inner and outer measure approximations. +We now briefly present three procedures that fits the requirements of Theorems 8 and 9: +(i) convex pooling [34, Section 2], in which the opinions of k agents (expressed via precise +probabilities Pj, j ∈ {1, . . . , k}) are first pooled in a convex way, thus forming a set +P = +� +P ∈ ∆(Ω, F) : P = +k +� +j=1 +ζjPj, ζj ≥ 0, ∀j ∈ {1, . . . , k}, +k +� +j=1 +ζj = 1 +� +, +from which a unique pooled opinion P ⋆ is selected; +(ii) Jaynes’ MaxEnt [24], in which, given a set of constraints C, the set of probabilities +of interest to the researcher is P = {P ∈ ∆(Ω, F) : P satisfies C}, and P ⋆ is selected +by maximizing the Shannon entropy in P; +(iii) generalized fiducial inference (GFI) [20], in which a set of data-dependent measures on +the parameter space Ω – called generalized fiducial distributions – is defined by care- +fully inverting a deterministic data-generating equation without the use of Bayes’ the- +orem. Mathematically, we can write P = {P ∈ ∆(Ω, F) : P satisfies [20, Eq. (2)]}. +As pointed out in [20, Remark 4], P ⋆ is then selected by choosing the appropriate norm +to endow the sample space. In [20, Section 1], the authors point out how, while GFI +is different philosophically from Dempster-Shafer theory [13] and inferential models +[29], the resulting solutions of these three methods are often mathematically closely +related to one another. +3. Constricting based on convex pooling +As an example of a procedure that allows to collect new evidence, but does not use condi- +tioning to update an agent’s beliefs, we present the famous model in [12]. There, the author +supposes that there are k individuals, each having their own subjective probability Fi for the + +Constriction for sets of probabilities +7 +unknown value of some parameter ω ∈ Ω.4 For agent i, the opinions of all the other k − 1 +agents represent new evidence. Instead of conditioning on those, agent i pools their own +opinion with that of the other agents. DeGroot shows that, repeating this process for all +agent i, the group reaches (asymptotically) an agreement on a common subjective probability +distribution. After updating their opinions, the probability distribution for every member of +the group belongs to the set +P = +� +F = +k +� +j=1 +ζjFj, ζj ∈ [0, 1], ∀j ∈ {1, . . . , k}, and +k +� +j=1 +ζj = 1 +� +. +(2) +In particular, for all i ∈ {1, . . . , k}, we write that after the first (pooling) iteration, the up- +dated probability measure for agent i, denoted by Fi1 is given by Fi1 = �k +j=1 pijFj. This +means that individual i weighs the opinion of all the agents, including themselves, via coef- +ficients pi1, . . . , pik representing the relative importance that agent i assigns to the opinion +of the other members of the group. Because this is true for all agents, we can give a linear +algebra notation to the updating process. Call P the k × k stochastic matrix whose rows are +given by probability vectors (pi1, . . . , pik), i ∈ {1, . . . , k}. Call then F = (F1, . . . , Fk)⊤; we +have that F(1) = PF, where F(1) := (F11, . . . , Fk1)⊤. Of course this holds for all iterations, +so in turn we have that +F(n) = PF(n−1) = PnF, +for all n ∈ N. The members continue to make these revisions indefinitely or until F(n) = +F(n−1), for all n ≥ N, for some N ∈ N, so further revisions would not change the opinions of +the members. The following is the main result of [12]. +Theorem 12. If there exists n ∈ N such that every element in at least one column of Pn is +positive, then a consensus is reached. +That is, if the condition in Theorem 12 is satisfied, then there exists a k × 1 dimensional +vector π = (π1, . . . , πk) (that is unique, as guaranteed by [12, Theorem 3]) whose elements +are non-negative and sum up to 1, and such that πP = π. In turn, this entails that if we +call Π the k × k stochastic matrix whose rows are all the same and equal to π, we have that +F⋆ = ΠF, where F⋆ = (F ⋆ +1 , . . . , F ⋆ +k )⊤ such that F ⋆ = F ⋆ +1 = · · · F ⋆ +k = �k +j=1 πjFj, where F ⋆ is +the common subjective distribution that is reached in the consensus. Notice that F ⋆ belongs +to P in (2). +Call Pn := Conv(F1n, . . . , Fkn), for all n ∈ N. In this example we have that Pn ⊂ Pn−1, for +all n ∈ Z+, where P0 is set P in equation (2). This means that the limit of sequence (Pn) is set +∩n∈Z+Pn. If the condition in Theorem 12 is satisfied, then ∩n∈Z+Pn = {F ⋆}. Given a generic +set A ∈ F, DeGroot procedure �ֹנ= DeGr may only weakly constrict A in general, for example +if F ⋆ belongs to the extrema of P ≡ P0. That is, F ⋆(A) ≥ F(A) and F ⋆(A) < F(A), or +F ⋆(A) > F(A) and F ⋆(A) ≤ F(A). Nevertheless, there may exist r, s ∈ Z+, r < s, such that +F s(A) > F r(A) and F s(A) < F r(A), where F s(A) = infF ∈Ps F(A), F s(A) = supF ∈Ps F(A), +and similarly for Pr. +4Usually the elements of the parameter space Θ are denoted by θ, while the elements of the state space Ω +by ω. Since the focus of DeGroot’s model is the parameter space only, we used – just in this section – the +ω ∈ Ω notation for the parameter space to maintain the notation consistent with other sections. + +8 +Michele Caprio and Teddy Seidenfeld +DeGroot model is one of the possible examples of an agent collecting evidence and then +revising their initial opinion using a rule that is different from conditioning on the gathered +data. We showed that there is at least one such procedure in which constriction can take +place if all the gathered information is used. +Notice that despite its similarities with deGroot’s model, there is no opportunity for con- +striction in Aumann’s famous “agreeing to disagree” model [3]. There, Bayes’ rule is utilized +to reach a consensus, and there is no exchange of information between the agents. Because +of this, Propositions 3 and 4 ensure us that not even weak constriction can take place. +4. Constricting based on non-Bayesian updating +Suppose the results of an experiment induce partition E = {Ej} of the state space of interest +Ω. +Then, if we retain the assumption that probability measures are countably additive, +conditioning on Ej does not allow for constriction, for all Ej ∈ E. So in general we have +that constriction for all Ej ∈ E can take place only if we intentionally forget the whole +experiment that induces partition E. But if we are able to make assumptions on the nature +of lower probability P associated with the set P of probabilities representing the agent’s +beliefs, and if we consider updating procedures that are alternative to Bayes’ conditioning, we +have opportunities for constriction. Finally, we have space for constriction when probability +measures are finitely additive, thanks to their property of non-conglomerability. +4.1. Background. In this section, we give some background concepts that are needed to +better understand the results that follow. We begin by noting that lower and upper prob- +abilities (LP and UP, respectively) of a set P are coherent if P is convex and closed [36]. +They are a particular type of Choquet capacities. +Definition 13. Given a measurable space (Ω, F) with Ω ̸= ∅, we say that a set function +ν : F → [0, 1] is a Choquet capacity if ν(∅) = 0, ν(Ω) = 1, and ν(A) ≤ ν(B) for all A, B ∈ F +such that A ⊂ B. +Denote by core(P) := {P ∈ ∆(Ω, F) : P(A) ≥ P(A), ∀A ∈ F} the core of lower prob- +ability P. It is the set of probability measures on (Ω, F) that setwise dominate P. It is +convex and weak⋆-compact [9]. P completely characterizes the core, that is, it is sufficient to +know P to be able to completely specify core(P). To emphasize this aspect, some authors +say that P is compatible with core(P) [17]. The following are special cases of coherent lower +probabilities. +Definition 14. Suppose a coherent LP P is such that core(P) is also weakly compact. +Then P is a Choquet capacity of order k, or k-monotone capacity, if for every collection +{A, A1, . . . , Ak} ⊂ F such that Ai ⊂ A, for all i ∈ {1, . . . , k}, we have +P(A) ≥ +� +∅̸=I⊂{1,...,k} +(−1)#I−1P(∩i∈IAi). +(3) +Its conjugate UP P is called a k-alternating capacity because it satisfies that for every col- +lection {A, A1, . . . , Ak} ⊂ F such that Ai ⊂ A, for all i ∈ {1, . . . , k}, +P(A) ≤ +� +∅̸=I⊂{1,...,k} +(−1)#I−1P(∪i∈IAi), +(4) + +Constriction for sets of probabilities +9 +A special case of coherent LP that we will use in the remainder of the paper are convex +LP’s, that are Choquet capacities of order 2; they satisfy +P(A ∪ B) ≥ P(A) + P(B) − P(A ∩ B), +for all A, B ∈ F. Another special case of coherent LP that we will use are belief functions. +Definition 15. A coherent LP P is called a belief function if it is a Choquet capacity of +order ∞, i.e., if (3) holds for every k. +Unique to a belief function is its intuitive interpretation as a random set object that realizes +itself as subsets of Ω. +Definition 16. If P is a belief function, its associated mass function is the non-negative set +function +m : F → [0, 1], +A �→ m(A) := +� +B⊂A +(−1)#(A−B)P(B), +(5) +where A − B ≡ A ∩ Bc, and the subsets B of A have to belong to F as well. +Properties of mass function m are the following +(a) m(∅) = 0; +(b) � +B⊂Ω m(B) = 1; +(c) P(A) = � +B⊂A m(B), and is unique to P. +Formula (5) is called the Möbius transform of P [37]. A mass function m induces a precise +probability distribution on F, as the distribution of a random set. These concepts are further +studied in [17]. +To update a set of probabilities P given a set E ∈ F is to replace set function P with +a version of the conditional set function P ×(· | E). The definition of P × is precisely the +job of the updating rule. Recall that we introduced generalized Bayes’ rule of conditioning +�ֹנ=(B, E) in section 1. +We now give the formal definitions of three additional updating rules for coherent lower +and upper probabilities. Generalized Bayes’, geometric, and Dempster’s rules are the ones +that are most commonly used and studied in the literature, while Gärdenfors’ rule is a very +general updating mechanism that subsumes many other methods of belief revision. +The +reasons for why an agent endorses one instead of another are explored in [17, 33]. +Definition 17. Let P ⊂ ∆(Ω, F) be closed and convex. Then, the conditional LP’s and +UP’s according to the geometric rule are set functions P G, P +G such that, for all A, E ∈ F +P (G,E)(A) ≡ P G(A | E) := P(A ∩ E) +P(E) +and +P +(G,E)(A) ≡ P +G(A | E) = P(A ∩ E) +P(E) +, +(6) +provided that P(E) > 0. +So the main difference between generalized Bayes’ and geometric updating procedures is +that the former considers the infimum of the ratio of P(A ∩ E) and P(E), while the latter +considers the ratio of the infima. We introduce next Dempter’s updating rule. + +10 +Michele Caprio and Teddy Seidenfeld +Definition 18. Call E ∈ F the collected evidence. +Assume that P is a belief function +having mass function m and such that P(E) > 0. Let P 0 be a separate belief function +whose associated mass function m0 is such that m0(E) = 1. The conditional belief function +P D(· | E) is defined as +P (D,E)(A) ≡ P D(A | E) := P(A) ⊕ P 0(E), +∀A ∈ F, +where combination operator ⊕ means that the mass function associated with P D(· | E) is +mD(A | E) = +� +C∩E=A m(C) +� +C′∩E̸=∅ m(C′), +∀A ∈ F. +Consequently, Dempster’s updating rule yields the following. If P = core(P), then the LP’s +and UP’s according to Dempster’s updating rule are set functions P D, P +D such that, for all +A, E ∈ F +P (D,E)(A) ≡ P D(A | E) = P(A ∩ E) +P(E) +and +P +(D,E)(A) ≡ P +D(A | E) := P(A ∩ E) +P(E) +, +(7) +provided that P(E) > 0. +If P is a belief function and P = core(P), the geometric rule appears to be a natural +dual to Dempster’s rule. Operationally, though, they differ, as pointed out in [17, Section +2.2]. The main difference is that Dempster’s rule requires P to be a belief function, while +geometric rule does not. Finally, we introduce the following. +Definition 19. Call E ∈ F the collected evidence. Assume that P is a belief function having +mass function m. Consider a function f : F × F → [0, 1] having constraints +(a) � +B∈F f(B, X) = 1, for all X ∈ F, +(b) B ⊂ Ec =⇒ f(B, X) = 0, +(c) f(∅, X) = 0, for all X ∈ F. +Then, the belief function P (I,E)(·) ≡ P I(· | E) obtained according to Gärdenfors’ generalized +imaging updating rule (GGI) is such that its associated mass function is given by +mI(A | E) = +� +X∈F +f(A, X)m(X), +∀A ∈ F. +(8) +Since E is the collected information, we have that E ⊔ Ec = Ω, where ⊔ denotes the +disjoint union. Equation (8) tells us that, upon learning that the evidence collected is not +in Ec, then the probabilities given by m to the (sub)events in Ec are transferred to the +“closest” (sub)events in E according to function f. Constraint (a) is needed to ensure that +the “probability bits” that are transferred are then normalized. Constraint (b) corresponds +to the closed world assumption, which can be expressed as the statement “if the evidence +collected is not in Ec, then it must belong to E”. Smets [33, Section C.6] drops requirement +(b) because he works under the open world assumption, which negates the previous statement +to symbolize that the agent may have specified the state space they work with incorrectly. +So if the evidence collected is not in Ec, it may be in E but also in a superset of E that the +agent did not consider at the beginning of the experiment. Constraint (c) is just a sanity +check. Function f is a version of a conditional probability; it was first introduced in [18] and + +Constriction for sets of probabilities +11 +then generalized by [33]. The choice of function f informs how having collected evidence E +influences our change of beliefs around A. GGI subsumes many other existing updating rules +[33]. +In the remainder of the paper, we write �ֹנ=(×, E), × ∈ {B, G, D, I}, to indicate the gen- +eralized Bayes’, geometric, Dempster’s, and Gärdenfors’ updating rules, respectively, given +collected evidence E. +We write that (×, E) ↫ A if given evidence E, rule × (strictly) +constricts A, and (×, E) ↫ A if rule × (strictly) constricts A for all elements of partition E. +4.2. Intentional forgetting. In this section we show that if we are not willing to assume +that P is at least convex – let alone a belief function – then we cannot obtain constriction +for all the elements E of a partition E representing the results of an experiment of interest. +In this very general case, we need to forget in order to constrict. We retain the assumptions +that P is closed and convex. +Lemma 20. Let E be a measurable and denumerable partition of Ω, and let × ∈ {B, G}. +Then, for any A ∈ F we have that +inf +E∈E P ×(A | E) ≤ P(A) +and +sup +E∈E +P +×(A | E) ≥ P(A). +An immediate consequence of Lemma 20 is the following. +Theorem 21. Let E be a measurable and denumerable partition of Ω and P = core(P). +Then for × ∈ {B, G}, we have that for all A ∈ F, there exists E ∈ E such that �ֹנ=(×, E) +does not weakly constrict A. +This result tells us that for any event A ∈ F of interest, we can always find at least one +element of E that dilates A. This means that there exist E1, E2 ∈ E, E1 possibly different +than E2, such that [P(A), P(A)] ⊊ [P B(A | E1), P +B(A | E1)] and [P(A), P(A)] ⊊ [P G(A | +E2), P +G(A | E2)]. As it appears clear, if the agent is unwilling to make any extra assumption +on the nature of P, then there is no opportunity for constriction to take place for all the +elements of partition E. +In [32, Theorem 2.3] the authors give sufficient conditions for dilation to take place for +all E ∈ E. Then, intentionally forgetting altogether the experiment that dilates A seems +the only viable option to reach constriction. As the name suggests, intentional forgetting +corresponds to an agent willingly forgetting pieces of information, for example because they +are redundant, because they may be harmful, or because they are instructed to do so. If after +collecting evidence E our current beliefs are encapsulated in lower and upper probabilities +P ×(· | E) and P +×(· | E), respectively, then by forgetting we mean reversing the learning +process so that our “updated” lower and upper probabilities becomes what used to be the +lower and upper “priors”, i.e. P(·) and P(·), respectively. +The topic of forgetting is studied in statistics. In [30], for example, forgetting is intended +in the sense of stabilized forgetting; with this we mean the following. Suppose that the agent +is operating in an environment that is susceptible to changes. Then, the agent’s response to +surprising events depends on their beliefs about how likely the environment is to change. If +it is volatile, a single unexpected event triggers forgetting of past beliefs and relearning of a +new contingency. So at time t, the agent collects evidence Et; they use it to infer whether the + +12 +Michele Caprio and Teddy Seidenfeld +environment has changed or not. In the former case, they erase their memory of past events +and reset their prior belief to their initial prior knowledge. In the latter, they can learn a new +posterior belief of the environment structure based on their previous belief. Another example +is given by limited memory procedures. In [4, 5], the authors study bandit problems based +on limited memory: working with restricted memory, data that are too old are forgotten. +Forgetting is studied in machine learning as well. In [7], for instance, the authors come up +with an algorithm that features a forgetting factor which balances the relative importance +of new data and past data and adjust the model to pay more attention to the new data +when the concept drift is detected. In this framework, forgetting is intended as in past data +progressively losing importance as new evidence is collected. Another example in machine +learning where forgetting is crucial is continual learning: as data gets discarded and has a +limited lifetime, the ability to forget what is not important and retain what matters for the +future are the main issues that continual learning targets and focuses on [25]. +Finally, and rather unsurprisingly, psychologists and cognitive scientists have thoroughly +inspected the phenomenon of forgetting. The reference textbook that investigates intentional +forgetting is [16]. The authors examine the effect on memory of instructions to forget in a wide +variety of contexts. They point out how with the enormous number of information available +nowadays, online forgetting of some information is necessary, and how often times replacing +existing information with new information is mandatory (think of a person changing their +phone number). Study on intentional forgetting stemmed from the phenomenon of directed +forgetting: we are able to deal more effectively with large amounts of information by following +instructions to treat some of the information as “to be forgotten” (e.g. evidence presented in +a courtroom that, being inadmissible, is asked to be disregarded). In this way, interference +is reduced and we are able to devote all of our resources to the remaining to-be-remembered +information. It is easy to see how stabilized forgetting is a particular case of intentional +forgetting, and so is the forgetting factor approach used in the machine learning literature. +Consider the problem of an agent that expresses their initial beliefs on (Ω, F) via a set of +probabilities P = core(P). For convenience, we write P ≡ PE0. As data become available, +they update their beliefs using Bayes’ rule of conditioning for every element of P. With this +we mean the following. Suppose we collect evidence in the form of E1 ⊂ F; then we update +the elements of P to obtain +PE1 := +� +PE1 ∈ ∆(Ω, F) : PE1(A) ≡ P(A | E1) = P(E1 | A)P(A) +P(E1) +∝ P(E1 | A)P(A), ∀A ∈ F, P ∈ P +� +, +where P(·) represents the prior and P(E1 | ·) represents the likelihood. More in general, let +the evidence collected up to time t > 0 be encapsulated in collection {Ek}t +k=1 ⊂ F. Then, +the agent’s updated opinion is given by set +PE1···Et := +� +PE1···Et ∈ ∆(Ω, F) : PE1···Et(A) ≡ PE1···Et−1(A | Et) +∝ P(Et | A, E1, . . . , Et−1)PE1···Et−1(A), ∀A ∈ F, PE1···Et−1 ∈ PE1···Et−1 +� +, + +Constriction for sets of probabilities +13 +where PE1···Et−1(·) ≡ P(· | E1 · · · Et−1) represents the “revised” prior (that is, the posterior +computed at time t − 1) and P(Et | ·) represents the likelihood. +Pick any k ∈ N such that 1 ≤ k ≤ t. Assume that PE1···Et−k and PE1···Et−k···Et are both +convex and closed. Fix an event A ∈ F of interest and define the following +P⋆E1···Et−k(A) := {PE1···Et−k ∈ PE1···Et−k : PE1···Et−k(A) = P E1···Et−k(A)}, +P⋆ +E1···Et−k(A) := {PE1···Et−k ∈ PE1···Et−k : PE1···Et−k(A) = P E1···Et−k(A)}. +For a generic P ∈ ∆(Ω, F), and generic A, B ∈ F, define notion of dependence dP and sets +induced by its value by +dP(A, B) := P(A ∩ B) − P(A)P(B), +Σ+(A, B) := {P ∈ ∆(Ω, F) : dP(A, B) > 0}, +Σ−(A, B) := {P ∈ ∆(Ω, F) : dP(A, B) < 0}. +Call now �ֹנ=(IF×, E ) the procedure of intentionally forgetting evidence E (where E can +be an element of F, a whole partition E of state space Ω, or a collection {E} of partitions) +after having updated endorsing rule ×. Then, the following gives sufficient conditions for +intentional forgetting to induce constriction. +Theorem 22. Fix an event A ∈ F of interest and let the agent endorse any rule × ∈ {B, G}. +If +P⋆E1···Et−k(A) ∩ Σ−(A, Et−k+1 ∩ · · · ∩ Et) ̸= ∅ +and +P⋆ +E1···Et−k(A) ∩ Σ+(A, Et−k+1 ∩ · · · ∩ Et) ̸= ∅, +then forgetting Et−k+1∩· · ·∩Et strictly constricts A, in symbols (IF×, Et−k+1∩· · ·∩Et) ↫ A. +If this holds for all elements Es of partition Es, s ∈ {t − k + 1, . . . , t}, we write +(IF×, Et−k+1, . . . , Et) ↫ A, +so we can forget all the experiments that took place after time t − k. +If we let k = t, +then we obtain stabilized forgetting as in [30]. Notice that for stabilized forgetting subscript +E1 · · · Et−k in Theorem 22 is substituted by E0. We also have the following. +Corollary 23. Fix an event A ∈ F of interest and let the agent endorse any rule × ∈ {B, G}. +If +P⋆Ek···Et(A) ∩ Σ−(A, E1 ∩ · · · ∩ Ek−1) ̸= ∅ +and +P⋆ +Ek···Et(A) ∩ Σ+(A, E1 ∩ · · · ∩ Ek−1) ̸= ∅, +then forgetting E1 ∩ · · · ∩ Ek−1 strictly constricts A, in symbols (IF×, E1 ∩ · · · ∩ Ek−1) ↫ A. +In this case, we obtain the machine learning version of forgetting [7]. The agent intention- +ally forgets data collected before time k. More formally, the authors weight the evidence Et +collected at each time t by a coefficient depending on t that goes to 0 the farther time t is +from present time T, that is, it goes to 0 as |t − T| grows to infinity. Evidence that is old + +14 +Michele Caprio and Teddy Seidenfeld +enough gets severely discounted, to the point that for practical purpose we can consider it +as being forgotten, and so the result in Corollary 23 applies. +Notice that intentional forgetting is always a viable way of inducing constriction, as long +as the selected updating rule induces dilation first. In this section we focused on × ∈ {B, G} +because generalized Bayes’ and geometric rules are the most general ones we presented (they +do not require the lower probability of interest to be a belief function). +4.2.1. Levi-inspired forgetting. A particular type of forgetting is the one inspired by Levi’s +work on corrigible infallibility, see e.g. [26]. Suppose that at time t an agent is equipped +with a body of beliefs Kt regarding the events in F, that is, a collection of logical predicates +that describe the beliefs of the agent at time t.5 For the purpose of studying constriction, +we focus on the (closed and convex) set PKt of probability measures representing the agent’s +beliefs at time t induced by Kt.6 Consider a generic event H ⊂ Ω; every element PKt of PKt +has the following two properties: +• if Kt rules out H, that is, if given the beliefs at time t event H is considered impossible +– written Kt �→ ¬H –, then PKt(A | H) is not well defined, for all A ∈ F; +• if instead Kt does not rule out H, then PKt(A | H) is well defined, for all A ∈ F, and +represents the agent’s beliefs around the plausibility of A if H obtains. +Lower probability P Kt completely characterizes PKt. Suppose then that at time t + 1 the +agent collects evidence E. Let Kt+1 be the new body of beliefs; abusing notation, we write +Kt ∪ {E}. Then, suppose that the agent endorses either of generalized Bayes’ or geometric +rules, × ∈ {B, G}; the updated beliefs of the agent are encapsulated in P × +Kt(· | E). +If +(according to ×) E dilates an event A′ of interest, then the agent can forget about E to obtain +constriction (the agent runs “reverse conditioning”). Their infallible beliefs, encapsulated in +Kt+1, are subject to being corrected, and we have Kt+2 = Kt. The body of beliefs contracts: +Kt+1 loses element E and goes back to what used to be at time t. We write (LF×, E) ↫ A′ +to denote that Levi-forgetting E – after having updated endorsing rule × – constricts A′. +If this holds for all elements of a partition E of Ω representing the possible outcomes of an +experiment of interest, we write (LF×, E) ↫ A′. +4.3. Assumptions on the nature of P. If the agent is willing to make some assumptions +on the type of lower probability P that represents their beliefs, then we can have constriction +for all E ∈ E without resorting to intentional forgetting. As the proverb goes, there is no +free lunch. The following is Theorem 5.9 in [17]. +Theorem 24. Let E = {E, Ec} be the partition associated with the outcomes of the experi- +ment of interest. Assume that P is a belief function such that P(E), P(Ec) > 0, and consider +any event A ∈ F. Then, if E dilates A under the Geometric rule, then it must constrict A +under Dempster’s rule. Similarly, if E dilates A under Dempster’s rule, then it must constrict +A under the Geometric rule. +5Not to be confused with belief functions. +6In particular, we assume PKt = core(P Kt), where P Kt(A) = infPKt∈PKt PKt(A), for all A ∈ F. + +Constriction for sets of probabilities +15 +The proof of Theorem 24 only requires that P is convex, but we need the assumption that +P is in fact a belief function otherwise we would not be able to use Dempster’s rule (see +Definition 18). As we can see, Dempster’s and geometric rule contradict each other. +Assuming that P is a belief function allows us to use updating rules that are otherwise +inaccessible. +Theorem 25. Let E be the evidence collected by the agent, and assume that P is a belief +function having mass function m such that P(E) > 0. Consider any event A ∈ F. We have +that (I, E) ↫ A if and only if +� +B⊂A +�� +X∈F +f(B, X)m(X) − m(B) +� +> 0 +and +� +B⊂Ac +�� +X∈F +f(B, X)m(X) − m(B) +� +> 0. +If the conditions in Theorem 25 hold for every elements of partition E, we write (I, E) ↫ A. +The main point of this section is that if we are willing to formulate an assumption on the +nature of lower probability P associated with set P representing our beliefs, then we are +able to find constriction by using updating rules that in general do not allow for constrict +for all E ∈ E, like the geometric rule. We can also use entirely new updating techniques like +Dempster’s rule or GGI that are otherwise inapplicable. +4.4. Dubins-deFinetti conditioning. In this section, we show how if we are willing to +depart from the classical Kolmogorovian paradigm of probabilities, then we have additional +opportunities for constriction. In particular, as we shall see, Bayes’ rule can induce constric- +tion if we allow probabilities to be merely finitely additive. +Suppose that we adopt Dubins-deFinetti conditioning (DdFC) framework; an in-depth +exposition of DdFC can be found in [10, 14, 31]. For the sake of the present work, the two +main differences with respect to the Kolmogorovian framework is that probabilities need not +be countably additive, and that conditioning does not happen on sigma-fields, but rather on +events, members of a partition of the state space. The following statements are true; we will +provide illustrations for the first one, and the second one is shown similarly. +(1) If probability measures are finitely but not countably additive, then constriction can +take place for all the elements of a countable partition; +(2) If probability measures are countably but not uncountably additive, then constriction +can take place for all the elements of an uncountable partition; +Notice also that if probability measures are uncountably additive, then they must be discrete. +Definition 26. We say that probability measure P is conglomerable in partition E when for +every event A such that P(A | E) is defined for all E ∈ E, and for all constants k1, k2, if +k1 ≤ P(A | E) ≤ k2 for all E ∈ E, then k1 ≤ P(A) ≤ k2. +Definition 26 asserts that for each event A, if all the conditional probabilities over a par- +tition E are bounded by two quantities, k1 and k2, then the unconditional probability for +that event is likewise bounded by these two quantities [23]. De Finetti [10] shows the non- +conglomerability of finitely additive probability measures (FAPMs) in denumerable parti- +tions. + +16 +Michele Caprio and Teddy Seidenfeld +Assume that instead of requiring P to be a set of countably additive probabilities, we +allow it to be a set of FAPMs. Then, weak and strict constriction can happen by Bayes +updating P thanks to the non-conglomerability property of FAPMs. The two illustrations +that we present in this section build on the example in [14, page 92], which we state here for +motivating their construction. +Example 27. (Dubins) Let Ω = {A, B} × {N = 1, 2, . . .}. Stipulate that +• P(A) = P(B) = 1/2, +• P(N = n | A) = 2−n, for n ∈ {1, 2, . . .}, a countably additive conditional probability, +• P(N = n | B) = 0, for n ∈ {1, 2, . . .}, a strongly finitely additive conditional proba- +bility.7 +Then, P(N = n) = 2−(n+1) > 0, for n ∈ {1, 2, . . .}, and (marginally) P is merely finitely +additive over the subalgebra generated by the partition EN = {{N = 1}, {N = 2}, . . .}. P +displays non-conglomerability for the event A in the partition EN = {{N = 1}, {N = 2}, . . .} +as P(A) = 1/2 and P(A | N = n) = 1, for n ∈ {1, 2, . . .}. +Illustration 1 (weak constriction). Use Example 27 as follows. +Consider a set P of +probabilities on Ω = {A, B} × {N = 1, 2, . . .} such that P = {Pα, 0 < α ≤ 1}, where +• Pα(A) = α, +• Pα(N = n | A) = 2−n, for n ∈ {1, 2, . . .}, a countably additive conditional probability, +• Pα(N = n | B) = 0, for n ∈ {1, 2, . . .}, a strongly finitely additive conditional +probability. +Note that neither Pα(N = n | A) nor Pα(N = n | B) depend upon α. With respect to P, +we have 0 < Pα(A) ≤ 1. For each 0 < α < 1, we have non-conglomerability of Pα for the +event A in the partition EN as Pα(A | N = n) = 1, for n ∈ {1, 2, . . .}. Observe then that +1 = P1(A) = P1(A | N = n), for n ∈ {1, 2, . . .}. Thus, Bayes-updating using the information +{N = n} from the partition EN weakly-constricts P = {Pα, 0 < α ≤ 1}. +Illustration 2 (strict constriction). Modify Example 27 as follows. Consider a set P of +probabilities on Ω = {A, B} × {N = 1, 2, . . .} such that P = {Pα, 0 < α < 1}, where +• Pα(A) = α, so that with respect to P, we have 0 < Pα(A) < 1, +• Pα(N = n | A) = (1 − α)2−n, for n ∈ {1, 2, . . .}, +• Pα(N = n | B) = α2−n, for n ∈ {1, 2, . . .}. +Note that each of these two conditional probabilities is a merely finitely additive probability +distribution over N that depends on α. In addition, observe that P(N = n) = 2α(1−α)2−n > +0, which (for each α) also is a merely finitely additive probability distribution over N. +By a Bayes’ updating, for each 0 < α < 1 and each n ∈ {1, 2, . . .}, Pα(A | N = n) = 1/2. +That is, for each α ̸= 1/2 with 0 < α < 1, there is non-conglomerability of Pα for the event A +in the partition EN. Whereas, 1/2 = P1/2(A) = P1/2(A | N = n). Thus, Bayes-updating using +the information {N = n} from the partition EN strictly-constricts P = {Pα, 0 < α < 1}. +Remark 28. These two illustrations help to explain why Propositions 3 and 4 are restricted +to countably additive probabilities. +7Recall that a FAPM is strongly finitely additive if it admits countable partitions by null sets [1]. + +Constriction for sets of probabilities +17 +5. Conclusion +In this paper, we showed that, when updating an agent’s opinions, there are at least three +settings for constricting sets of probabilities (representing the beliefs), namely when belief +revision is performed without evidence, with evidence but without conditioning, and with +evidence using different updating rules. We also provided interesting examples of procedures +for every such framework. +This is just the first step towards a deeper study of the constricting phenomenon, that we +will carry over in the next future. In particular, we plan to find more instances in which +constricting is possible, and to find a trait d’union linking these settings. +Acknowledgements +Michele Caprio would like to acknowledge funding from ARO MURI W911NF2010080. +Appendix A. Proofs +Proof of Proposition 3. Let P be a probability function that satisfies P(A) = minP ∈P P(A). +By Lemma 2, if P(XA+ +P ) > 0, then P(XA− +P ) > 0 and (B, E) does not strictly constrict A. That +is, for each P1i ∈ P, P(A) ≤ P1i(A), and we saw in (1) that (B, E) does not strictly constrict +A when P(A) = P1i(A). So if (B, E) weakly constricts A, we have that P(XA+ +P ) = 0, and then +P({x ∈ X : P B(A | X = x) = P(A)}) = 1. Let now P be a probability function that satisfies +P(A) = maxP ∈P P(A). By the same reasoning, P({x ∈ X : P +B(A | X = x) = P(A)}) = 1, +and then (B, E) does not weakly constrict A either. +□ +Proof of Proposition 4. Assume (for a reductio proof) that on a set of X values with P- +measure 1 (i.e. with P-probability 1, for all P ∈ P), for each xi there exist P1i and P2i +in P such that for each P ∈ P(A | xi), either P1i(A) < P(A | xi) ≤ P2i(A) or P1i(A) ≤ +P(A | xi) < P2i(A). Since X is a simple random variable, define P1(A) := mini P1i(A) and +P2(A) := maxi P2i(A). Given Proposition 3, assume P(A) is not a closed set. Without loss +of generality, assume it is open below (the reasoning is parallel if P(A) is open above). So, +P(A) < P1(A). Then, there exists P0 ∈ P with P(A) < P0(A) < P1(A). Since for each i, +P1i(A) < P0(A | xi) or P1i(A) ≤ P0(A | xi), we also have that for each i, P0(A) < P1(A) ≤ +P0(A | xi). But then P0(XA+ +P0 ) = 1, which is a contradiction according to Lemma 2. +□ +Proof of Theorem 7. Immediate from Definition 1 and Theorem 6. +□ +Proof of Theorem 8. We first show that the lower probability (LP) infP ∈Conv(P) P(·) of the +convex hull of P and the LP infP ∈ex[Conv(P)] P(·) of the extrema of the convex hull of P +coincide. To see this, pick any A ∈ F. Since ex[Conv(P)] ⊂ Conv(P), we have that +inf +P ∈Conv(P) P(A) ≤ +inf +P ∈ex[Conv(P)] P(A). +(9) +Then, let ex[Conv(P)] = {P ex +j }K +j=1, K ∈ N ∪ {∞}. For all P ∈ Conv(P) and all A ∈ F, we +have that +P(A) = +K +� +j=1 +αjP ex +j (A) ≥ +K +� +j=1 +αjP ex(A) = P ex(A) := +inf +P ∈ex[Conv(P)] P(A), + +18 +Michele Caprio and Teddy Seidenfeld +where {αj}K +j=1 is a collection of positive reals such that �K +j=1 αj = 1, which implies that +inf +P ∈Conv(P) P(A) ≥ +inf +P ∈ex[Conv(P)] P(A). +(10) +By combining together (9) and (10) we obtained the desired equality. +Now, if P ⋆ ∈ ex[Conv(P)], then there might be a collection { ˜A} ⊂ F for which P ⋆( ˜A) = +P( ˜A) or P ⋆( ˜A) = P( ˜A), so the constriction is weak for the elements of the collection, while +P ⋆(A) > P( ˜A) and P ⋆(A) < P( ˜A), for all A ∈ F \ { ˜A}. +If instead P ⋆ ∈ Conv(P) \ +ex[Conv(P)], then P ⋆(A) ∈ (P(A), P(A)), for all A ∈ F, so we have �ֹנ↫ A, for all A ∈ F. +□ +Proof of Theorem 9. If P(A) is closed in the Euclidean topology and P ⋆(A) ∈ ∂B([0,1])P(A), +then P ⋆(A) = P(A) or P ⋆(A) = P(A), so the constriction is weak. If instead P ⋆(A) ∈ +intB([0,1])P(A), then P ⋆(A) ∈ (P(A), P(A)), so we have �ֹנ↫ A. +□ +Proof of Lemma 20. This proof draws on that of [17, Lemma 5.1]. +Fix any A ∈ F and +suppose for the sake of contradiction that infE∈E P ×(A | E) > P(A). Because P is closed, +there exists P(A) ∈ P such that P(A)(A) = P(A). Notice that subscript (A) reminds us that +this probability measure can vary with the choice of A. Then, we have that +P(A) = P(A)(A) = +� +E∈E +P(A)(A | E)P(A)(E) ≥ +� +E∈E +P ×(A | E)P(A)(E) +≥ +� +E∈E +inf +E∈E P ×(A | E)P(A)(E) = inf +E∈E P ×(A | E) +� +E∈E +P(A)(E) += inf +E∈E P ×(A | E) > P(A), +where the last inequality comes from our assumption. As we can see, this leads to a contra- +diction. The same argument applies for the upper probability of A, that is, if we assume for +the sake of contradiction that supE∈E P +×(A | E) < P(A) and we pick P ′ +(A) ∈ P such that +P ′ +(A)(A) = P(A), P ′ +(A) possibly different from P(A), then +P(A) ≤ +� +E∈E +P +×(A | E)P ′ +(A)(E) < P(A) +� +E∈E +P ′ +(A)(E) = P(A), +reaching again a contradiction. +□ +Proof of Theorem 21. Immediate from Lemma 20. +□ +Proof of Theorem 22. This proof comes from that of [32, Theorem 2.3]. Fix an event A ∈ F of +interest, and let × ∈ {B, G}. Pick any PE1···Et−k ∈ P⋆E1···Et−k(A) ∩ Σ−(A, Et−k+1 ∩ · · · ∩ Et). +Then, we have that PE1···Et−k(A) = P E1···Et−k(A) because PE1···Et−k ∈ P⋆E1···Et−k(A) and +PE1···Et−k(A∩Et−k+1 ∩· · ·∩Et) < PE1···Et−k(A)PE1···Et−k(Et−k+1∩· · ·∩Et) because PE1···Et−k ∈ +Σ−(A, Et−k+1 ∩ · · · ∩ Et). Then, +P E1···Et−k(A) = PE1···Et−k(A) > PE1···Et−k(A ∩ Et−k+1 ∩ · · · ∩ Et) +PE1···Et−k(Et−k+1 ∩ · · · ∩ Et) += PE1···Et−k(A | Et−k+1 ∩ · · · ∩ Et) ≥ P × +E1···Et−k(A | Et−k+1 ∩ · · · ∩ Et). + +Constriction for sets of probabilities +19 +A similar argument gives us that P E1···Et−k(A) < P +× +E1···Et−k(A | Et−k+1 ∩ · · · ∩ Et). +So +Et−k+1 ∩ · · · ∩ Et dilates A regardless of which updating rule × ∈ {B, G} the agent endorses. +In turn, forgetting Et−k+1∩· · ·∩Et constricts A, in symbols (IF×, Et−k+1∩· · ·∩Et) ↫ A. +□ +Proof of Corollary 23. Analogous to the proof of Theorem 22. +□ +Proof of Theorem 25. Fix any A ∈ F. +Recall that by (8), we have that mI(A | E) = +� +X∈F f(A, X)m(X), for all A ∈ F. Also, by property (c) of mass function m associated to +LP P (see Definition 5), we have that P(A) = � +B⊂A m(B). So, +P I(A | E) = +� +B⊂A +mI(B | E) = +� +B⊂A +� +X∈F +f(B, X)m(X). +Only if Suppose (I, E) ↫ A. Then, by definition of constriction, P I(A | E) > P(A) and +P +I(A | E) < P(A). This happens if and only if +� +B⊂A +� +X∈F +f(B, X)m(X) > +� +B⊂A +m(B) ⇐⇒ +� +B⊂A +�� +X∈F +f(B, X)m(X) − m(B) +� +> 0 +and +� +B⊂Ac +� +X∈F +f(B, X)m(X) > +� +B⊂Ac +m(B) ⇐⇒ +� +B⊂Ac +�� +X∈F +f(B, X)m(X) − m(B) +� +> 0. +This latter is true because +P +I(A | E) < P(A) ⇐⇒ 1 − P I(Ac | E) < 1 − P(Ac) ⇐⇒ P I(Ac | E) > P(Ac). +If Assume that +� +B⊂A +�� +X∈F +f(B, X)m(X) − m(B) +� +> 0 +and +� +B⊂Ac +�� +X∈F +f(B, X)m(X) − m(B) +� +> 0. +Then, we have +P I(A | E) = +� +B⊂A +� +X∈F +f(B, X)m(X) > +� +B⊂A +m(B) = P(A) +and +P I(Ac | E) = +� +B⊂Ac +� +X∈F +f(B, X)m(X) > +� +B⊂Ac +m(B) = P(Ac), +which implies P +I(A | E) < P(A). This concludes the proof. +□ +References +[1] Thomas E. Armstrong. Strong singularity, disjointness, and strong finite additivity of finitely additive +measures. Journal of Mathematical Analysis and Applications, 131(2):565–587, 1988. +[2] Thomas Augustin, Frank P. A. Coolen, Gert de Cooman, and Matthias C. M. Troffaes (editors). +Introduction to Imprecise Probabilities. Wiley Series in Probability and Statistics. Chichester, UK : +Wiley, 2014. +[3] Robert J. Aumann. Agreeing to disagree. The Annals of Statistics, 4(6):1236–1239, 1976. + +20 +Michele Caprio and Teddy Seidenfeld +[4] Dorian Baudry, Yoan Russac, and Olivier Cappé. On limited-memory subsampling strategies for bandits. +In Marina Meila and Tong Zhang, editors, ICML 2021: Proceedings of the 38th International Conference +on Machine Learning, pages 727–737. PMLR, 2021. +[5] Donald A. Berry and Bert Fristedt. Bandit problems – Sequential Allocation of Experiments. Monographs +on Statistics and Applied Probability. Dordrecht : Springer, 1985. +[6] Patrick Billingsley. Probability and Measure. New York, NY : Wiley, second edition, 1995. +[7] Weipeng Cao, Zhong Ming, Zhiwu Xu, Jiyong Zhang, and Qiang Wang. Online sequential extreme +learning machine with dynamic forgetting factor. IEEE Access, 7:179746–179757, 2019. +[8] Michele Caprio and Ruobin Gong. Dynamic precise and imprecise probability kinematics. Available on +arXiv:2110.04382, 2022. +[9] Simone Cerreia-Vioglio, Fabio Maccheroni, and Massimo Marinacci. Ergodic theorems for lower proba- +bilities. Proceedings of the American Mathematical Society, 144:3381–3396, 2015. +[10] Bruno de Finetti. Probability, Induction and Statistics. New York : Wiley, 1972. +[11] Bruno de Finetti. Theory of Probability, volume 1. New York : Wiley, 1974. +[12] Morris H. DeGroot. Reaching a consensus. Journal of the American Statistical Association, 69(345):118– +121, 1974. +[13] Arthur P. Dempster. The Dempster–Shafer calculus for statisticians. International Journal of +Approximate Reasoning, 48(2):365–377, 2008. +[14] Lester E. Dubins. Finitely Additive Conditional Probabilities, Conglomerability and Disintegrations. +The Annals of Probability, 3(1):89 – 99, 1975. +[15] Daniel Ellsberg. Risk, ambiguity, and the Savage axioms. The Quarterly Journal of Economics, +75(4):643–669, 1961. +[16] Jonathan M. Golding and Colin M. MacLeod (editors). Intentional forgetting: Interdisciplinary ap- +proaches. Mahwah, NJ : Lawrence Erlbaum Associates, 1998. +[17] Ruobin Gong and Xiao-Li Meng. Judicious judgment meets unsettling updating: dilation, sure loss, and +Simpson’s paradox. Statistical Science, 36(2):169–190, 2021. +[18] Peter Gärdenfors. Knowledge in flux: Modeling the dynamics of epistemic states. Boston, MA : The MIT +press, 1988. +[19] Paul R. Halmos. Measure Theory. Graduate Texts in Mathematics. New York, NY : Springer, 1950. +[20] Jan Hannig, Hari Iyer, Randy C. S. Lai, and Thomas C. M. Lee. Generalized fiducial inference: A review +and new results. Journal of the American Statistical Association, 111(515):1346–1361, 2016. +[21] Sven Ove Hansson. Logic of Belief Revision. In Edward N. Zalta, editor, The Stanford Encyclopedia of +Philosophy. Palo Alto, CA : Metaphysics Research Lab, Stanford University, Spring 2022 edition, 2022. +[22] Timothy Herron, Teddy Seidenfeld, and Larry Wasserman. Divisive conditioning: Further results on +dilation. Philosophy of Science, 64(3):411–444, 1997. +[23] Joseph B. Kadane, Mark J. Schervish, and Teddy Seidenfeld. Statistical Implications of Finitely +Additive Probability, pages 211–232. Cambridge Studies in Probability, Induction and Decision Theory. +Cambridge, UK : Cambridge University Press, 1999. +[24] Hiremagalur K. Kesavan. Jaynes’ maximum entropy principle. In Christodoulos A. Floudas and Panos M. +Pardalos, editors, Encyclopedia of Optimization. Boston, MA : Springer, 2008. +[25] Timothée Lesort, Vincenzo Lomonaco, Andrei Stoian, Davide Maltoni, David Filliat, and Natalia Díaz- +Rodríguez. Continual learning for robotics: Definition, framework, learning strategies, opportunities and +challenges. Information Fusion, 58:52–68, 2020. +[26] Isaac Levi. Why indeterminate probability is rational. Journal of Applied Logic, 7(4):364–376, 2009. +[27] Isaac Levi. How infallible but corrigible full belief is possible. In Horacio Arló-Costa, Vincent F. +Hendricks, and Johan van Benthem, editors, Readings in Formal Epistemology: Sourcebook, pages 247– +267. Cham, CH : Springer International Publishing, 2016. +[28] Massimo Marinacci and Luigi Montrucchio. Introduction to the Mathematics of Ambiguity. Uncertainty +in Economic Theory. New York, NY : Routledge, 2004. +[29] Ryan Martin and Chuanhai Liu. Inferential Models: +Reasoning with Uncertainty, volume 145 of +Monograph on Statistics and Applied Probability. Boca Raton, FL : CRC Press, 2015. + +Constriction for sets of probabilities +21 +[30] Vincent Moens and Alexandre Zénon. Learning and forgetting using reinforced Bayesian change detec- +tion. PLoS Computational Biology, 4(15):1–41, 2019. +[31] Eugenio Regazzini. De Finetti’s coherence and statistical inference. The Annals of Statistics, 15(2):845– +864, 1987. +[32] Teddy Seidenfeld and Larry Wasserman. Dilation for sets of probabilities. The Annals of Statistics, +21(3):1139–1154, 1993. +[33] Philippe Smets. About updating. In Proceedings of the Seventh international conference on Uncertainty +in Artificial Intelligence, pages 378–385, 1991. +[34] Rush T. Stewart and Ignacio Ojea Quintana. Probabilistic opinion pooling with imprecise probabilities. +Journal of Philosophical Logic, 47(1):17–45, 2018. +[35] Peter Walley. Statistical Reasoning with Imprecise Probabilities, volume 42 of Monographs on Statistics +and Applied Probability. London, UK : Chapman and Hall, 1991. +[36] Peter Walley. Towards a unified theory of imprecise probability. International Journal of Approximate +Reasoning, 24(2-3):125–148, 2000. +[37] Roland R. Yager and Liping Liu (editors). Classic works of the Dempster-Shafer theory of belief functions, +volume 39 of Studies in Fuzziness and Soft Computing. New York, NY : Springer, 2008. +PRECISE Center, Department of Computer and Information Science, University of Pennsylvania, +3330 Walnut Street, Philadelphia, PA 19104 +Email address: caprio@seas.upenn.edu +URL: https://precise.seas.upenn.edu/people/doctoral-students +Departments of Philosophy and Statistics, Carnegie Mellon University, 5000 Forbes Ave, +Pittsburgh, PA 15213 +Email address: teddy@stat.cmu.edu +URL: https://www.cmu.edu/dietrich/philosophy/people/faculty/seidenfeld.html + diff --git a/nNE5T4oBgHgl3EQfjQ_b/content/tmp_files/load_file.txt b/nNE5T4oBgHgl3EQfjQ_b/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bc41c65443f7fbafb7c9b972458becbeadbd7119 --- /dev/null +++ b/nNE5T4oBgHgl3EQfjQ_b/content/tmp_files/load_file.txt @@ -0,0 +1,816 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf,len=815 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='05655v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='ST] 13 Jan 2023 CONSTRICTION FOR SETS OF PROBABILITIES MICHELE CAPRIO AND TEDDY SEIDENFELD Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Given a set of probability measures P representing an agent’s knowledge on the elements of a sigma-algebra F, we can compute upper and lower bounds for the probability of any event A ∈ F of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' A procedure generating a new assessment of beliefs is said to constrict A if the bounds on the probability of A after the procedure are contained in those before the procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' It is well documented that (generalized) Bayes’ updating does not allow for constriction, for all A ∈ F [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In this work, we show that constriction can take place with and without evidence being observed, and we characterize these possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Introduction Call ∆(Ω, F) the space of countably additive probability measures on a measurable space (Ω, F) of interest and let P ⊂ ∆(Ω, F) be a set of probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then P(A) = infP ∈P P(A) is called the lower probability of A, and its conjugate P(A) = 1−infP ∈P P(Ac) = supP ∈P P(A) is called the upper probability of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' They are two of the main building blocks of the literature known as imprecise probability theory [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Like in measure theory, where if outer and inner measures of a set coincide, then we say that the set has a measure, if upper and lower probabilities coincide, then they are an ordinary probability measure, and P is a singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The parallel we just drew between imprecise probability theory and measure theory is not merely heuristic in nature: [35, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='5] shows that if F is a sigma-algebra, then the lower probability P(A) of any set A ∈ F is its inner measure, while P(A) is its outer measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The reasons for studying imprecise probabilities are discussed at length in [2, 35] and references therein;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' in this work we focus especially on the motivations expressed in [15, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' There, the authors point out how specifying sets of probabilities – and thus their “boundary elements”, namely lower and upper probabilities – accounts for the ambiguity faced by the agent carrying out the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' This means that since the agent does not know the true data generating process governing the experiment of interest, they may want to take advantage of the flexibility of IP theory and specify a set of probability measures to represent their ignorance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The set will be “wider”, that is, the difference between P(A) and P(A) will be larger for all A ∈ F, the higher the uncertainty faced by the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The aim of this paper is to study the constriction phenomenon that takes place after a given procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Consider an event of interest A ∈ F, a generic set of probability measures P ⊂ ∆(Ω, F), and denote by P and P the lower and upper probabilities associated with P, 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Primary: 62A01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Secondary: 60A10, 60A99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Constriction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' dilation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' sets of probabilities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' evidence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' conditioning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' 2 Michele Caprio and Teddy Seidenfeld respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Call �ֹנ a generic procedure that produces a new assessment of beliefs, and denote by P �ֹנ and P �ֹנ the lower and upper probabilities resulting from such procedure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='1 Then, we say that procedure �ֹנ(strictly) constricts A, in symbols �ֹנ↫ A, if P �ֹנ(A) > P(A) and P �ֹנ(A) < P(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We say that �ֹנ weakly constricts A if one of the two inequalities is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Trivially, if �ֹנ strictly constricts A, then �ֹנ weakly constricts A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Constriction is sometimes referred to as contraction [8, 17, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We prefer constriction – as denoted in [22] – because contraction is used in the belief revision literature to denote an instance of corrigibility for full beliefs, which happens when an agent gives up some current evidence by moving to a logically weaker body of evidence [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In addition when we say that �ֹנ produces a new assessment of beliefs, we mean that procedure �ֹנ outputs (a set of) probabilities that represent the belief of the agent around the elements of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' This should not be confused with AGM theory [21] where procedures generate a new set of full beliefs (sets of sentences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We keep the same terminology as, given the context, no confusion arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Our interest for constriction stems from surprising results involving the opposite phe- nomenon, called dilation, which was first observed in the context of (generalized) Bayes’ updating of P [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We remark that P need not be closed or convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Pick any A ∈ F, and call P(A) := {P(A) : P ∈ P}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Let X : Ω → R be a P-measurable random variable, that is, let it be P-measurable for all P ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Call then I a generic index set, and let X := {X = xi}i∈I be the sample space of measurable events associated with X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Denote by P(A | xi) := {P(A | X = xi) : P ∈ P} the set of conditional probabilities of event A, given X = xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In order to avoid issues with conditional probability given a P-null event, P ∈ P, we assume that the elements of P are mutually absolutely continuous with respect to random variable X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Let us denote by E = X−1(x) ⊂ Ω the evidence collected after experiment X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The (generalized) Bayes’ updating procedure, denoted by �ֹנ=(B, E),2 strictly constricts A if, for each xi in a set of P-probability 1 (that is, a set of P-probability 1, for all P ∈ P), there exist P1i, P2i ∈ P such that, for all P ∈ P(A | xi), P1i(A) < P(A | xi) < P2i(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' (1) Notice that by putting P (B,E)(A) ≡ P B(A | xi) = infP ∈P P(A | xi), xi ∈ X, equation (1) implies that the condition in Definition 1 for strict constriction is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' �ֹנ=(B, E) weakly constricts A if one of the two inequalities in (1) is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Define now, for all P ∈ P, XA+ P := {x ∈ X : P(A | X = x) > P(A)} and XA− P := {x ∈ X : P(A | X = x) < P(A)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The following lemma comes immediately from the law of conditional expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Pick any A ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, for all P ∈ P, P(XA+ P ) > 0 if and only if P(XA− P ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We then have two important propositions which give two instances in which collecting evidence in the form of experiment X makes us less certain around the probability of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' 1We use Hebrew letter �ֹנ to denote the procedure because the Hebrew word for procedure, �לַהֹנ(pronounced nohal), begins with �ֹנ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In addition, Latin, Greek, and Cyrillic letters p, P, π, п, П – that could be associated with the word “procedure” – are usually associated with probabilities and partitions, while Greek letter ̟ can be easily confused with ω, which we will use to denote an element of the state space Ω of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' 2In �ֹנ=(B, E), letter B denotes generalized Bayes’ updating, and E = X−1(x) is the conditioning set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Throughout the paper, we refer to generalized Bayes’ updating simply as “contitioning”, while other techniques are referred to as “updating rules”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Constriction for sets of probabilities 3 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Pick any A ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' If P(A) is closed in the Euclidean topology, then no experiment X is such that (B, E) weakly constricts A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Pick any A ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' No simple experiment X is such that (B, E) weakly constricts A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' That is, if X is a simple random variable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' if the index set I for the sample space X is finite with P-probability 1, for all P ∈ P), then (B, E) does not weakly constrict A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Let us give an example, borrowed from [32], which illustrates that no matter the outcome of an experiment, we are going to be less certain about an event A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Suppose we flip a fair coin twice so that the flips may be dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Denote by Hi and Ti outcome “heads” and “tails”, respectively, in tosses i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Let P := � P : P(H1) = P(H2) = 1 2, P(H1 ∩ H2) = p � p∈[0, 1 2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Now, suppose we flip the coin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' we have P(H2) = 1/2, but 0 = P B(H2 | H1) < P(H2) = 1 2 = P(H2) < P B(H2 | H1) = 1 and 0 = P B(H2 | T1) < P(H2) = 1 2 = P(H2) < P B(H2 | T1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' As we can see, we start with a precise belief about the second toss and, no matter what the outcome of the first toss is, we end up having a vacuous beliefs about the second toss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The fact that Bayes’ rule of conditioning – arguably the most popular beliefs updating procedure – can give rise to dilation is one motivation for exploring updating techniques that instead admit constriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' That is the focus of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The paper is divided as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Section 2 studies procedures that allow constriction to take place when no new evidence is collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Theorems 8 and 9 are the main results and give very general conditions for procedures to give the opportunity for constriction in the absence of new collected evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In section 3, each individual in a group applies a (convex) personal pooling rule with “precise” inputs from the others in order to form their revised opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The process iterates until the individual opinions merge to a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Because the pooling rules are convex, the fixed point is a constriction of the original set of opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Section 4 studies constriction when evidence is collected and non-Bayesian updating rules are used to revise the agent’s beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' For a countably additive probability, given a generic partition E of Ω, conditioning does not allow constriction for all E ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' So the only way of obtaining constriction for all E ∈ E is to intentionally forget the experiment associated with E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' But if we are able to make assumptions about the nature of P, we can give conditions for constriction to take place for all E ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Section 5 concludes our work, and we prove our results in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Constricting without evidence In this section, we study procedures that give the opportunity for constriction when no data are collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' 4 Michele Caprio and Teddy Seidenfeld 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Coherent extension of a precise probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Recall that, for de Finetti, a proba- bility measure P is coherent if for any finite collection {Ai}n i=1 of nonempty subsets of a state space Ω of interest, we have that sup ω∈Ω n � i=1 ci[IAi(ω) − P(Ai)] ≥ 0, for all c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , cn ∈ R, where IAi denotes the indicator function for set Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' De Finetti’s Fundamental Theorem of Probability [11, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='10] is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Call Ω the state space of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Given the probabilities P(Ai) of a finite number of events A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , An ⊂ Ω, the probability P(An+1) of a further event An+1 (1) either turns out to be determined if An+1 is linearly dependent on the Ai’s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' (2) or can be assigned, coherently, any value in a closed interval [p′, p′′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' More precisely, p′ is the greatest lower bound (GLB) sup P(X) of the evaluations from below of the P(X) given by the random quantities X linearly dependent on the Ai’s for which we certainly have X ≤ An+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='3 The same can be said for p′′ (replacing sup by inf, maximum by minimum, A′ n+1 by A′′ n+1, and changing the direction of the inequalities, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' It is the least upper bound of evaluations from above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Notice that [p′, p′′] can be an illusory restriction, for example if p′ = 0 and p′′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The interpretation to this result is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Suppose we express our subjective beliefs around events A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , An via a precise probability distribution P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The fact that P is precise is a crucial tenet of de Finetti’s subjective probability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, if we want to coherently extend our beliefs to a new event An+1 of interest, either we can do that “for free” if An+1 is a linear combination of the other events, or we have an interval [p′, p′′] within which to select the value to assign to P(An+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' De Finetti himself does not say specifically how to choose a value within [p′, p′′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The takeaway seems to be along the lines of “you should be able to think hard enough to come up with a precise number p ∈ [p′, p′′] to attach to P(An+1)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Denote by �ֹנ= deFin the procedure of choosing any value in [p′, p′′] to assign to the probability of event An+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Suppose – in the notation of Theorem 6 – that p′ ̸= p′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, deFin ↫ An+1 if P deFin(An+1) ∈ (p′, p′′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' the constriction is weak if P deFin(An+1) ∈ {p′, p′′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Recall that, for Walley, a lower probability measure P is coherent if for any finite collection {Ai}n i=0 of nonempty subsets of a state space Ω of interest, we have that sup ω∈Ω n � i=1 [IAi(ω) − P(Ai) − s(IA0(ω) − P(A0))] ≥ 0, for all s, n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In [35, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='1] the author gives the imprecise probabilities (IP) counterpart of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' That is, Walley presents a method to extend coherently lower and upper probabilities P(Ai), P(Ai) from a finite collection of sets {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , An} ⊂ 2Ω to any other An+1 ⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' This result does not give opportunity for constriction, though, because it does not prescribe to select a precise value in [P(An+1), P(An+1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The whole point of the 3This inequality has to be interpreted as X(ω) ≤ IAn+1(ω), for all ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Constriction for sets of probabilities 5 IP literature is to be “comfortable” with working with sets of probabilities, and not being forced to select a precise value inside the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Let # denote the cardinality operator, Conv(H) the convex hull of a generic set H, and ex[K] the extreme points of a generic convex set K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We can generalize Theorem 7 to the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Suppose a generic procedure �ֹנ generates a set P ⊂ ∆(Ω, F) of probabilities on (Ω, F) such that #P ≥ 2, and then prescribes a way of selecting one element P ⋆ = P �ֹנ= P �ֹנ from Conv(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We have that if P ⋆ ∈ ex[Conv(P)], then there may exist a collection { ˜A} ⊂ F for which �ֹנ weakly constricts ˜A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In addition, �ֹנ↫ A, for all A ∈ F \\ { ˜A};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' if instead P ⋆ ∈ Conv(P) \\ ex[Conv(P)], then �ֹנ↫ A, for all A ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We can also give a topological version of Theorem 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' call ∂XH and intXH the boundary and the interior of a generic set H in X, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Endow [0, 1] with the Euclidean topology, and call B([0, 1]) the Borel sigma- algebra on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Fix a generic A ∈ F, and assume that P(A) := {P(A) : P ∈ P} ⊂ B([0, 1]) and that #P(A) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, if P(A) is closed in the Euclidean topology and P ⋆(A) ∈ ∂B([0,1])P(A), then �ֹנ weakly constricts A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' if instead P ⋆(A) ∈ intB([0,1])P(A), then �ֹנ↫ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Notice that if P has countably many extreme points, then the assumption that P(A) ⊂ B([0, 1]) is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' It is also verified in the case that P is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' It is immediate to see how Theorem 7 is a special case of Theorems 8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Another procedure that fits the requirement of Theorems 8 and 9 is Halmos’ extension [19, Exercise 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='4], [6, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Consider two generic measurable spaces (X, X ) and (Y, Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Let µ ∈ ∆(X, X ) and, for all x ∈ X, νx ∈ ∆(Y, Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Suppose further that ∀B ∈ Y, ν•(B) : X → [0, 1] is X -measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, (i) map E �→ νx({y ∈ Y : (x, y) ∈ E}) is X -measurable, for all E ∈ X × Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' (ii) map π : X × Y → [0, 1], E �→ π(E) := � X νx ({y ∈ Y : (x, y) ∈ E})µ(dx) is a probability measure on X × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Suppose now that there exists a set A ⊂ Y such that its inner and outer measures do not coincide, that is, for all x ∈ X, ν⋆x(A) ̸= ν⋆ x(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, consider A′ = X × A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We have that πA′ ≡ π(A′) = � X νx ({y ∈ Y : (x, y) ∈ A′})µ(dx) , and πA′ ∈ [ν⋆x(A), ν⋆ x(A)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' So Halmos’ extension prescribes a way to extend a countable additive probability measure on Y to another countably additive probability measure on 6 Michele Caprio and Teddy Seidenfeld X × Y that gives a well defined measure to a Y-non-measurable set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' This value belongs to the interval whose endpoints are the inner and outer measures of A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' It is immediate to see, then, how Halmos’ extension satisfies the conditions of Theorems 8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Remark 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Before going on, we need to mention a noteworthy difference between �ֹנ= deFin and extension theorems from measure theory (à la Halmos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The relevant contrast is that for the Fundamental Theorem (applied to probability), de Finetti uses as his domains linear spans of, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' indicator functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' And for the measure theorists, the extension of probabilities to a larger ring of sets uses (countable) sums of indicators defined: in the finite case from an algebra, and in the infinite case from a sigma-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Two additional meaningful differences are the following 1) for finite structures, de Finetti does not require that probabilities are defined over an algebra, whereas, the others do;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' 2) for infinite structures, as de Finetti does not require countable additivity, his inner and outer approximations are by finite sums of indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' By contrast, the measure theorists require countably additive probabilities, and so they use countable sums for constructing inner and outer measure approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We now briefly present three procedures that fits the requirements of Theorems 8 and 9: (i) convex pooling [34, Section 2], in which the opinions of k agents (expressed via precise probabilities Pj, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , k}) are first pooled in a convex way, thus forming a set P = � P ∈ ∆(Ω, F) : P = k � j=1 ζjPj, ζj ≥ 0, ∀j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , k}, k � j=1 ζj = 1 � , from which a unique pooled opinion P ⋆ is selected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' (ii) Jaynes’ MaxEnt [24], in which, given a set of constraints C, the set of probabilities of interest to the researcher is P = {P ∈ ∆(Ω, F) : P satisfies C}, and P ⋆ is selected by maximizing the Shannon entropy in P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' (iii) generalized fiducial inference (GFI) [20], in which a set of data-dependent measures on the parameter space Ω – called generalized fiducial distributions – is defined by care- fully inverting a deterministic data-generating equation without the use of Bayes’ the- orem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Mathematically, we can write P = {P ∈ ∆(Ω, F) : P satisfies [20, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' (2)]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' As pointed out in [20, Remark 4], P ⋆ is then selected by choosing the appropriate norm to endow the sample space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In [20, Section 1], the authors point out how, while GFI is different philosophically from Dempster-Shafer theory [13] and inferential models [29], the resulting solutions of these three methods are often mathematically closely related to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Constricting based on convex pooling As an example of a procedure that allows to collect new evidence, but does not use condi- tioning to update an agent’s beliefs, we present the famous model in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' There, the author supposes that there are k individuals, each having their own subjective probability Fi for the Constriction for sets of probabilities 7 unknown value of some parameter ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='4 For agent i, the opinions of all the other k − 1 agents represent new evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Instead of conditioning on those, agent i pools their own opinion with that of the other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' DeGroot shows that, repeating this process for all agent i, the group reaches (asymptotically) an agreement on a common subjective probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' After updating their opinions, the probability distribution for every member of the group belongs to the set P = � F = k � j=1 ζjFj, ζj ∈ [0, 1], ∀j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , k}, and k � j=1 ζj = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' (2) In particular, for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , k}, we write that after the first (pooling) iteration, the up- dated probability measure for agent i, denoted by Fi1 is given by Fi1 = �k j=1 pijFj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' This means that individual i weighs the opinion of all the agents, including themselves, via coef- ficients pi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , pik representing the relative importance that agent i assigns to the opinion of the other members of the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Because this is true for all agents, we can give a linear algebra notation to the updating process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Call P the k × k stochastic matrix whose rows are given by probability vectors (pi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , pik), i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Call then F = (F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , Fk)⊤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' we have that F(1) = PF, where F(1) := (F11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , Fk1)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Of course this holds for all iterations, so in turn we have that F(n) = PF(n−1) = PnF, for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The members continue to make these revisions indefinitely or until F(n) = F(n−1), for all n ≥ N, for some N ∈ N, so further revisions would not change the opinions of the members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The following is the main result of [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' If there exists n ∈ N such that every element in at least one column of Pn is positive, then a consensus is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' That is, if the condition in Theorem 12 is satisfied, then there exists a k × 1 dimensional vector π = (π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , πk) (that is unique, as guaranteed by [12, Theorem 3]) whose elements are non-negative and sum up to 1, and such that πP = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In turn, this entails that if we call Π the k × k stochastic matrix whose rows are all the same and equal to π, we have that F⋆ = ΠF, where F⋆ = (F ⋆ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , F ⋆ k )⊤ such that F ⋆ = F ⋆ 1 = · · · F ⋆ k = �k j=1 πjFj, where F ⋆ is the common subjective distribution that is reached in the consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Notice that F ⋆ belongs to P in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Call Pn := Conv(F1n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , Fkn), for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In this example we have that Pn ⊂ Pn−1, for all n ∈ Z+, where P0 is set P in equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' This means that the limit of sequence (Pn) is set ∩n∈Z+Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' If the condition in Theorem 12 is satisfied, then ∩n∈Z+Pn = {F ⋆}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Given a generic set A ∈ F, DeGroot procedure �ֹנ= DeGr may only weakly constrict A in general, for example if F ⋆ belongs to the extrema of P ≡ P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' That is, F ⋆(A) ≥ F(A) and F ⋆(A) < F(A), or F ⋆(A) > F(A) and F ⋆(A) ≤ F(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Nevertheless, there may exist r, s ∈ Z+, r < s, such that F s(A) > F r(A) and F s(A) < F r(A), where F s(A) = infF ∈Ps F(A), F s(A) = supF ∈Ps F(A), and similarly for Pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' 4Usually the elements of the parameter space Θ are denoted by θ, while the elements of the state space Ω by ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Since the focus of DeGroot’s model is the parameter space only, we used – just in this section – the ω ∈ Ω notation for the parameter space to maintain the notation consistent with other sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' 8 Michele Caprio and Teddy Seidenfeld DeGroot model is one of the possible examples of an agent collecting evidence and then revising their initial opinion using a rule that is different from conditioning on the gathered data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We showed that there is at least one such procedure in which constriction can take place if all the gathered information is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Notice that despite its similarities with deGroot’s model, there is no opportunity for con- striction in Aumann’s famous “agreeing to disagree” model [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' There, Bayes’ rule is utilized to reach a consensus, and there is no exchange of information between the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Because of this, Propositions 3 and 4 ensure us that not even weak constriction can take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Constricting based on non-Bayesian updating Suppose the results of an experiment induce partition E = {Ej} of the state space of interest Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, if we retain the assumption that probability measures are countably additive, conditioning on Ej does not allow for constriction, for all Ej ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' So in general we have that constriction for all Ej ∈ E can take place only if we intentionally forget the whole experiment that induces partition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' But if we are able to make assumptions on the nature of lower probability P associated with the set P of probabilities representing the agent’s beliefs, and if we consider updating procedures that are alternative to Bayes’ conditioning, we have opportunities for constriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Finally, we have space for constriction when probability measures are finitely additive, thanks to their property of non-conglomerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In this section, we give some background concepts that are needed to better understand the results that follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We begin by noting that lower and upper prob- abilities (LP and UP, respectively) of a set P are coherent if P is convex and closed [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' They are a particular type of Choquet capacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Definition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Given a measurable space (Ω, F) with Ω ̸= ∅, we say that a set function ν : F → [0, 1] is a Choquet capacity if ν(∅) = 0, ν(Ω) = 1, and ν(A) ≤ ν(B) for all A, B ∈ F such that A ⊂ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Denote by core(P) := {P ∈ ∆(Ω, F) : P(A) ≥ P(A), ∀A ∈ F} the core of lower prob- ability P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' It is the set of probability measures on (Ω, F) that setwise dominate P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' It is convex and weak⋆-compact [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' P completely characterizes the core, that is, it is sufficient to know P to be able to completely specify core(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' To emphasize this aspect, some authors say that P is compatible with core(P) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The following are special cases of coherent lower probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Definition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Suppose a coherent LP P is such that core(P) is also weakly compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then P is a Choquet capacity of order k, or k-monotone capacity, if for every collection {A, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , Ak} ⊂ F such that Ai ⊂ A, for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , k}, we have P(A) ≥ � ∅̸=I⊂{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=',k} (−1)#I−1P(∩i∈IAi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' (3) Its conjugate UP P is called a k-alternating capacity because it satisfies that for every col- lection {A, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , Ak} ⊂ F such that Ai ⊂ A, for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , k}, P(A) ≤ � ∅̸=I⊂{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=',k} (−1)#I−1P(∪i∈IAi), (4) Constriction for sets of probabilities 9 A special case of coherent LP that we will use in the remainder of the paper are convex LP’s, that are Choquet capacities of order 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' they satisfy P(A ∪ B) ≥ P(A) + P(B) − P(A ∩ B), for all A, B ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Another special case of coherent LP that we will use are belief functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Definition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' A coherent LP P is called a belief function if it is a Choquet capacity of order ∞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=', if (3) holds for every k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Unique to a belief function is its intuitive interpretation as a random set object that realizes itself as subsets of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Definition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' If P is a belief function, its associated mass function is the non-negative set function m : F → [0, 1], A �→ m(A) := � B⊂A (−1)#(A−B)P(B), (5) where A − B ≡ A ∩ Bc, and the subsets B of A have to belong to F as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Properties of mass function m are the following (a) m(∅) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' (b) � B⊂Ω m(B) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' (c) P(A) = � B⊂A m(B), and is unique to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Formula (5) is called the Möbius transform of P [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' A mass function m induces a precise probability distribution on F, as the distribution of a random set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' These concepts are further studied in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' To update a set of probabilities P given a set E ∈ F is to replace set function P with a version of the conditional set function P ×(· | E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The definition of P × is precisely the job of the updating rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Recall that we introduced generalized Bayes’ rule of conditioning �ֹנ=(B, E) in section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We now give the formal definitions of three additional updating rules for coherent lower and upper probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Generalized Bayes’, geometric, and Dempster’s rules are the ones that are most commonly used and studied in the literature, while Gärdenfors’ rule is a very general updating mechanism that subsumes many other methods of belief revision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The reasons for why an agent endorses one instead of another are explored in [17, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Definition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Let P ⊂ ∆(Ω, F) be closed and convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, the conditional LP’s and UP’s according to the geometric rule are set functions P G, P G such that, for all A, E ∈ F P (G,E)(A) ≡ P G(A | E) := P(A ∩ E) P(E) and P (G,E)(A) ≡ P G(A | E) = P(A ∩ E) P(E) , (6) provided that P(E) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' So the main difference between generalized Bayes’ and geometric updating procedures is that the former considers the infimum of the ratio of P(A ∩ E) and P(E), while the latter considers the ratio of the infima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We introduce next Dempter’s updating rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' 10 Michele Caprio and Teddy Seidenfeld Definition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Call E ∈ F the collected evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Assume that P is a belief function having mass function m and such that P(E) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Let P 0 be a separate belief function whose associated mass function m0 is such that m0(E) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The conditional belief function P D(· | E) is defined as P (D,E)(A) ≡ P D(A | E) := P(A) ⊕ P 0(E), ∀A ∈ F, where combination operator ⊕ means that the mass function associated with P D(· | E) is mD(A | E) = � C∩E=A m(C) � C′∩E̸=∅ m(C′), ∀A ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Consequently, Dempster’s updating rule yields the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' If P = core(P), then the LP’s and UP’s according to Dempster’s updating rule are set functions P D, P D such that, for all A, E ∈ F P (D,E)(A) ≡ P D(A | E) = P(A ∩ E) P(E) and P (D,E)(A) ≡ P D(A | E) := P(A ∩ E) P(E) , (7) provided that P(E) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' If P is a belief function and P = core(P), the geometric rule appears to be a natural dual to Dempster’s rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Operationally, though, they differ, as pointed out in [17, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The main difference is that Dempster’s rule requires P to be a belief function, while geometric rule does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Finally, we introduce the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Definition 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Call E ∈ F the collected evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Assume that P is a belief function having mass function m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Consider a function f : F × F → [0, 1] having constraints (a) � B∈F f(B, X) = 1, for all X ∈ F, (b) B ⊂ Ec =⇒ f(B, X) = 0, (c) f(∅, X) = 0, for all X ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, the belief function P (I,E)(·) ≡ P I(· | E) obtained according to Gärdenfors’ generalized imaging updating rule (GGI) is such that its associated mass function is given by mI(A | E) = � X∈F f(A, X)m(X), ∀A ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' (8) Since E is the collected information, we have that E ⊔ Ec = Ω, where ⊔ denotes the disjoint union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Equation (8) tells us that, upon learning that the evidence collected is not in Ec, then the probabilities given by m to the (sub)events in Ec are transferred to the “closest” (sub)events in E according to function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Constraint (a) is needed to ensure that the “probability bits” that are transferred are then normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Constraint (b) corresponds to the closed world assumption, which can be expressed as the statement “if the evidence collected is not in Ec, then it must belong to E”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Smets [33, Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='6] drops requirement (b) because he works under the open world assumption, which negates the previous statement to symbolize that the agent may have specified the state space they work with incorrectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' So if the evidence collected is not in Ec, it may be in E but also in a superset of E that the agent did not consider at the beginning of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Constraint (c) is just a sanity check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Function f is a version of a conditional probability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' it was first introduced in [18] and Constriction for sets of probabilities 11 then generalized by [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The choice of function f informs how having collected evidence E influences our change of beliefs around A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' GGI subsumes many other existing updating rules [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In the remainder of the paper, we write �ֹנ=(×, E), × ∈ {B, G, D, I}, to indicate the gen- eralized Bayes’, geometric, Dempster’s, and Gärdenfors’ updating rules, respectively, given collected evidence E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We write that (×, E) ↫ A if given evidence E, rule × (strictly) constricts A, and (×, E) ↫ A if rule × (strictly) constricts A for all elements of partition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Intentional forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In this section we show that if we are not willing to assume that P is at least convex – let alone a belief function – then we cannot obtain constriction for all the elements E of a partition E representing the results of an experiment of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In this very general case, we need to forget in order to constrict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We retain the assumptions that P is closed and convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Let E be a measurable and denumerable partition of Ω, and let × ∈ {B, G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, for any A ∈ F we have that inf E∈E P ×(A | E) ≤ P(A) and sup E∈E P ×(A | E) ≥ P(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' An immediate consequence of Lemma 20 is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Theorem 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Let E be a measurable and denumerable partition of Ω and P = core(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then for × ∈ {B, G}, we have that for all A ∈ F, there exists E ∈ E such that �ֹנ=(×, E) does not weakly constrict A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' This result tells us that for any event A ∈ F of interest, we can always find at least one element of E that dilates A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' This means that there exist E1, E2 ∈ E, E1 possibly different than E2, such that [P(A), P(A)] ⊊ [P B(A | E1), P B(A | E1)] and [P(A), P(A)] ⊊ [P G(A | E2), P G(A | E2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' As it appears clear, if the agent is unwilling to make any extra assumption on the nature of P, then there is no opportunity for constriction to take place for all the elements of partition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In [32, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='3] the authors give sufficient conditions for dilation to take place for all E ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, intentionally forgetting altogether the experiment that dilates A seems the only viable option to reach constriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' As the name suggests, intentional forgetting corresponds to an agent willingly forgetting pieces of information, for example because they are redundant, because they may be harmful, or because they are instructed to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' If after collecting evidence E our current beliefs are encapsulated in lower and upper probabilities P ×(· | E) and P ×(· | E), respectively, then by forgetting we mean reversing the learning process so that our “updated” lower and upper probabilities becomes what used to be the lower and upper “priors”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' P(·) and P(·), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The topic of forgetting is studied in statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In [30], for example, forgetting is intended in the sense of stabilized forgetting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' with this we mean the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Suppose that the agent is operating in an environment that is susceptible to changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, the agent’s response to surprising events depends on their beliefs about how likely the environment is to change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' If it is volatile, a single unexpected event triggers forgetting of past beliefs and relearning of a new contingency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' So at time t, the agent collects evidence Et;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' they use it to infer whether the 12 Michele Caprio and Teddy Seidenfeld environment has changed or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In the former case, they erase their memory of past events and reset their prior belief to their initial prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In the latter, they can learn a new posterior belief of the environment structure based on their previous belief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Another example is given by limited memory procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In [4, 5], the authors study bandit problems based on limited memory: working with restricted memory, data that are too old are forgotten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Forgetting is studied in machine learning as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In [7], for instance, the authors come up with an algorithm that features a forgetting factor which balances the relative importance of new data and past data and adjust the model to pay more attention to the new data when the concept drift is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In this framework, forgetting is intended as in past data progressively losing importance as new evidence is collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Another example in machine learning where forgetting is crucial is continual learning: as data gets discarded and has a limited lifetime, the ability to forget what is not important and retain what matters for the future are the main issues that continual learning targets and focuses on [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Finally, and rather unsurprisingly, psychologists and cognitive scientists have thoroughly inspected the phenomenon of forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The reference textbook that investigates intentional forgetting is [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The authors examine the effect on memory of instructions to forget in a wide variety of contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' They point out how with the enormous number of information available nowadays, online forgetting of some information is necessary, and how often times replacing existing information with new information is mandatory (think of a person changing their phone number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Study on intentional forgetting stemmed from the phenomenon of directed forgetting: we are able to deal more effectively with large amounts of information by following instructions to treat some of the information as “to be forgotten” (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' evidence presented in a courtroom that, being inadmissible, is asked to be disregarded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In this way, interference is reduced and we are able to devote all of our resources to the remaining to-be-remembered information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' It is easy to see how stabilized forgetting is a particular case of intentional forgetting, and so is the forgetting factor approach used in the machine learning literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Consider the problem of an agent that expresses their initial beliefs on (Ω, F) via a set of probabilities P = core(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' For convenience, we write P ≡ PE0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' As data become available, they update their beliefs using Bayes’ rule of conditioning for every element of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' With this we mean the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Suppose we collect evidence in the form of E1 ⊂ F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' then we update the elements of P to obtain PE1 := � PE1 ∈ ∆(Ω, F) : PE1(A) ≡ P(A | E1) = P(E1 | A)P(A) P(E1) ∝ P(E1 | A)P(A), ∀A ∈ F, P ∈ P � , where P(·) represents the prior and P(E1 | ·) represents the likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' More in general, let the evidence collected up to time t > 0 be encapsulated in collection {Ek}t k=1 ⊂ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, the agent’s updated opinion is given by set PE1···Et := � PE1···Et ∈ ∆(Ω, F) : PE1···Et(A) ≡ PE1···Et−1(A | Et) ∝ P(Et | A, E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , Et−1)PE1···Et−1(A), ∀A ∈ F, PE1···Et−1 ∈ PE1···Et−1 � , Constriction for sets of probabilities 13 where PE1···Et−1(·) ≡ P(· | E1 · · · Et−1) represents the “revised” prior (that is, the posterior computed at time t − 1) and P(Et | ·) represents the likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Pick any k ∈ N such that 1 ≤ k ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Assume that PE1···Et−k and PE1···Et−k···Et are both convex and closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Fix an event A ∈ F of interest and define the following P⋆E1···Et−k(A) := {PE1···Et−k ∈ PE1···Et−k : PE1···Et−k(A) = P E1···Et−k(A)}, P⋆ E1···Et−k(A) := {PE1···Et−k ∈ PE1···Et−k : PE1···Et−k(A) = P E1···Et−k(A)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' For a generic P ∈ ∆(Ω, F), and generic A, B ∈ F, define notion of dependence dP and sets induced by its value by dP(A, B) := P(A ∩ B) − P(A)P(B), Σ+(A, B) := {P ∈ ∆(Ω, F) : dP(A, B) > 0}, Σ−(A, B) := {P ∈ ∆(Ω, F) : dP(A, B) < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Call now �ֹנ=(IF×, E ) the procedure of intentionally forgetting evidence E (where E can be an element of F, a whole partition E of state space Ω, or a collection {E} of partitions) after having updated endorsing rule ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, the following gives sufficient conditions for intentional forgetting to induce constriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Theorem 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Fix an event A ∈ F of interest and let the agent endorse any rule × ∈ {B, G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' If P⋆E1···Et−k(A) ∩ Σ−(A, Et−k+1 ∩ · · · ∩ Et) ̸= ∅ and P⋆ E1···Et−k(A) ∩ Σ+(A, Et−k+1 ∩ · · · ∩ Et) ̸= ∅, then forgetting Et−k+1∩· · ·∩Et strictly constricts A, in symbols (IF×, Et−k+1∩· · ·∩Et) ↫ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' If this holds for all elements Es of partition Es, s ∈ {t − k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , t}, we write (IF×, Et−k+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' , Et) ↫ A, so we can forget all the experiments that took place after time t − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' If we let k = t, then we obtain stabilized forgetting as in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Notice that for stabilized forgetting subscript E1 · · · Et−k in Theorem 22 is substituted by E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We also have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Corollary 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Fix an event A ∈ F of interest and let the agent endorse any rule × ∈ {B, G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' If P⋆Ek···Et(A) ∩ Σ−(A, E1 ∩ · · · ∩ Ek−1) ̸= ∅ and P⋆ Ek···Et(A) ∩ Σ+(A, E1 ∩ · · · ∩ Ek−1) ̸= ∅, then forgetting E1 ∩ · · · ∩ Ek−1 strictly constricts A, in symbols (IF×, E1 ∩ · · · ∩ Ek−1) ↫ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In this case, we obtain the machine learning version of forgetting [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The agent intention- ally forgets data collected before time k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' More formally, the authors weight the evidence Et collected at each time t by a coefficient depending on t that goes to 0 the farther time t is from present time T, that is, it goes to 0 as |t − T| grows to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Evidence that is old 14 Michele Caprio and Teddy Seidenfeld enough gets severely discounted, to the point that for practical purpose we can consider it as being forgotten, and so the result in Corollary 23 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Notice that intentional forgetting is always a viable way of inducing constriction, as long as the selected updating rule induces dilation first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In this section we focused on × ∈ {B, G} because generalized Bayes’ and geometric rules are the most general ones we presented (they do not require the lower probability of interest to be a belief function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Levi-inspired forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' A particular type of forgetting is the one inspired by Levi’s work on corrigible infallibility, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Suppose that at time t an agent is equipped with a body of beliefs Kt regarding the events in F, that is, a collection of logical predicates that describe the beliefs of the agent at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='5 For the purpose of studying constriction, we focus on the (closed and convex) set PKt of probability measures representing the agent’s beliefs at time t induced by Kt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='6 Consider a generic event H ⊂ Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' every element PKt of PKt has the following two properties: if Kt rules out H, that is, if given the beliefs at time t event H is considered impossible – written Kt �→ ¬H –, then PKt(A | H) is not well defined, for all A ∈ F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' if instead Kt does not rule out H, then PKt(A | H) is well defined, for all A ∈ F, and represents the agent’s beliefs around the plausibility of A if H obtains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Lower probability P Kt completely characterizes PKt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Suppose then that at time t + 1 the agent collects evidence E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Let Kt+1 be the new body of beliefs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' abusing notation, we write Kt ∪ {E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, suppose that the agent endorses either of generalized Bayes’ or geometric rules, × ∈ {B, G};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' the updated beliefs of the agent are encapsulated in P × Kt(· | E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' If (according to ×) E dilates an event A′ of interest, then the agent can forget about E to obtain constriction (the agent runs “reverse conditioning”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Their infallible beliefs, encapsulated in Kt+1, are subject to being corrected, and we have Kt+2 = Kt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The body of beliefs contracts: Kt+1 loses element E and goes back to what used to be at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We write (LF×, E) ↫ A′ to denote that Levi-forgetting E – after having updated endorsing rule × – constricts A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' If this holds for all elements of a partition E of Ω representing the possible outcomes of an experiment of interest, we write (LF×, E) ↫ A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Assumptions on the nature of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' If the agent is willing to make some assumptions on the type of lower probability P that represents their beliefs, then we can have constriction for all E ∈ E without resorting to intentional forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' As the proverb goes, there is no free lunch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The following is Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='9 in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Theorem 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Let E = {E, Ec} be the partition associated with the outcomes of the experi- ment of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Assume that P is a belief function such that P(E), P(Ec) > 0, and consider any event A ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, if E dilates A under the Geometric rule, then it must constrict A under Dempster’s rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Similarly, if E dilates A under Dempster’s rule, then it must constrict A under the Geometric rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' 5Not to be confused with belief functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' 6In particular, we assume PKt = core(P Kt), where P Kt(A) = infPKt∈PKt PKt(A), for all A ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Constriction for sets of probabilities 15 The proof of Theorem 24 only requires that P is convex, but we need the assumption that P is in fact a belief function otherwise we would not be able to use Dempster’s rule (see Definition 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' As we can see, Dempster’s and geometric rule contradict each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Assuming that P is a belief function allows us to use updating rules that are otherwise inaccessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Theorem 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Let E be the evidence collected by the agent, and assume that P is a belief function having mass function m such that P(E) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Consider any event A ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We have that (I, E) ↫ A if and only if � B⊂A �� X∈F f(B, X)m(X) − m(B) � > 0 and � B⊂Ac �� X∈F f(B, X)m(X) − m(B) � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' If the conditions in Theorem 25 hold for every elements of partition E, we write (I, E) ↫ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The main point of this section is that if we are willing to formulate an assumption on the nature of lower probability P associated with set P representing our beliefs, then we are able to find constriction by using updating rules that in general do not allow for constrict for all E ∈ E, like the geometric rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We can also use entirely new updating techniques like Dempster’s rule or GGI that are otherwise inapplicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Dubins-deFinetti conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In this section, we show how if we are willing to depart from the classical Kolmogorovian paradigm of probabilities, then we have additional opportunities for constriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In particular, as we shall see, Bayes’ rule can induce constric- tion if we allow probabilities to be merely finitely additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Suppose that we adopt Dubins-deFinetti conditioning (DdFC) framework;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' an in-depth exposition of DdFC can be found in [10, 14, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' For the sake of the present work, the two main differences with respect to the Kolmogorovian framework is that probabilities need not be countably additive, and that conditioning does not happen on sigma-fields, but rather on events, members of a partition of the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The following statements are true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' we will provide illustrations for the first one, and the second one is shown similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' (1) If probability measures are finitely but not countably additive, then constriction can take place for all the elements of a countable partition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' (2) If probability measures are countably but not uncountably additive, then constriction can take place for all the elements of an uncountable partition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Notice also that if probability measures are uncountably additive, then they must be discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Definition 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We say that probability measure P is conglomerable in partition E when for every event A such that P(A | E) is defined for all E ∈ E, and for all constants k1, k2, if k1 ≤ P(A | E) ≤ k2 for all E ∈ E, then k1 ≤ P(A) ≤ k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Definition 26 asserts that for each event A, if all the conditional probabilities over a par- tition E are bounded by two quantities, k1 and k2, then the unconditional probability for that event is likewise bounded by these two quantities [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' De Finetti [10] shows the non- conglomerability of finitely additive probability measures (FAPMs) in denumerable parti- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' 16 Michele Caprio and Teddy Seidenfeld Assume that instead of requiring P to be a set of countably additive probabilities, we allow it to be a set of FAPMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, weak and strict constriction can happen by Bayes updating P thanks to the non-conglomerability property of FAPMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The two illustrations that we present in this section build on the example in [14, page 92], which we state here for motivating their construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Example 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' (Dubins) Let Ω = {A, B} × {N = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Stipulate that P(A) = P(B) = 1/2, P(N = n | A) = 2−n, for n ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' }, a countably additive conditional probability, P(N = n | B) = 0, for n ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' }, a strongly finitely additive conditional proba- bility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='7 Then, P(N = n) = 2−(n+1) > 0, for n ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' }, and (marginally) P is merely finitely additive over the subalgebra generated by the partition EN = {{N = 1}, {N = 2}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' P displays non-conglomerability for the event A in the partition EN = {{N = 1}, {N = 2}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='} as P(A) = 1/2 and P(A | N = n) = 1, for n ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Illustration 1 (weak constriction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Use Example 27 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Consider a set P of probabilities on Ω = {A, B} × {N = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='} such that P = {Pα, 0 < α ≤ 1}, where Pα(A) = α, Pα(N = n | A) = 2−n, for n ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' }, a countably additive conditional probability, Pα(N = n | B) = 0, for n ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' }, a strongly finitely additive conditional probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Note that neither Pα(N = n | A) nor Pα(N = n | B) depend upon α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' With respect to P, we have 0 < Pα(A) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' For each 0 < α < 1, we have non-conglomerability of Pα for the event A in the partition EN as Pα(A | N = n) = 1, for n ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Observe then that 1 = P1(A) = P1(A | N = n), for n ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Thus, Bayes-updating using the information {N = n} from the partition EN weakly-constricts P = {Pα, 0 < α ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Illustration 2 (strict constriction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Modify Example 27 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Consider a set P of probabilities on Ω = {A, B} × {N = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='} such that P = {Pα, 0 < α < 1}, where Pα(A) = α, so that with respect to P, we have 0 < Pα(A) < 1, Pα(N = n | A) = (1 − α)2−n, for n ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' }, Pα(N = n | B) = α2−n, for n ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Note that each of these two conditional probabilities is a merely finitely additive probability distribution over N that depends on α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In addition, observe that P(N = n) = 2α(1−α)2−n > 0, which (for each α) also is a merely finitely additive probability distribution over N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' By a Bayes’ updating, for each 0 < α < 1 and each n ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' }, Pα(A | N = n) = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' That is, for each α ̸= 1/2 with 0 < α < 1, there is non-conglomerability of Pα for the event A in the partition EN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Whereas, 1/2 = P1/2(A) = P1/2(A | N = n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Thus, Bayes-updating using the information {N = n} from the partition EN strictly-constricts P = {Pα, 0 < α < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Remark 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' These two illustrations help to explain why Propositions 3 and 4 are restricted to countably additive probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' 7Recall that a FAPM is strongly finitely additive if it admits countable partitions by null sets [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Constriction for sets of probabilities 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Conclusion In this paper, we showed that, when updating an agent’s opinions, there are at least three settings for constricting sets of probabilities (representing the beliefs), namely when belief revision is performed without evidence, with evidence but without conditioning, and with evidence using different updating rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We also provided interesting examples of procedures for every such framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' This is just the first step towards a deeper study of the constricting phenomenon, that we will carry over in the next future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In particular, we plan to find more instances in which constricting is possible, and to find a trait d’union linking these settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Acknowledgements Michele Caprio would like to acknowledge funding from ARO MURI W911NF2010080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Proofs Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Let P be a probability function that satisfies P(A) = minP ∈P P(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' By Lemma 2, if P(XA+ P ) > 0, then P(XA− P ) > 0 and (B, E) does not strictly constrict A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' That is, for each P1i ∈ P, P(A) ≤ P1i(A), and we saw in (1) that (B, E) does not strictly constrict A when P(A) = P1i(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' So if (B, E) weakly constricts A, we have that P(XA+ P ) = 0, and then P({x ∈ X : P B(A | X = x) = P(A)}) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Let now P be a probability function that satisfies P(A) = maxP ∈P P(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' By the same reasoning, P({x ∈ X : P B(A | X = x) = P(A)}) = 1, and then (B, E) does not weakly constrict A either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' □ Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Assume (for a reductio proof) that on a set of X values with P- measure 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' with P-probability 1, for all P ∈ P), for each xi there exist P1i and P2i in P such that for each P ∈ P(A | xi), either P1i(A) < P(A | xi) ≤ P2i(A) or P1i(A) ≤ P(A | xi) < P2i(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Since X is a simple random variable, define P1(A) := mini P1i(A) and P2(A) := maxi P2i(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Given Proposition 3, assume P(A) is not a closed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Without loss of generality, assume it is open below (the reasoning is parallel if P(A) is open above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' So, P(A) < P1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, there exists P0 ∈ P with P(A) < P0(A) < P1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Since for each i, P1i(A) < P0(A | xi) or P1i(A) ≤ P0(A | xi), we also have that for each i, P0(A) < P1(A) ≤ P0(A | xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' But then P0(XA+ P0 ) = 1, which is a contradiction according to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' □ Proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Immediate from Definition 1 and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' □ Proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' We first show that the lower probability (LP) infP ∈Conv(P) P(·) of the convex hull of P and the LP infP ∈ex[Conv(P)] P(·) of the extrema of the convex hull of P coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' To see this, pick any A ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Since ex[Conv(P)] ⊂ Conv(P), we have that inf P ∈Conv(P) P(A) ≤ inf P ∈ex[Conv(P)] P(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' (9) Then, let ex[Conv(P)] = {P ex j }K j=1, K ∈ N ∪ {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' For all P ∈ Conv(P) and all A ∈ F, we have that P(A) = K � j=1 αjP ex j (A) ≥ K � j=1 αjP ex(A) = P ex(A) := inf P ∈ex[Conv(P)] P(A), 18 Michele Caprio and Teddy Seidenfeld where {αj}K j=1 is a collection of positive reals such that �K j=1 αj = 1, which implies that inf P ∈Conv(P) P(A) ≥ inf P ∈ex[Conv(P)] P(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' (10) By combining together (9) and (10) we obtained the desired equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Now, if P ⋆ ∈ ex[Conv(P)], then there might be a collection { ˜A} ⊂ F for which P ⋆( ˜A) = P( ˜A) or P ⋆( ˜A) = P( ˜A), so the constriction is weak for the elements of the collection, while P ⋆(A) > P( ˜A) and P ⋆(A) < P( ˜A), for all A ∈ F \\ { ˜A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' If instead P ⋆ ∈ Conv(P) \\ ex[Conv(P)], then P ⋆(A) ∈ (P(A), P(A)), for all A ∈ F, so we have �ֹנ↫ A, for all A ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' □ Proof of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' If P(A) is closed in the Euclidean topology and P ⋆(A) ∈ ∂B([0,1])P(A), then P ⋆(A) = P(A) or P ⋆(A) = P(A), so the constriction is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' If instead P ⋆(A) ∈ intB([0,1])P(A), then P ⋆(A) ∈ (P(A), P(A)), so we have �ֹנ↫ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' □ Proof of Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' This proof draws on that of [17, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Fix any A ∈ F and suppose for the sake of contradiction that infE∈E P ×(A | E) > P(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Because P is closed, there exists P(A) ∈ P such that P(A)(A) = P(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Notice that subscript (A) reminds us that this probability measure can vary with the choice of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, we have that P(A) = P(A)(A) = � E∈E P(A)(A | E)P(A)(E) ≥ � E∈E P ×(A | E)P(A)(E) ≥ � E∈E inf E∈E P ×(A | E)P(A)(E) = inf E∈E P ×(A | E) � E∈E P(A)(E) = inf E∈E P ×(A | E) > P(A), where the last inequality comes from our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' As we can see, this leads to a contra- diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The same argument applies for the upper probability of A, that is, if we assume for the sake of contradiction that supE∈E P ×(A | E) < P(A) and we pick P ′ (A) ∈ P such that P ′ (A)(A) = P(A), P ′ (A) possibly different from P(A), then P(A) ≤ � E∈E P ×(A | E)P ′ (A)(E) < P(A) � E∈E P ′ (A)(E) = P(A), reaching again a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' □ Proof of Theorem 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Immediate from Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' □ Proof of Theorem 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' This proof comes from that of [32, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Fix an event A ∈ F of interest, and let × ∈ {B, G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Pick any PE1···Et−k ∈ P⋆E1···Et−k(A) ∩ Σ−(A, Et−k+1 ∩ · · · ∩ Et).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, we have that PE1···Et−k(A) = P E1···Et−k(A) because PE1···Et−k ∈ P⋆E1···Et−k(A) and PE1···Et−k(A∩Et−k+1 ∩· · ·∩Et) < PE1···Et−k(A)PE1···Et−k(Et−k+1∩· · ·∩Et) because PE1···Et−k ∈ Σ−(A, Et−k+1 ∩ · · · ∩ Et).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, P E1···Et−k(A) = PE1···Et−k(A) > PE1···Et−k(A ∩ Et−k+1 ∩ · · · ∩ Et) PE1···Et−k(Et−k+1 ∩ · · · ∩ Et) = PE1···Et−k(A | Et−k+1 ∩ · · · ∩ Et) ≥ P × E1···Et−k(A | Et−k+1 ∩ · · · ∩ Et).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Constriction for sets of probabilities 19 A similar argument gives us that P E1···Et−k(A) < P × E1···Et−k(A | Et−k+1 ∩ · · · ∩ Et).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' So Et−k+1 ∩ · · · ∩ Et dilates A regardless of which updating rule × ∈ {B, G} the agent endorses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In turn, forgetting Et−k+1∩· · ·∩Et constricts A, in symbols (IF×, Et−k+1∩· · ·∩Et) ↫ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' □ Proof of Corollary 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Analogous to the proof of Theorem 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' □ Proof of Theorem 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Fix any A ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Recall that by (8), we have that mI(A | E) = � X∈F f(A, X)m(X), for all A ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Also, by property (c) of mass function m associated to LP P (see Definition 5), we have that P(A) = � B⊂A m(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' So, P I(A | E) = � B⊂A mI(B | E) = � B⊂A � X∈F f(B, X)m(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Only if Suppose (I, E) ↫ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, by definition of constriction, P I(A | E) > P(A) and P I(A | E) < P(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' This happens if and only if � B⊂A � X∈F f(B, X)m(X) > � B⊂A m(B) ⇐⇒ � B⊂A �� X∈F f(B, X)m(X) − m(B) � > 0 and � B⊂Ac � X∈F f(B, X)m(X) > � B⊂Ac m(B) ⇐⇒ � B⊂Ac �� X∈F f(B, X)m(X) − m(B) � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' This latter is true because P I(A | E) < P(A) ⇐⇒ 1 − P I(Ac | E) < 1 − P(Ac) ⇐⇒ P I(Ac | E) > P(Ac).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' If Assume that � B⊂A �� X∈F f(B, X)m(X) − m(B) � > 0 and � B⊂Ac �� X∈F f(B, X)m(X) − m(B) � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Then, we have P I(A | E) = � B⊂A � X∈F f(B, X)m(X) > � B⊂A m(B) = P(A) and P I(Ac | E) = � B⊂Ac � X∈F f(B, X)m(X) > � B⊂Ac m(B) = P(Ac), which implies P I(A | E) < P(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' □ References [1] Thomas E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Armstrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Strong singularity, disjointness, and strong finite additivity of finitely additive measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Journal of Mathematical Analysis and Applications, 131(2):565–587, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [2] Thomas Augustin, Frank P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Coolen, Gert de Cooman, and Matthias C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Troffaes (editors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Introduction to Imprecise Probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Wiley Series in Probability and Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Chichester, UK : Wiley, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [3] Robert J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Aumann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Agreeing to disagree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The Annals of Statistics, 4(6):1236–1239, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' 20 Michele Caprio and Teddy Seidenfeld [4] Dorian Baudry, Yoan Russac, and Olivier Cappé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' On limited-memory subsampling strategies for bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In Marina Meila and Tong Zhang, editors, ICML 2021: Proceedings of the 38th International Conference on Machine Learning, pages 727–737.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [5] Donald A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Berry and Bert Fristedt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Bandit problems – Sequential Allocation of Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Monographs on Statistics and Applied Probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Dordrecht : Springer, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [6] Patrick Billingsley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Probability and Measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' New York, NY : Wiley, second edition, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [7] Weipeng Cao, Zhong Ming, Zhiwu Xu, Jiyong Zhang, and Qiang Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Online sequential extreme learning machine with dynamic forgetting factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' IEEE Access, 7:179746–179757, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [8] Michele Caprio and Ruobin Gong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Dynamic precise and imprecise probability kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Available on arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='04382, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [9] Simone Cerreia-Vioglio, Fabio Maccheroni, and Massimo Marinacci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Ergodic theorems for lower proba- bilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Proceedings of the American Mathematical Society, 144:3381–3396, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [10] Bruno de Finetti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Probability, Induction and Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' New York : Wiley, 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [11] Bruno de Finetti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Theory of Probability, volume 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' New York : Wiley, 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [12] Morris H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' DeGroot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Reaching a consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Journal of the American Statistical Association, 69(345):118– 121, 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [13] Arthur P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Dempster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The Dempster–Shafer calculus for statisticians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' International Journal of Approximate Reasoning, 48(2):365–377, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [14] Lester E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Dubins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Finitely Additive Conditional Probabilities, Conglomerability and Disintegrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The Annals of Probability, 3(1):89 – 99, 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [15] Daniel Ellsberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Risk, ambiguity, and the Savage axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The Quarterly Journal of Economics, 75(4):643–669, 1961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [16] Jonathan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Golding and Colin M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' MacLeod (editors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Intentional forgetting: Interdisciplinary ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Mahwah, NJ : Lawrence Erlbaum Associates, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [17] Ruobin Gong and Xiao-Li Meng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Judicious judgment meets unsettling updating: dilation, sure loss, and Simpson’s paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Statistical Science, 36(2):169–190, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [18] Peter Gärdenfors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Knowledge in flux: Modeling the dynamics of epistemic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Boston, MA : The MIT press, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [19] Paul R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Halmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Measure Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Graduate Texts in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' New York, NY : Springer, 1950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [20] Jan Hannig, Hari Iyer, Randy C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Lai, and Thomas C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Generalized fiducial inference: A review and new results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Journal of the American Statistical Association, 111(515):1346–1361, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [21] Sven Ove Hansson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Logic of Belief Revision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In Edward N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Zalta, editor, The Stanford Encyclopedia of Philosophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Palo Alto, CA : Metaphysics Research Lab, Stanford University, Spring 2022 edition, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [22] Timothy Herron, Teddy Seidenfeld, and Larry Wasserman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Divisive conditioning: Further results on dilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Philosophy of Science, 64(3):411–444, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [23] Joseph B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Kadane, Mark J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Schervish, and Teddy Seidenfeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Statistical Implications of Finitely Additive Probability, pages 211–232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Cambridge Studies in Probability, Induction and Decision Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Cambridge, UK : Cambridge University Press, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [24] Hiremagalur K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Kesavan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Jaynes’ maximum entropy principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In Christodoulos A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Floudas and Panos M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Pardalos, editors, Encyclopedia of Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Boston, MA : Springer, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [25] Timothée Lesort, Vincenzo Lomonaco, Andrei Stoian, Davide Maltoni, David Filliat, and Natalia Díaz- Rodríguez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Continual learning for robotics: Definition, framework, learning strategies, opportunities and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Information Fusion, 58:52–68, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [26] Isaac Levi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Why indeterminate probability is rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Journal of Applied Logic, 7(4):364–376, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [27] Isaac Levi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' How infallible but corrigible full belief is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In Horacio Arló-Costa, Vincent F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Hendricks, and Johan van Benthem, editors, Readings in Formal Epistemology: Sourcebook, pages 247– 267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Cham, CH : Springer International Publishing, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [28] Massimo Marinacci and Luigi Montrucchio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Introduction to the Mathematics of Ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Uncertainty in Economic Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' New York, NY : Routledge, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [29] Ryan Martin and Chuanhai Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Inferential Models: Reasoning with Uncertainty, volume 145 of Monograph on Statistics and Applied Probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Boca Raton, FL : CRC Press, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Constriction for sets of probabilities 21 [30] Vincent Moens and Alexandre Zénon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Learning and forgetting using reinforced Bayesian change detec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' PLoS Computational Biology, 4(15):1–41, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [31] Eugenio Regazzini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' De Finetti’s coherence and statistical inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The Annals of Statistics, 15(2):845– 864, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [32] Teddy Seidenfeld and Larry Wasserman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Dilation for sets of probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' The Annals of Statistics, 21(3):1139–1154, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [33] Philippe Smets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' About updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' In Proceedings of the Seventh international conference on Uncertainty in Artificial Intelligence, pages 378–385, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [34] Rush T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Stewart and Ignacio Ojea Quintana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Probabilistic opinion pooling with imprecise probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Journal of Philosophical Logic, 47(1):17–45, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [35] Peter Walley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Statistical Reasoning with Imprecise Probabilities, volume 42 of Monographs on Statistics and Applied Probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' London, UK : Chapman and Hall, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [36] Peter Walley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Towards a unified theory of imprecise probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' International Journal of Approximate Reasoning, 24(2-3):125–148, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' [37] Roland R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Yager and Liping Liu (editors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' Classic works of the Dempster-Shafer theory of belief functions, volume 39 of Studies in Fuzziness and Soft Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' New York, NY : Springer, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content=' PRECISE Center, Department of Computer and Information Science, University of Pennsylvania, 3330 Walnut Street, Philadelphia, PA 19104 Email address: caprio@seas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='upenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='edu URL: https://precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='seas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='upenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='edu/people/doctoral-students Departments of Philosophy and Statistics, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA 15213 Email address: teddy@stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='edu URL: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='edu/dietrich/philosophy/people/faculty/seidenfeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} +page_content='html' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE5T4oBgHgl3EQfjQ_b/content/2301.05655v1.pdf'} diff --git a/ndE2T4oBgHgl3EQfJga_/vector_store/index.pkl b/ndE2T4oBgHgl3EQfJga_/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..7992b759513aadb23f9b9ac23b86a3042edff7fb --- /dev/null +++ b/ndE2T4oBgHgl3EQfJga_/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c9c22b36743f8af777054d4f6adf46f590e98bccae32f8d0db92430472f1cae3 +size 220043 diff --git a/ndE4T4oBgHgl3EQfug0k/content/tmp_files/2301.05232v1.pdf.txt b/ndE4T4oBgHgl3EQfug0k/content/tmp_files/2301.05232v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e163ab7cae672a0f50cfa7103a34d32539b4aae0 --- /dev/null +++ b/ndE4T4oBgHgl3EQfug0k/content/tmp_files/2301.05232v1.pdf.txt @@ -0,0 +1,1187 @@ +arXiv:2301.05232v1 [math.CO] 12 Jan 2023 +On forced periodicity of perfect colorings +Pyry Herva and Jarkko Kari +Abstract +We study forced periodicity of two-dimensional configurations under certain constraints +and use an algebraic approach to multidimensional symbolic dynamics in which d- +dimensional configurations and finite patterns are presented as formal power series +and Laurent polynomials, respectively, in d variables. We consider perfect colorings +that are configurations such that the number of points of a given color in the neigh- +borhood of any point depends only on the color of the point for some fixed relative +neighborhood, and we show that by choosing the alphabet suitably any perfect coloring +has a non-trivial annihilator, that is, there exists a Laurent polynomial whose formal +product with the power series presenting the perfect coloring is zero. Using known +results we obtain a sufficient condition for forced periodicity of two-dimensional per- +fect colorings. As corollaries of this result we get simple new proofs for known results +of forced periodicity on the square and the triangular grids. Moreover, we obtain a +new result concerning forced periodicity of perfect colorings in the king grid. We also +consider perfect colorings of a particularly simple type: configurations that have low +abelian complexity with respect to some shape, and we generalize a result that gives +a sufficient condition for such configurations to be necessarily periodic. Also, some +algorithmic aspects are considered. +1 +Introduction +We say that a d-dimensional configuration c ∈ AZd, that is, a coloring of the d-dimensional +integer grid Zd using colors from a finite set A is a perfect coloring with respect to some finite +relative neighborhood D ⊆ Zd if the number of any given color of A in the pattern c|u+D +depends only on the color c(u) for any u ∈ Zd. There is a similar version of this definition +for general graphs: a vertex coloring ϕ: V → A of a graph G = (V, E) with a finite set A of +colors is a perfect coloring of radius r if the number of any given color in the r-neighborhood +of a vertex u ∈ V depends only on the color ϕ(u) of u [28, 29]. More generally, the definition +of perfect colorings is a special case of the definition of equitable partitions [8]. +If ϕ: V → {0, 1} is a binary vertex coloring of a graph G = (V, E) then we can define a +subset C ⊆ V of the vertex set – a code – such that it contains all the vertices with color 1. +If ϕ is a perfect coloring of radius r, then the code C has the property that the number of +codewords of C in the r-neighborhood of a vertex u ∈ V is a if u ̸∈ C and b if u ∈ C for some +fixed non-negative integers a and b. This kind of code is called a perfect (r, b, a)-covering +1 + +or simply just a perfect multiple covering [1, 5]. This definition is related to domination in +graphs and covering codes [11, 5]. +Let D ⊆ Zd be a finite set and A a finite set of colors. Two finite patterns p, q ∈ AD are +abelian equivalent if the number of occurrences of each symbol in A is the same in them. The +abelian complexity of a configuration c ∈ AZd with respect to a finite shape D is the number +of abelian equivalence classes of patterns of shape D in c [30]. We note that if c ∈ AZd +is a perfect coloring with respect to D and |A| = n, then the abelian complexity of c with +respect to D is at most n. Abelian complexity is a widely studied concept in one-dimensional +symbolic dynamics and combinatorics on words [22]. +In this paper we study forced periodicity of two-dimensional perfect colorings, that is, we +study conditions under which all the colorings are necessarily periodic. We give a general +condition for forced periodicity. As corollaries of this result we get new proofs for known +results [1, 28, 29] concerning forced periodicity of perfect colorings in the square and the +triangular grid and a new result for forced periodicity of perfect colorings in the king grid. +Moreover, we study two-dimensional configurations of low abelian complexity, that is, con- +figurations that have abelian complexity 1 with respect to some shape: we generalize a +statement of forced periodicity concerning this type of configurations. We use an algebraic +approach [17] to multidimensional symbolic dynamics, i.e., we present configurations as for- +mal power series and finite patterns as Laurent polynomials. This approach was developed +to make progress in a famous open problem in symbolic dynamics – Nivat’s conjecture [27] – +concerning forced periodicity of two-dimensional configurations that have a sufficiently low +number of m × n rectangular patterns for some m, n. The Nivat’s conjecture thus claims a +two-dimensional generalization of the Morse-Hedlund theorem [24]. +This article is an extended version of the conference paper [12] where we considered forced +periodicity of perfect coverings, that is, perfect colorings with only two colors. +The structure of the paper +We begin in Section 2 by introducing the basic concepts of symbolic dynamics, cellular +automata and graphs, and defining perfect colorings formally. In Section 3 we present the +relevant algebraic concepts and the algebraic approach to multidimensional symbolic dynam- +ics, and in Section 4 we describe an algorithm to find the line polynomial factors of a given +two-dimensional Laurent polynomial. In Section 5 we consider forced periodicity of perfect +coverings, i.e., perfect colorings with only two colors and then in Section 6 we extend the re- +sults from the previous section to concern perfect colorings using arbitrarily large alphabets. +After this we prove a statement concerning forced periodicity of two-dimensional configu- +rations of low abelian complexity in Section 7. In Section 8 we consider some algorithmic +questions concerning perfect colorings. +2 + +2 +Preliminaries +Basics on symbolic dynamics +Let us review briefly some basic concepts of symbolic dynamics relevant to us. For a reference +see e.g. [4, 19, 21]. Although our results concern mostly two-dimensional configurations, we +state our definitions in an arbitrary dimension. +Let A be a finite set (the alphabet) and let d be a positive integer (the dimension). A +d-dimensional configuration over A is a coloring of the infinite grid Zd using colors from +A, that is, an element of AZd – the d-dimensional configuration space over the alphabet A. +We denote by cu = c(u) the symbol or color that a configuration c ∈ AZd has in cell u. +The translation τ t by a vector t ∈ Zd shifts a configuration c such that τ t(c)u = cu−t for all +u ∈ Zd. A configuration c is t-periodic if τ t(c) = c, and it is periodic if it is t-periodic for some +non-zero t ∈ Zd. Moreover, we say that a configuration is periodic in direction v ∈ Qd \ {0} +if it is kv-periodic for some k ∈ Z. A d-dimensional configuration c is strongly periodic +if it has d linearly independent vectors of periodicity. A strongly periodic configuration is +periodic in every rational direction. Two-dimensional strongly periodic configurations are +called two-periodic. +A finite pattern is an assignment of symbols on some finite shape D ⊆ Zd, that is, an +element of AD. In particular, the finite patterns in AD are called D-patterns. Let us denote +by A∗ the set of all finite patterns over A where the dimension d is known from the context. +We say that a finite pattern p ∈ AD appears in a configuration c ∈ AZd or that c contains p +if τ t(c)|D = p for some t ∈ Zd. For a fixed shape D, the set of all D-patterns of c is the set +LD(c) = {τ t(c)|D | t ∈ Zd} and the set of all finite patterns of c is denoted by L(c) which is +called the language of c. For a set S ⊆ AZd of configurations we define LD(S) and L(S) as +the unions of LD(c) and L(c) over all c ∈ S, respectively. +The pattern complexity P(c, D) of a configuration c ∈ AZd with respect to a shape D +is the number of distinct D-patterns that c contains. For any a ∈ A we denote by |p|a the +number of occurrences of the color a in a finite pattern p. Two finite patterns p, q ∈ AD are +called abelian equivalent if |p|a = |q|a for all a ∈ A, that is, if the number of occurrences of +each color is the same in both p and q. The abelian complexity A(c, D) of a configuration +c ∈ AZ2 with respect to a finite shape D is the number of different D-patterns in c up to +abelian equivalence [30]. Clearly A(c, D) ≤ P(c, D). We say that c has low complexity with +respect to D if +P(c, D) ≤ |D| +and that c has low abelian complexity with respect to D if +A(c, D) = 1. +The configuration space AZd can be made a compact topological space by endowing A +with the discrete topology and considering the product topology it induces on AZd – the +prodiscrete topology. This topology is induced by a metric where two configurations are close +if they agree on a large area around the origin. So, AZd is a compact metric space. +3 + +A subset S ⊆ AZd of the configuration space is a subshift if it is topologically closed and +translation-invariant meaning that if c ∈ S, then for all t ∈ Zd also τ t(c) ∈ S. Equivalently, +subshifts can be defined using forbidden patterns: Given a set F ⊆ A∗ of forbidden finite +patterns, the set +XF = {c ∈ AZd | L(c) ∩ F = ∅} +of configurations that avoid all forbidden patterns is a subshift. Moreover, every subshift is +obtained by forbidding some set of finite patterns. If F ⊆ A∗ is finite, then we say that XF +is a subshift of finite type (SFT). +The orbit of a configuration c is the set O(c) = {τ t(c) | t ∈ Zd} of its every translate. +The orbit closure O(c) is the topological closure of its orbit under the prodiscrete topology. +The orbit closure of a configuration c is the smallest subshift that contains c. It consists of +all configurations c′ such that L(c′) ⊆ L(c). +Cellular automata +Let us describe briefly an old result of cellular automata theory that we use in Section 6. +See [13] for a more thorough survey on the topic. +A d-dimensional cellular automaton or a CA for short over a finite alphabet A is a map +F : AZd −→ AZd determined by a neighborhood vector N = (t1, . . . , tn) and a local rule +f : An −→ A such that +F(c)(u) = f(c(u + t1), . . . , c(u + tn)). +A CA is additive or linear if its local rule is of the form +f(x1, . . . , xn) = a1x1 + . . . + anxn +where a1, . . . , an ∈ R are elements of some finite ring R and A is an R-module. +In Section 6 we consider the surjectivity of cellular automata and use a classic result called +the Garden-of-Eden theorem proved by Moore and Myhil that gives a characterization for +surjectivity in terms of injectivity on “finite” configurations. Two configurations c1 and c2 +are called asymptotic if the set diff(c1, c2) = {u | c1(u) ̸= c2(u)} of cells where they differ is +finite. A cellular automaton F is pre-injective if F(c1) ̸= F(c2) for any distinct asymptotic +configurations c1 and c2. Clearly injective CA are pre-injective. The Garden-of-Eden theorem +states that pre-injectivity of a CA is equivalent to surjectivity: +Theorem (Garden-of-Eden theorem, [23, 25]). A CA is surjective if and only if it is pre- +injective. +In the one-dimensional setting the Garden-of-Eden theorem yields the following corollary: +Corollary. For a one-dimensional surjective CA every configuration has only a finite number +of pre-images. +4 + +Figure 1: The relative 2-neighborhoods of the square grid, the triangular grid and the king +grid, respectively. +Graphs +In this paper we consider graphs that are simple, undirected and connected. A graph G +that has vertex set V and edge set E is denoted by G = (V, E). The distance d(u, v) of +two vertices u ∈ V and v ∈ V of a graph G = (V, E) is the length of a shortest path +between them in G. +The r-neighborhood of u ∈ V in a graph G = (V, E) is the set +Nr(u) = {v ∈ V | d(v, u) ≤ r}. The graphs we consider has vertex set V = Z2 and a +translation invariant edge set E ⊆ {{u, v} | u, v ∈ Z2, u ̸= v}. This implies that for all +r and for any two points u ∈ Z2 and v ∈ Z2 their r-neighborhoods are the same up to +translation, that is, Nr(u) = Nr(v) + u − v. Moreover, we assume that all the vertices of +G have only finitely many neighbors, i.e., we assume that the degree of G is finite. We call +these graphs two-dimensional (infinite) grid graphs or just (infinite) grids. In a grid graph +G, let us call the r-neighborhood of 0 the relative r-neighborhood of G since it determines +the r-neighborhood of any vertex in G. Indeed, for all u ∈ Z2 we have Nr(u) = Nr + u +where Nr is the relative r-neighborhood of G. +Given the edge set of a grid graph, the +relative r-neighborhood is determined for every r. We specify three 2-dimensional infinite +grid graphs: +• The square grid is the infinite grid graph (Z2, ES) with +ES = {{u, v} | u − v ∈ {(±1, 0), (0, ±1)}}. +• The triangular grid is the infinite grid graph (Z2, ET ) with +ET = {{u, v} | u − v ∈ {(±1, 0), (0, ±1), (1, 1), (−1, −1)}}. +• The king grid is the infinite grid graph (Z2, EK) with +EK = {{u, v} | u − v ∈ {(±1, 0), (0, ±1), (±1, ±1)}}. +The relative 2-neighborhoods of these grid graphs are pictured in Figure 1. +5 + +Perfect colorings +Let A = {a1, . . . , an} be a finite alphabet of n colors and let D ⊆ Zd be a finite shape. A +configuration c ∈ AZd is a perfect coloring with respect to D ⊆ Zd or a D-perfect coloring if for +all i, j ∈ {1, . . . , n} there exist numbers bij such that for all u ∈ Zd with cu = aj the number +of occurrences of color ai in the D-neighborhood of u, i.e., in the pattern c|u+D is exactly +bij. The matrix of a D-perfect coloring c is the matrix B = (bij)n×n where the numbers bij +are as above. A D-perfect coloring with matrix B is called a (perfect) (D, B)-coloring. Any +D-perfect coloring is called simply a perfect coloring. In other words, a configuration is a +perfect coloring if the number of cells of a given color in the given neighborhood of a vertex +u depends only on the color of u. +Perfect colorings are defined also for arbitrary graphs G = (V, E). +Again, let A = +{a1, . . . , an} be a finite set of n colors. A vertex coloring ϕ: V → A of G is an r-perfect +coloring with matrix B = (bij)n×n if the number of vertices of color ai in the r-neighborhood +of a vertex of color aj is exactly bij. Clearly if G is a translation invariant graph with vertex +set Zd, then the r-perfect colorings of G are exactly the D-perfect colorings in AZd where D +is the relative r-neighborhood of the graph G. +3 +Algebraic concepts +We review the basic concepts and some results relevant to us concerning an algebraic ap- +proach to multidimensional symbolic dynamics introduced and studied in [17]. See also [14] +for a short survey of the topic. +Let c ∈ AZd be a d-dimensional configuration. The power series presenting c is the formal +power series +c(X) = c(x1, . . . , xd) = +� +u=(u1,...,ud)∈Zd +cuxu1 +1 · · ·xud +d = +� +u∈Zd +cuXu +in d variables X = (x1, . . . , xd). We denote the set of all formal power series in d variables +X = (x1, . . . , xd) over a domain M by M[[X±1]] = M[[x±1 +1 , . . . , x±1 +d ]]. If d = 1 or d = 2, +we denote x = x1 and y = x2. +A power series is finitary if it has only finitely many +distinct coefficients and integral if its coefficients are all integers, i.e., if it belongs to the set +Z[[X±1]]. A configuration is always presented by a finitary power series and a finitary power +series always presents a configuration. So, from now on we may call any finitary power series +a configuration. +We consider also Laurent polynomials which we may call simply just polynomials. We +denote the set of Laurent polynomials in d variables X = (x1, . . . , xd) over a ring R by +R[X±1] = R[x±1 +1 , . . . , x±1 +d ]. The term “proper” is used when we talk about proper (i.e., +non-Laurent) polynomials and denote the proper polynomial ring over R by R[X] as usual. +We say that two Laurent polynomials have no common factors if all their common factors +are units in the polynomial ring under consideration and that they have a common factor +if they have a non–unit common factor. For example, in C[X±1] two polynomials have no +6 + +common factors if all their common factors are constants or monomials, and two proper +polynomials in C[X] have no common factors if all their common factors are constants. The +support of a power series c = c(X) = � +u∈Zd cuXu is the set supp(c) = {u ∈ Zd | cu ̸= 0}. +Clearly a polynomial is a power series with a finite support. The kth dilation of a polynomial +f(X) is the polynomial f(Xk). See Figure 2 for an illustration of dilations. +Figure 2: The supports of the polynomial f(X) = 1 + x−1y−1 + x−1y1 + x1y−1 + x1y1 and +its dilations f(X2) and f(X3). +The xi-resultant Resxi(f, g) of two proper polynomials f, g ∈ R[x1, . . . , xd] is the deter- +minant of the Sylvester matrix of f and g with respect to variable xi. We omit the details +which the reader can check from [6], and instead we consider the resultant Resxi(f, g) ∈ +R[x1, . . . , xi−1, xi+1, . . . , xd] for every i ∈ {1, . . . , d} as a certain proper polynomial that has +the following two properties: +• Resxi(f, g) is in the ideal generated by f and g, i.e., there exist proper polynomials h +and l such that +hf + lg = Resxi(f, g). +• If two proper polynomials f and g have no common factors in R[x1, . . . , xd], then +Resxi(f, g) ̸= 0. +Let R be a ring and M a (left) R-module. The formal product of a polynomial f = +f(X) = �m +i=1 aiXui ∈ R[X±1] and a power series c = c(X) = � +u∈Zd cuXu ∈ M[X±1] is +well-defined as the formal power series +fc = f(X)c(X) = +� +u∈Zd +(fc)uXu ∈ M[X±1] +where +(fc)u = +m +� +i=1 +aicu−ui. +We say that a polynomial f = f(X) annihilates (or is an annihilator of ) a power series +c = c(X) if fc = 0, that is, if their product is the zero power series. +In a typical setting, we assume that A ⊆ Z and hence consider any configuration c ∈ AZd +as a finitary and integral power series c(X). Since multiplying c(X) by the monomial Xu +7 + +produces the power series presenting the translation τ u(c) of c by u, we have that c is +u-periodic if and only if c(X) is annihilated by the difference polynomial Xu − 1. (By a +difference polynomial we mean a polynomial Xu − 1 for any u ̸= 0.) This means that it is +natural to consider multiplication of c by polynomials in C[X±1]. However, note that the +product of c and a polynomial f ∈ C[X±1] may not be integral, but it is still finitary, hence a +configuration. We say that a polynomial f periodizes (or is a periodizer of ) a configuration +c if fc is strongly periodic, that is, periodic in d linearly independent directions. We denote +the set of all periodizers with complex coefficients of a configuration c by Per(c) which is +an ideal of C[X±1] and hence we call it the periodizer ideal of c. Note that annihilators +are periodizers. Note also that if c has a periodizer f, then (Xu − 1)f is an annihilator +of c for some u. Thus, c has a non-trivial (= non-zero) annihilator if and only if it has a +non-trivial periodizer. The following theorem states that if a configuration has a non-trivial +periodizer, then it has in fact an annihilator of a particular simple form – a product of +difference polynomials. +Theorem 1 ([17]). Let c ∈ Z[[X±1]] be a configuration in any dimension and assume that +it has a non-trivial periodizer. Then there exist m ≥ 1 and pairwise linearly independent +vectors t1, . . . , tm such that +(Xt1 − 1) · · ·(Xtm − 1) +annihilates c. +A line polynomial is a polynomial whose support contains at least two points and the +points of the support lie on a unique line. In other words, a polynomial f is a line polynomial +if it is not a monomial and there exist vectors u, v ∈ Zd such that supp(f) ⊆ u + Qv. In +this case we say that f is a line polynomial in direction v. We say that non-zero vectors +v, v′ ∈ Zd are parallel if v′ ∈ Qv, and clearly then a line polynomial in direction v is also a +line polynomial in any parallel direction. A vector v ∈ Zd is primitive if its components are +pairwise relatively prime. If v is primitive, then Qv ∩ Zd = Zv. For any non-zero v ∈ Zd +there exists a parallel primitive vector v′ ∈ Zd. Thus, we may assume the vector v in the +definition of a line polynomial f to be primitive so that supp(f) ⊆ u + Zv. In the following +our preferred presentations of directions are in terms of primitive vectors. +Any line polynomial φ in a (primitive) direction v can be written uniquely in the form +φ = Xu(a0 + a1Xv + . . . + anXnv) = Xu(a0 + a1t + . . . + antn) +where u ∈ Zd, n ≥ 1, a0 ̸= 0, an ̸= 0 and t = Xv. Let us call the single variable proper +polynomial a0 + a1t + . . . + antn ∈ C[t] the normal form of φ. Moreover, for a monomial +aXu we define its normal form to be a. So, two line polynomials in the direction v have the +same normal form if and only if they are the same polynomial up to multiplication by Xu, +for some u ∈ Zd. +Difference polynomials are line polynomials and hence the annihilator provided by The- +orem 1 is a product of line polynomials. Annihilation by a difference polynomial means +periodicity. +More generally, annihilation of a configuration c by a line polynomial in a +primitive direction v can be understood as the annihilation of the one-dimensional v-fibers +8 + +� +k∈Z cu+kvXu+kv of c in direction v, and since annihilation in the one-dimensional setting +implies periodicity with a bounded period, we conclude that a configuration is periodic if +and only if it is annihilated by a line polynomial. It is known that if c has a periodizer with +line polynomial factors in at most one primitive direction, then c is periodic: +Theorem 2 ([18]). Let c ∈ Z[[x±1, y±1]] be a two-dimensional configuration and let f be a +periodizer of c. Then the following conditions hold. +• If f does not have any line polynomial factors, then c is two-periodic. +• If all line polynomial factors of f are in the same primitive direction, then c is periodic +in this direction. +Proof sketch. The periodizer ideal Per(c) = {g ∈ C[x±1, y±1] | gc is two-periodic} of c is a +principal ideal generated by a polynomial g = φ1 · · · φm where φ1, . . . , φm are line polynomials +in pairwise non-parallel directions [18]. Because f ∈ Per(c), we know that g divides f. If +f does not have any line polynomial factors, then g = 1 and hence c = gc is two-periodic. +If f has line polynomial factors, and they are in the same primitive direction v, then g is +a line polynomial in this direction. Since gc is two-periodic, it is annihilated by (Xkv − 1) +for some k ∈ Z. This implies that the configuration c is annihilated by the line polynomial +(Xkv − 1)g in direction v. We conclude that c is periodic in direction v. +The proof of the previous theorem sketched above relies heavily on the structure of the +ideal Per(c) developed in [17]. We give an alternative proof sketch that mimics the usage of +resultants in [16]: +Second proof sketch of Theorem 2. The existence of a non-trivial periodizer f implies by +Theorem 1 that c has a special annihilator g = φ1 · · ·φm that is a product of (difference) +line polynomials φ1, . . . , φm in pairwise non-parallel directions. All irreducible factors of g +are line polynomials. If f does not have any line polynomial factors, then the periodizers +f and g do not have common factors. We can assume that both are proper polynomials as +they can be multiplied by a suitable monomial if needed. Because f, g ∈ Per(c), also their +resultant Resx(f, g) ∈ Per(c), implying that c has a non-trivial annihilator containing only +variable y since Resx(f, g) ̸= 0 because f and g have no common factors. This means that c +is periodic in the vertical direction. Analogously, the y-resultant Resy(f, g) shows that c is +horizontally periodic, and hence two-periodic. +The proof for the case that f has line polynomial factors only in one direction v goes +analogously by considering φc instead of c, where φ is the greatest common line polynomial +factor of f and g in the direction v. We get that φc is two-periodic, implying that c is +periodic in direction v. +In this paper we also consider configurations over alphabets A that are finite subsets +of Zn, that is, the set of length n integer vectors, and hence study finitary formal power +series from the set Zn[[X±1]] for n ≥ 2. In particular, we call this kind of configurations +integral vector configurations. Also in this setting we consider multiplication of power series +by polynomials. The coefficients of the polynomials are n×n integer matrices, i.e., elements +9 + +of the ring Zn×n. Since Zn is a (left) Zn×n-module where we consider the vectors of Zn as +column vectors, the product of a polynomial f = f(X) ∈ Zn×n[X±1] and a power series +c = c(X) ∈ Zn[[X±1]] is well-defined. Consequently, we say that c(X) ∈ Zn[[X±1]] is t- +periodic if it is annihilated by the polynomial IXt−I and that it is periodic if it is t-periodic +for some non-zero t. +There is a natural way to present configurations over arbitrary alphabets as integral +vector configurations. Let A = {a1, . . . , an} be a finite alphabet with n elements. The vector +presentation of a configuration c ∈ AZd is the configuration c′ ∈ {e1, . . . , en}Zd (or the power +series c′(X) ∈ Zn[[X±1]] presenting c′) defined such that c′ +u = ei if and only if cu = ai. +Here by ei ∈ Zn we denote the ith natural base vector, i.e., the vector whose ith component +is 1 while all the other components are 0. Clearly c is t-periodic if and only if its vector +presentation is t-periodic. Thus, to study the periodicity of a configuration it is sufficient to +study the periodicity of its vector presentation. +The ith layer of c = � cuXu ∈ Zn[[X±1]] is the power series +layeri(c) = +� +c(i) +u Xu ∈ Z[[X±1]] +where c(i) +u is the ith component of cu. Clearly c ∈ Zn[[X±1]] is periodic in direction v if and +only if for all i ∈ {1, . . . , n} the ith layer of c is periodic in direction v. +Finally, let R be a finite ring and A a finite R-module. A polynomial f(X) = �n +i=1 aiX−ui ∈ +R[x±1 +1 , . . . , x±1 +d ] defines an additive CA that has neighborhood vector (u1, . . . , un) and local +rule f ′(y1, . . . , yn) = a1y1 + . . . + anyn. More precisely, the image of a configuration c under +the CA determined by f is the configuration fc. +4 +Finding the line polynomial factors of a given two- +variate Laurent polynomial +In this section we have d = 2 and hence all our polynomials are in two variables x and y. +The open and closed discrete half planes determined by a non-zero vector v ∈ Z2 are the +sets Hv = {u ∈ Z2 | ⟨u, v⊥⟩ > 0} and Hv = {u ∈ Z2 | ⟨u, v⊥⟩ ≥ 0}, respectively, where +v⊥ = (v2, −v1) is orthogonal to v = (v1, v2). Let us also denote by lv = Hv \ Hv the discrete +line parallel to v that goes through the origin. In other words, the half plane determined by +v is the half plane “to the right” of the line lv when moving along the line in the direction of +v. We say that a finite set D ⊆ Z2 has an outer edge in direction v if there exists a vector +t ∈ Z2 such that D ⊆ Hv + t and |D ∩ (lv + t)| ≥ 2. We call D ∩ (lv + t) the outer edge of +D in direction v. An outer edge corresponding to v means that the convex hull of D has an +edge in direction v in the clockwise orientation around D. +If a finite non-empty set D does not have an outer edge in direction v, then there exists +a vector t ∈ Z2 such that D ⊆ Hv + t and |D ∩ (lv + t)| = 1, and then we say that D has +a vertex in direction v. We call D ∩ (lv + t) the vertex of D in direction v. We say that a +polynomial f has an outer edge or a vertex in direction v if its support has an outer edge or +a vertex in direction v, respectively. Note that every non-empty finite shape D has either +10 + +an edge or a vertex in any non-zero direction. Note also that in this context directions v +and −v are not the same: a shape may have an outer edge in direction v but no outer edge +in direction −v. The following lemma shows that a polynomial can have line polynomial +factors only in the directions of its outer edges. +Lemma 3 ([16]). Let f be a non-zero polynomial with a line polynomial factor in direction +v. Then f has outer edges in directions v and −v. +Let v ∈ Z2 \ {0} be a non-zero primitive vector and let f = � fuXu be a polynomial. +Recall that a v-fiber of f is a polynomial of the form +� +k∈Z +fu+kvXu+kv +for some u ∈ Z2. Thus, a non-zero v-fiber of a polynomial is either a line polynomial or a +monomial. Let us denote by Fv(f) the set of different normal forms of all non-zero v-fibers of +a polynomial f, which is hence a finite set of one-variate proper polynomials. The following +simple example illustrates the concept of fibers and their normal forms. +Example 4. Let us determine the set Fv(f) for f = f(X) = f(x, y) = 3x + y + xy2 + xy + +x3y3 + x4y4 and v = (1, 1). By grouping the terms we can write +f = 3x + y(1 + xy) + xy(1 + x2y2 + x3y3) = X(1,0) · 3 + X(0,1)(1 + t) + X(1,1)(1 + t2 + t3) +where t = X(1,1) = xy. Hence, Fv(f) = {3, 1 + t, 1 + t2 + t3}. See Figure 3 for a pictorial +illustration. +3x +y +xy2 +xy +x3y3 +x4y4 +Figure 3: The support of f = 3x + y + xy2 + xy + x3y3 + x4y4 and its different (1, 1)-fibers. +As noticed in the example above, polynomials are linear combinations of their fibers: for any +polynomial f and any non-zero primitive vector v we can write +f = Xu1ψ1 + . . . + Xunψn +for some u1, . . . , un ∈ Z2 where ψ1, . . . , ψn ∈ Fv(f). We use this in the proof of the next +theorem. +Theorem 5. A polynomial f has a line polynomial factor in direction v if and only if the +polynomials in Fv(f) have a common factor. +11 + +Proof. For any line polynomial φ in direction v, and for any polynomial g, the v-fibers +of the product φg have a common factor φ. In other words, if a polynomial f has a line +polynomial factor φ in direction v, then the polynomials in Fv(f) have the normal form of +φ as a common factor. +For the converse direction, assume that the polynomials in Fv(f) have a common factor +φ. Then there exist vectors u1, . . . , un ∈ Z2 and polynomials φψ1, . . . , φψn ∈ Fv(f) such +that +f = Xu1φψ1 + . . . + Xunφψn. +Hence, φ is a line polynomial factor of f in direction v. +Note that Lemma 3 actually follows immediately from Theorem 5: A vertex instead of an +outer edge in direction v or −v provides a non-zero monomial v-fiber, which implies that +the polynomials in Fv(f) have no common factors. +So, to find out the line polynomial factors of f we first need to find out the possible +directions of the line polynomials, that is, the directions of the (finitely many) outer edges +of f, and then we need to check for which of these possible directions v the polynomials in +Fv(f) have a common factor. There are clearly algorithms to find the outer edges of a given +polynomial and to determine whether finitely many line polynomials have a common factor. +If such a factor exists, then by Theorem 5 the polynomial f has a line polynomial factor in +this direction. We have proved the following theorem. +Theorem 6. There is an algorithm to find the line polynomial factors of a given (Laurent) +polynomial in two variables. +5 +Forced periodicity of perfect colorings with two col- +ors +In this section we consider forced periodicity of two-dimensional perfect colorings with only +two colors. Without loss of generality we may assume that A = {a1, a2} = {0, 1} (a1 = +0, a2 = 1) and consider perfect colorings c ∈ AZ2 since the names of the colors do not matter +in our considerations. So, let c ∈ {0, 1}Z2 be a perfect coloring with respect to D ⊆ Z2 and +let B = (bij)2×2 be the matrix of c. Let us define a set C = {u ∈ Z2 | cu = 1}. This set has +the property that the neighborhood u+D of a point u contains exactly a = b21 points of color +1 if u ̸∈ C and exactly b = b22 points of color 1 if u ∈ C. In fact, C is a perfect (multiple) +covering of the infinite grid G determined by the relative neighborhood D. More precisely, +the set C is a (perfect) (D, b, a)-covering of G. This is a variant of the following definition: +in any graph a subset C of its vertex set is an (r, b, a)-covering if the number of vertices of +C in the r-neighborhood of a vertex u is a if u ̸∈ C and b if u ∈ C. See [1] for a reference. +Clearly in translation invariant graphs the (r, b, a)-coverings correspond to (D, b, a)-coverings +where D is the relative r-neighborhood of the graph. Thus, it is natural to call any perfect +coloring with only two colors a perfect covering. Note that a (D, b, a)-covering is a D-perfect +12 + +coloring with the matrix +B = +� +|D| − a +|D| − b +a +b +� +. +The following theorem by Axenovich states that “almost every” (1, b, a)-covering in the +square grid is two-periodic. +Theorem 7 ([1]). If b−a ̸= 1, then every (1, b, a)-covering in the square grid is two-periodic. +For a finite set D ⊆ Z2 we define its characteristic polynomial to be the polynomial fD(X) = +� +u∈D X−u. We denote by +1(X) the constant power series � +u∈Z2 Xu. If c ∈ {0, 1}Z2 is a +(D, b, a)-covering, then from the definition we get that fD(X)c(X) = (b − a)c(X) + a1(X) +which is equivalent to (fD(X) − (b − a)) c(X) = a1(X). Thus, if c is a (D, b, a)-covering, +then fD(X) − (b − a) is a periodizer of c. Hence, by Theorem 2 the condition that the +polynomial fD(X) − (b − a) has no line polynomial factors is a sufficient condition for forced +periodicity of a (D, b, a)-covering. Hence, we have the following corollary of Theorem 2: +Corollary 8. Let D ⊆ Z2 be a finite shape and let b and b be non-negative integers. If +g = fD −(b−a) has no line polynomial factors, then every (D, b, a)-covering is two-periodic. +Using our formulation and the algebraic approach we get a simple proof for Theorem 7: +Reformulation of Theorem 7. Let D be the relative 1-neighborhood of the square grid and +assume that b − a ̸= 1. Then every (D, b, a)-covering is two-periodic. +Proof. Let c be an arbitrary (D, b, a)-covering. The outer edges of g = fD − (b − a) = +x−1 + y−1 + 1 − (b − a) + x + y are in directions (1, 1), (−1, −1), (1, −1) and (−1, 1) and +hence by Lemma 3 any line polynomial factor of g is either in direction (1, 1) or (1, −1). For +v ∈ {(1, 1), (1, −1)} we have Fv(g) = {1 + t, 1 − (b − a)}. See Figure 4 for an illustration. +Since 1 − (b − a) is a non-trivial monomial, by Theorem 5 the periodizer g ∈ Per(c) has no +line polynomial factors and hence the claim follows by corollary 8. +We also get a similar proof for the following known result concerning the forced periodicity +perfect coverings in the square grid with radius r ≥ 2. +Theorem 9 ([29]). Let r ≥ 2 and let D be the relative r-neighborhood of the square grid. +Then every (D, b, a)-covering is two-periodic. In other words, all (r, b, a)-coverings in the +square grid are two-periodic for all r ≥ 2. +Proof. Let c be an arbitrary (D, b, a)-covering. By Lemma 3 any line polynomial factor of +g = fD − (b − a) has direction (1, 1) or (1, −1). So, assume that v ∈ {(1, 1), (1, −1)}. We +have φ1 = 1 + t + . . . + tr ∈ Fv(g) and φ2 = 1 + t + . . . + tr−1 ∈ Fv(g). See Figure 4 for +an illustration in the case r = 2. Since φ1 − φ2 = tr, the polynomials φ1 and φ2 have no +common factors, and hence by Theorem 5 the periodizer g has no line polynomial factors. +Corollary 8 gives the claim. +There are analogous results in the triangular grid, and we can prove them similarly using +Corollary 8. +13 + +1 + t +1 − (b − a) +1 + t + t2 +1 + t +1 + t + t2 + t3 + t4 +1 + t + (1 − (b − a))t2 + t3 + t4 +1 + t +1 + (1 − (b − a))t + t2 +1 + t + t2 +1 + t + t2 + t3 +Figure 4: Pictorial illustrations for the proofs of Theorems 7, 9, 10, 11 and 12. The constel- +lation on the left of the upper row illustrates the proof of Theorem 7. The constellation in +the center of the upper row illustrates the proof of Theorem 9 with r = 2. The constellation +on the right of the upper row illustrates the proof of Theorem 12 with r = 2. The constel- +lation on the left of the lower row illustrates the proof of Theorem 10. The constellation on +the right of the lower row illustrates the proof of Theorem 11 with r = 2. In each of the +constellations we have pointed out two normal forms with no common factors in Fv(g) from +the points of supp(g) for one of the outer edges v of supp(g). +Theorem 10 ([29]). Let D be the relative 1-neighborhood of the triangular grid and assume +that b−a ̸= −1. Then every (D, b, a)-covering in the triangular grid is two-periodic. In other +words, all (1, b, a)-coverings in the triangular grid are two-periodic whenever b − a ̸= −1. +Proof. Let c be an arbitrary (D, b, a)-covering. The outer edges of g = fD − (b − a) = +x−1y−1 + x−1 + y−1 + 1 − (b − a) + x + y + xy have directions (1, 1), (−1, −1), (1, 0), (−1, 0), +(0, 1) and (0, −1) and hence by Lemma 3 any line polynomial factor of g has direction (1, 1), +(1, 0) or (0, 1). So, let v ∈ {(1, 1), (1, 0), (0, 1)}. We have Fv(g) = {1+t, 1+(1−(b−a))t+t2}. +See Figure 4 for an illustration. Polynomials φ1 = 1 + t and φ2 = 1 + (1 − (b − a))t + t2 +satisfy φ2 +1 − φ2 = (1 + b − a)t. Thus, they do not have any common factors if b − a ̸= −1 and +hence by Theorem 5 the polynomial g has no line polynomial factors. The claim follows by +Corollary 8. +Theorem 11 ([29]). Let r ≥ 2 and let D be the relative r-neighborhood of the triangular +grid. Then every (D, b, a)-covering is two-periodic. In other words, every (r, b, a)-covering +in the triangular grid is two-periodic for all r ≥ 2. +Proof. Let c be an arbitrary (D, b, a)-covering. The outer edges of g = fD − (b − a) have +directions (1, 1), (−1, −1), (1, 0), (−1, 0), (0, 1) and (0, −1), and hence by Lemma 3 any line +polynomial factor of g has direction (1, 1), (1, 0) or (0, 1). So, let v ∈ {(1, 1), (1, 0), (0, 1)}. +There exists n ≥ 1 such that 1+t+. . .+tn ∈ Fv(g) and 1+t+. . .+tn+1 ∈ Fv(g). See Figure +4 for an illustration with r = 2. Since these two polynomials have no common factors, by +14 + +Theorem 5 the polynomial g has no line polynomial factors. Again, Corollary 8 yields the +claim. +If a ̸= b, then for all r ≥ 1 any (r, b, a)-covering in the king grid is two-periodic: +Theorem 12. Let r ≥ 1 be arbitrary and let D be the relative r-neighborhood of the king +grid and assume that a ̸= b. Then any (D, b, a)-covering is two-periodic. In other words, all +(r, b, a)-coverings in the king grid are two-periodic whenever a ̸= b. +Proof. Let c be an arbitrary (D, b, a)-covering. The outer edges of g = fD − (b − a) are in +directions (1, 0), (−1, 0), (0, 1) and (0, −1). Hence, by Lemma 3 any line polynomial factor +of g has direction (1, 0) or (0, 1). Let v ∈ {(1, 0), (0, 1)}. We have φ1 = 1 + t + . . . + tr−1 + +(1 − (b − a))tr + tr+1 + . . . + t2r ∈ Fv(g) and φ2 = 1 + t + . . . + t2r ∈ Fv(g). See Figure 4 for +an illustration in the case r = 2. Since φ2 − φ1 = (b − a)tr is a non-trivial monomial, φ1 and +φ2 have no common factors. Thus, by Theorem 5 the polynomial g has no line polynomial +factors and the claim follows by Corollary 8. +In the above proofs we used the fact that two Laurent polynomials in one variable have no +common factors if and only if they generate the entire ideal C[t±1], and they do this if and +only if they generate a non-zero monomial. This is known as the weak Nullstellensatz [6]. +A shape D ⊆ Z2 is convex if it is the intersection D = conv(D)∩Z2 where conv(D) ⊆ R2 +is the real convex hull of D. Above all our shapes were convex. Next we generalize the +above theorems and give a sufficient condition for forced periodicity of (D, b, a)-coverings for +convex D. +So, let D ⊆ Z2 be a finite convex shape. Any (D, b, a)-covering has a periodizer g = +fD − (b − a). As earlier, we study whether g has any line polynomial factors since if it does +not, then Corollary 8 guarantees forced periodicity. For any v ̸= 0 the set Fv(fD) contains +only polynomials φn = 1 + . . . + tn−1 for different n ≥ 1 since D is convex: if D contains two +points, then D contains every point between them. Thus, Fv(g) contains only polynomials +φn for different n ≥ 1 and, if b − a ̸= 0, it may also contain a polynomial φn0 − (b − a)tm0 +for some n0 ≥ 1 such that φn0 ∈ Fv(fD) and for some m0 ≥ 0. If b − a = 0, then g = fD +and thus Fv(g) = Fv(fD). +Two polynomials φm and φn have a common factor if and only if gcd(m, n) > 1. More gen- +erally, the polynomials φn1, . . . , φnr have a common factor if and only if d = gcd(n1, . . . , nr) > +1 and, in fact, their greatest common factor is the dth cyclotomic polynomial +� +1≤k≤d +gcd(k,d)=1 +(t − ei· 2πk +d ). +Let us introduce the following notation. For any polynomial f, we denote by F ′ +v(f) the +set of normal forms of the non-zero fibers � +k∈Z fu+kvXu+kv for all u ̸∈ Zv. In other words, +we exclude the fiber through the origin. Let us also denote fibv(f) for the normal form of +the fiber � +k∈Z fkvXkv through the origin. We have Fv(f) = F ′ +v(f)∪{fibv(f)} if fibv(f) ̸= 0 +and Fv(f) = F ′ +v(f) if fibv(f) = 0. +15 + +Applying Theorems 2 and 5 we have the following theorem that gives sufficient conditions +for every (D, b, a)-covering to be periodic for a finite and convex D. This theorem generalizes +the results proved above. In fact, they are corollaries of the theorem. The first part of the +theorem was also mentioned in [7] in a slightly different context and in a more general form. +Theorem 13. Let D be a finite convex shape, g = fD − (b − a) and let E be the set of the +outer edge directions of g. +• Assume that b − a = 0. For any v ∈ E denote dv = gcd(n1, . . . , nr) where Fv(g) = +{φn1, . . . , φnr}. If dv = 1 holds for all v ∈ E, then every (D, b, a)-covering is two- +periodic. If dv = 1 holds for all but some parallel v ∈ E, then every (D, b, a)-covering +is periodic. +• Assume that b − a ̸= 0. For any v ∈ E denote dv = gcd(n1, . . . , nr) where F ′ +v(g) = +{φn1, . . . , φnr}. If the dv’th cyclotomic polynomial and fibv(g) have no common factors +for any v ∈ E, then every (D, b, a)-covering is two-periodic. If the condition holds for +all but some parallel v ∈ E, then every (D, b, a)-covering is periodic. (Note that the +condition is satisfied, in particular, if dv = 1.) +Proof. Assume first that b − a = 0. If dv = 1 for all v ∈ E, then the v-fibers of g have +no common factors and hence by Theorem 5 g has no line polynomial factors. If dv = 1 +holds for all but some parallel v ∈ E, then all the line polynomial factors of g are in parallel +directions. Thus, the claim follows by Theorem 2. +Assume then that b − a ̸= 0. If the dv’th cyclotomic polynomial and fibv(g) have no +common factors for all v ∈ E, then by Theorem 5 g has no line polynomial factors. If the +condition holds for all but some parallel v ∈ E, then all the line polynomial factors of g are +in parallel directions. Thus, by Theorem 2 the claim holds also in this case. +6 +Forced periodicity of perfect colorings over arbitrar- +ily large alphabets +In this section we prove a theorem that gives a sufficient condition for forced periodicity of +two-dimensional perfect colorings over an arbitrarily large alphabet. As corollaries of the +theorem and theorems from the previous section we obtain conditions for forced periodicity +of perfect colorings in two-dimensional infinite grid graphs. +We start by proving some lemmas that work in any dimension. We consider the vector +presentations of perfect colorings because this way we get a non-trivial annihilator for any +such vector presentation: +Lemma 14. Let c be the vector presentation of a D-perfect coloring over an alphabet of size +n with matrix B = (bij)n×n. Then c is annihilated by the polynomial +f(X) = +� +u∈D +IX−u − B. +16 + +Remark. Note the similarity of the above annihilator to the periodizer � +u∈D X−u − (b − a) +of a (D, b, a)-covering. +Proof. Let v ∈ Zd be arbitrary and assume that cv = ej. Then (Bc)v = Bej is the jth +column of B. On the other hand, from the definition of B we have ((� +u∈D IX−u)c)v = +� +u∈D cv+u = �n +i=1 bijei which is also the jth column of B. Thus, (fc)v = 0 and hence +fc = 0 since v was arbitrary. +The following lemma shows that as in the case of integral configurations with non-trivial +annihilators, also the vector presentation of a perfect coloring has a special annihilator +which is a product of difference polynomials. By congruence of two polynomials with integer +matrices as coefficients (mod p) we mean that their corresponding coefficients are congruent +(mod p) and by congruence of two integer matrices (mod p) we mean that their corresponding +components are congruent (mod p). +Lemma 15. Let c be the vector presentation of a D-perfect coloring over an alphabet of size +n with matrix B = (bij)n×n. Then c is annihilated by the polynomial +g(X) = (IXv1 − I) · · ·(IXvm − I) +for some vectors v1, . . . , vm. +Proof. By Lemma 14 the power series c is annihilated by f(X) = � +u∈D IX−u − B. Let p +be a prime larger than ncmax where cmax is the maximum absolute value of the components +of the coefficients of c. Since the coefficients of f commute with each other, we have for any +positive integer k using the binomial theorem that +f pk = f pk(X) ≡ +� +u∈D +IX−pku − Bpk (mod p). +We have f pk(X)c(X) ≡ 0 (mod p). There are only finitely many distinct matrices Bpk (mod p). +So, let k and k′ be distinct and such that Bpk ≡ Bpk′ +(mod p). Then the coefficients of +f ′ = f pk−f pk′ +(mod p) are among I and −I. Since f pkc ≡ 0 (mod p) and f pk′ +c ≡ 0 (mod p), +also +f ′c ≡ 0 (mod p). +The components of the configuration f ′c are bounded in absolute value by ncmax. Since we +chose p larger than ncmax, this implies that +f ′c = 0. +Because f ′ = � +u∈P1 IXu − � +u∈P2 IXu for some finite subsets P1 and P2 of Zd, the +annihilation of c by f ′ is equivalent to the annihilation of every layer of c by f ′′ = � +u∈P1 Xu− +� +u∈P2 Xu. Thus, every layer of c has a non-trivial annihilator and hence by Theorem 1 every +layer of c has a special annihilator which is a product of difference polynomials. Let +g′ = (Xv1 − 1) · · ·(Xvm − 1) +17 + +be the product of all these special annihilators. Since g′ annihilates every layer of c, the +polynomial +g = (IXv1 − I) · · ·(IXvm − I) +annihilates c. +Lemma 16. Let p be a prime and let H be an additive CA over Zn +p determined by a polyno- +mial h = �k +i=0 AiXui ∈ Zn×n +p +[X±1] whose coefficients Ai commute with each other. Assume +that there exist M ∈ Zp \ {0} and matrices C0, . . . , Ck that commute with each other and +with every Ai such that +C0A0 + . . . + CkAk = M · I +holds in Zk×k +p +. Then H is surjective. +Proof. Assume the contrary that H is not surjective. By the Garden-of-Eden theorem H +is not pre-injective and hence there exist two distinct asymptotic configurations c1 and c2 +such that H(c1) = H(c2), that is, h(X)c1(X) = h(X)c2(X). +Thus, h is an annihilator +of e = c1 − c2. Without loss of generality we may assume that c1(0) ̸= c2(0), i.e., that +e(0) = v ̸= 0. Let l be such that the support supp(e) = {u ∈ Zd | e(u) ̸= 0} of e is +contained in a d-dimensional pl × . . . × pl hypercube. Note that in Zk×k +p +we have +f pl = +k +� +i=0 +Apl +i Xplui +which is also an annihilator of e. +Hence, by the choice of l we have Apl +i v = 0 for all +i ∈ {1, . . . , k}. By raising the identity +C0A0 + . . . + CkAk = M · I +to power pl and multiplying the result by the vector v from the right we get +Mpl · v = Cpl +0 Apl +0 v + . . . + Cpl +k Apl +k v = 0 + . . . + 0 = 0. +However, this is a contradiction because Mplv ̸= 0. Thus, H must be surjective as claimed. +Theorem 17. Let D ⊆ Z2 be a finite shape and assume that there exists an integer t0 such +that the polynomial fD − t = � +u∈D X−u − t has no line polynomial factors whenever t ̸= t0. +Then any D-perfect coloring with matrix B is two-periodic whenever det(B − t0I) ̸= 0. If +fD −t has no line polynomial factors for any t, then every D-perfect coloring is two-periodic. +Proof. Let c be the vector presentation of a D-perfect coloring with matrix B. By Lemmas 14 +and 15 it has two distinct annihilators: f = � +u∈D IX−u −B and g = (IXv1 −I) · · · (IXvm − +I). Let us replace I by 1 and B by a variable t and consider the corresponding integral +polynomials f ′ = � +u∈D X−u −t = fD −t and g′ = (Xv1 −1) · · ·(Xvm −1) in C[x, y, t]. Here +X = (x, y). +18 + +Without loss of generality we may assume that f ′ and g′ are proper polynomials. Indeed, +we can multiply f ′ and g′ by monomials such that the obtained polynomials f ′′ and g′′ are +proper polynomials and that they have a common factor if and only if f ′ and g′ have a +common factor. So, we may consider f ′′ and g′′ instead of f ′ and g′ if they are not proper +polynomials. +We consider the y-resultant Resy(f ′, g′) of f ′ and g′, and write +Resy(f ′, g′) = f0(t) + f1(t)x + . . . + fk(t)xk. +By the properties of resultants Resy(f ′, g′) is in the ideal generated by f ′ and g′, and it can +be the zero polynomial only if f ′ and g′ have a common factor. Since g′ is a product of +line polynomials, any common factor of f ′ and g′ is also a product of line polynomials. In +particular, if f ′ and g′ have a common factor, then they have a common line polynomial +factor. However, by the assumption f ′ has no line polynomial factors if t ̸= t0. Thus, f ′ and +g′ may have a common factor only if t = t0 and hence Resy(f ′, g′) can be zero only if t = t0. +On the other hand, Resy(f ′, g′) = 0 if and only if f0(t) = . . . = fk(t) = 0. We conclude that +gcd(f0(t), . . . , fk(t)) = (t − t0)m for some m ≥ 0. Thus, +Resy(f ′, g′) = (t − t0)m(f ′ +0(t) + f ′ +1(t)x + . . . + f ′ +k(t)xk) +where the polynomials f ′ +0(t), . . . , f ′ +k(t) have no common factors. +By the Euclidean algorithm there are polynomials a0(t), . . . , ak(t) such that +a0(t)f ′ +0(t) + . . . + ak(t)f ′ +k(t) = 1. +(1) +Moreover, the coefficients of the polynomials a0(t), . . . , ak(t) are rational numbers because +the polynomials f ′ +0(t), . . . , f ′ +k(t) are integral. Note that if f ′ has no line polynomial factors +for any t, then m = 0 and hence f ′ +i(t) = fi(t) for every i ∈ {1, . . . , k}. +Let us now consider the polynomial +(B − t0I)m(f ′ +0(B) + f ′ +1(B)x + . . . + f ′ +k(B)xk) +which is obtained from Resy(f ′, g′) by plugging back I and B in the place of 1 and t, re- +spectively. Since Resy(f ′, g′) is in the ideal generated by f ′ and g′, the above polynomial is +in the ideal generated by f and g. Thus, it is an annihilator of c because both f and g are +annihilators of c. +Assume that det(B − t0I) ̸= 0 or that m = 0. Now also +h = f ′ +0(B) + f ′ +1(B)x + . . . + f ′ +k(B)xk +is an annihilator of c. Since f ′ +0(t), . . . , f ′ +k(t) have no common factors, h is non-zero, because +otherwise it would be f ′ +0(B) = . . . = f ′ +k(B) = 0 and the minimal polynomial of B would be +a common factor of f ′ +0(t), . . . , f ′ +k(t), a contradiction. +Plugging t = B to Equation 1 we get +a0(B)f ′ +0(B) + . . . + ak(B)f ′ +k(B) = I. +19 + +Let us multiply the above equation by a common multiple M of all the denominators of the +rational numbers appearing in the equation and let us consider it (mod p) where p is a prime +that does not divide M. We obtain the following identity +a′ +0(B)f ′ +0(B) + . . . + a′ +k(B)f ′ +k(B) = M · I ̸≡ 0 (mod p) +where all the coefficients in the equation are integer matrices. +By Lemma 16 the additive CA determined by h = �k +i=0 f ′ +i(B)xi is surjective. Since h +is a polynomial in variable x only, it defines a 1-dimensional CA H which is surjective and +which maps every horizontal fiber of c to 0. Hence, every horizontal fiber of c is a pre-image +of 0. Let c′ be a horizontal fiber of c. The Garden-of-Eden theorem implies that 0 has finitely +many, say N, pre-images under H. Since also every translation of c′ is a pre-image of 0, we +conclude that c′ = τ i(c′) for some i ∈ {0, . . . , N − 1}. Thus, (N − 1)! is a common period of +all the horizontal fibers of c and hence c is horizontally periodic. +Repeating the same argumentation for the x-resultant of f ′ and g′ we can show that c is +also vertically periodic. Thus, c is two-periodic. +As corollaries of the above theorem and theorems from the previous section, we obtain new +proofs for forced periodicity of perfect colorings in the square and the triangular grids, and +a new result for forced periodicity of perfect colorings in the king grid: +Corollary 18 ([29]). Let D be the relative 1-neighborhood of the square grid. Then any D- +perfect coloring with matrix B is two-periodic whenever det(B − I) ̸= 0. In other words, any +1-perfect coloring with matrix B in the square grid is two-periodic whenever det(B − I) ̸= 0. +Proof. In our proof of Theorem 7 it was shown that the polynomial fD − t has no line +polynomial factors if t ̸= 1. +Thus, by Theorem 17 any (D, B)-coloring is two-periodic +whenever det(B − I) ̸= 0. +Corollary 19 ([29]). Let D be the relative 1-neighborhood of the triangular grid. +Then +any D-perfect coloring with matrix B is two-periodic whenever det(B + I) ̸= 0. In other +words, any 1-perfect coloring with matrix B in the triangular grid is two-periodic whenever +det(B + I) ̸= 0. +Proof. In the proof of Theorem 10 it was shown that the polynomial fD − t has no line +polynomial factors if t ̸= −1. Thus, by Theorem 17 any (D, B)-coloring is two-periodic +whenever det(B + I) ̸= 0. +Corollary 20 ([29]). Let r ≥ 2 and let D be the relative r-neighborhood of the square grid. +Then every D-perfect coloring is two-periodic. In other words, any r-perfect coloring in the +square grid is two-periodic for all r ≥ 2. +Proof. In the proof of Theorem 9 it was shown that the polynomial fD − t has no line +polynomial factors for any t. Thus, by Theorem 17 every D-perfect coloring is two-periodic. +20 + +Corollary 21 ([29]). Let r ≥ 2 and let D be the relative r-neighborhood of the triangular +grid. Then every D-perfect coloring is two-periodic. In other words, any r-perfect coloring +in the triangular grid is two-periodic for all r ≥ 2. +Proof. In the proof of Theorem 11 it was shown that the polynomial fD − t has no line +polynomial factors for any t. Thus, by Theorem 17 every D-perfect coloring is two-periodic. +Corollary 22. Let r ≥ 1 and let D be the relative r-neighborhood of the king grid. Then +every D-perfect coloring with matrix B is two-periodic whenever det(B) ̸= 0. In other words, +every r-perfect coloring with matrix B in the king grid is two-periodic whenever det(B) ̸= 0. +Proof. In the proof of Theorem 12 we showed that the polynomial fD − t has no line poly- +nomial factors if t ̸= 0. Thus, by Theorem 17 any (D, B)-coloring is two-periodic whenever +det(B) ̸= 0. +Remark. Note that the results in Corollaries 18, 19, 20 and 21 were stated and proved in [29] +in a slightly more general form. Indeed, in [29] it was proved that if a configuration c ∈ AZ2 +is annihilated by +� +u∈D +IX−u − B +where B ∈ Zn×n is an arbitrary integer matrix whose determinant satisfies the conditions +in the four corollaries and D is as in the corollaries, then c is necessarily periodic. This +kind of configuration was called a generalized centered function. However, in Lemma 14 we +proved that the vector presentation of any D-perfect coloring with matrix B is annihilated +by this polynomial, that is, we proved that the vector presentation of a perfect coloring +is a generalized centered function. By analyzing the proof of Theorem 17 we see that the +theorem holds also for generalized centered functions and hence the corollaries following it +hold also for generalized centered functions, and thus we have the same results as in [29]. +7 +Forced periodicity of configurations of low abelian +complexity +In this section we prove a statement concerning forced periodicity of two-dimensional con- +figurations of low abelian complexity which generalizes a result in [7]. In fact, as in [7] we +generalize the definition of abelian complexity from finite patterns to polynomials and prove +a statement of forced periodicity under this more general definition of abelian complexity. +Let c ∈ {e1, . . . , en}Zd and let D ⊆ Zd be a finite shape. +Consider the polynomial +f = I · fD(X) = � +u∈D IX−u ∈ Zn×n[X±1]. The ith coefficient of (fc)v = � +u∈D I · cv+u +tells the number of cells of color ei in the D-neighborhood of v in c and hence the abelian +complexity of c with respect to D is exactly the number of distinct coefficients of fc. +21 + +More generally, we define the abelian complexity A(c, f) of an integral vector con- +figuration c ∈ AZd where A is finite set of integer vectors with respect to a polynomial +f ∈ Zn×n[X±1] as +A(c, f) = |{(fc)v | v ∈ Zd}|. +This definition can be extended to integral configurations and polynomials. +Indeed, we +define the abelian complexity A(c, f) of a configuration c ∈ AZd where A ⊆ Z with respect +to a polynomial f = � fiXui ∈ Z[X±1] to be the abelian complexity A(c′, f ′) of the vector +presentation c′ of c with respect to the polynomial f ′ = I·f = � fi·I·Xui. Consequently, we +say that c has low abelian complexity with respect to a polynomial f if A(c, f) = 1. Clearly +this definition is consistent with the definition of low abelian complexity of a configuration +with respect to a finite shape since if c is an integral configuration, then A(c, D) = 1 if and +only if A(c, fD) = 1, and if c is an integral vector configuration, then A(c, D) = 1 if and only +if A(c, I · fD) = 1. +We study forced periodicity of two-dimensional configurations of low abelian complexity. +Note that a configuration of low abelian complexity is not necessarily periodic. Indeed, in +[30] it was shown that there exist non-periodic two-dimensional configurations that have +abelian complexity A(c, D) = 1 for some finite shape D. However, in [7] it was shown that +if A(c, f) = 1 and if the polynomial f has no line polynomial factors, then c is two-periodic +assuming that the support of f is convex. The following theorem strengthens this result and +shows that the convexity assumption of the support of the polynomial is not needed. We +obtain this result as a corollary of Theorem 2. +Theorem 23. Let c be a two-dimensional integral configuration over an alphabet of size n +and assume that it has low abelian complexity with respect to a polynomial f ∈ Z[x±1, y±1]. If +f has no line polynomial factors, then c is two-periodic. If f has line polynomial factors in a +unique primitive direction v, then c is v-periodic. Thus, if fD has no line polynomial factors +or its line polynomial factors are in a unique primitive direction, then any configuration that +has low abelian complexity with respect to D is two-periodic or periodic, respectively. +Proof. By the assumption that A(c, f) = 1 we have f ′c′ = c0 +1 for some c0 ∈ Zn where c′ +is the vector presentation of c and f ′ = I · f. Thus, f periodizes every layer of c′. If f has +no line polynomial factors, then by Theorem 2 every layer of c′ is two-periodic and hence c′ +is two-periodic. If f has line polynomial factors in a unique primitive direction v, then by +Theorem 2 every layer of c′ is v-periodic and hence also c′ is v-periodic. Since c is periodic +if and only if its vector presentation c′ is periodic, the claim follows. +Remark. In [7] a polynomial f ∈ Z[X±1] is called abelian rigid if an integral configuration c +having low abelian complexity with respect to f implies that c is strongly periodic. In the +above theorem we proved that if a polynomial f ∈ Z[x±1, y±1] has no line polynomial factors +then it is abelian rigid. Also, the converse holds as proved in [7], that is, if a polynomial +f ∈ Z[x±1, y±1] has a line polynomial factor then it is not abelian rigid. This means that +if f has a line polynomial factor then there exists a configuration which is not two-periodic +but has low abelian complexity with respect to f. In fact this direction holds for all d, not +just for d = 2 as reported in [7]. +22 + +In the following example we introduce an open problem related to configurations of low +abelian complexity. +Example 24 (Periodic tiling problem). This example concerns translational tilings by a +single tile. In this context by a tile we mean any finite subset F ⊆ Zd and by a tiling by the +tile F we mean such subset C ⊆ Zd that every point of the grid Zd has a unique presentation +as a sum of an element of F and an element of C. Presenting the tiling C as its indicator +function we obtain a d-dimensional binary configuration c ∈ {0, 1}Zd defined by +cu = +� +1, if u ∈ C +0, if u ̸∈ C +. +The configuration c has exactly |F| different patterns of shape −F, namely the patterns +with exactly one symbol 1. In other words, it has low complexity with respect to −F. Let +f = fF = � +u∈F X−u be the characteristic polynomial of F. Since C is a tiling by F, we +have fc = +1. In fact, c has low abelian complexity with respect to f and −F. Thus, by +Theorem 23 any tiling by F ⊂ Z2 is two-periodic if fF has no line polynomial factors. +The periodic tiling problem claims that if there exists a tiling by a tile F ⊆ Zd, then there +exists also a periodic tiling by F [20, 31]. By a simple pigeonholing argument it can be seen +that in dimension d = 1 all translational tilings by a single tile are periodic and hence the +periodic tiling problem holds in dimension 1 [26]. For d ≥ 2 the conjecture is much trickier +and only recently it was proved by Bhattacharya that it holds for d = 2 [3]. In [9] it was +presented a slightly different proof in the case d = 2 with some generalizations. For d ≥ 3 +the conjecture is still partly open. However, very recently it has been proved that for some +sufficiently large d the periodic tiling conjecture is false [10]. +8 +Algorithmic aspects +All configurations in a subshift are periodic, in particular, if there are no configurations in +the subshift at all! It is useful to be able to detect such trivial cases. +The set +S(D, b, a) = {c ∈ {0, 1}Z2 | (fD − (b − a))c = a1(X)} +of all (D, b, a)-coverings is an SFT for any given finite shape D and non-negative integers +b and a. Hence, the question whether there exist any (D, b, a)-coverings for a given neigh- +borhood D and covering constants b and a is equivalent to the question whether the SFT +S(D, b, a) is non-empty. The question of emptiness of a given SFT is undecidable in general, +but if the SFT is known to be not aperiodic, then the problem becomes decidable as a classic +argumentation by Hao Wang shows: +Lemma 25 ([32]). If an SFT is either the empty set or it contains a strongly periodic config- +uration, then its emptiness problem is decidable, that is, there is an algorithm to determine +whether there exist any configurations in the SFT. +23 + +In particular, if g = fD − (b − a) has line polynomial factors in at most one direction, then +the question whether there exist any (D, b, a)-coverings is decidable: +Theorem 26. Let a finite D ⊆ Z2 and non-negative integers b and a be given such that +the polynomial g = fD − (b − a) ∈ Z[x±1, y±1] has line polynomial factors in at most one +primitive direction. Then there exists an algorithm to determine whether there exist any +(D, b, a)-coverings. +Proof. Let S = S(D, b, a) be the SFT of all (D, b, a)-coverings. Since g has line polynomial +factors in at most one primitive direction, by Theorem 2 every element of S is periodic. +Any two-dimensional SFT that contains periodic configurations contains also two-periodic +configurations. Thus, S is either empty or contains a two-periodic configuration and hence +by Lemma 25 there is an algorithm to determine whether S is non-empty. +One may also want to design a perfect (D, b, a)-covering for given D, b and a. This can be +effectively done under the assumptions of Theorem 26: As we have seen, if S = S(D, b, a) is +non-empty, it contains a two-periodic configuration. For any two-periodic configuration c it is +easy to check if c contains a forbidden pattern. By enumerating two-periodic configurations +one-by-one one is guaranteed to find eventually one that is in S. +If the polynomial g has no line polynomial factors, then the following stronger result +holds: +Theorem 27. If the polynomial g = fD − (b − a) has no line polynomial factors for given +finite shape D ⊆ Z2 and non-negative integers b and a, then the SFT S = S(D, b, a) is finite. +One can then effectively construct all the finitely many elements of S. +The proof of the first part of above theorem relies on the fact that a two-dimensional subshift +is finite if and only if it contains only two-periodic configurations +[2]. +If g has no line +polynomial factors, then every configuration it periodizes (including every configuration in +S) is two-periodic by Theorem 2, and hence S is finite. The second part of the theorem, i.e., +the fact that one can effectively produce all the finitely many elements of S holds generally +for finite SFTs in any dimension: +Lemma 28. Given a finite F ⊆ A∗ such that XF is finite, one can effectively construct the +elements of XF. +Proof. Given a finite F ⊆ A∗ and a pattern p ∈ AD, assuming that strongly periodic +configurations are dense in XF, one can effectively check whether p ∈ L(XF). Indeed, we +have a semi-algorithm for the positive instances that guesses a strongly periodic configuration +c and verifies that c ∈ XF and p ∈ L(c). A semi-algorithm for the negative instances exists +for any SFT XF and is a standard compactness argument: guess a finite E ⊆ Zd such that +D ⊆ E and verify that every q ∈ AE such that q|D = p contains a forbidden subpattern. +Consequently, given finite F, G ⊆ A∗, assuming that strongly periodic configurations are +dense in XF and XG, one can effectively determine whether XF = XG. Indeed, XF ⊆ XG +if and only if no p ∈ G is in L(XF), a condition that we have shown above to be decidable. +Analogously we can test XG ⊆ XF. +24 + +Finally, let a finite F ⊆ A∗ be given such that XF is known to be finite. All elements +of XF are strongly periodic so that strongly periodic configurations are certainly dense in +XF. One can effectively enumerate all finite sets P of strongly periodic configurations. For +each P that is translation invariant (and hence a finite SFT) one can construct a finite set +G ⊆ A∗ of forbidden patterns such that XG = P. As shown above, there is an algorithm +to test whether XF = XG = P. Since XF is finite, a set P is eventually found such that +XF = P. +Let us now turn to the more general question of existence of perfect colorings over alpha- +bets of arbitrary size. Let D ⊆ Z2 be a finite shape and let B be an n × n integer matrix. +To determine whether there exist any (D, B)-colorings is equivalent to asking whether the +SFT +S(D, B) = {c ∈ {e1, . . . , en}Z2 | gc = 0} +is non-empty where g = � +u∈D IX−u−B since it is exactly the set of the vector presentations +of all (D, B)-colorings. +Theorem 29. Let a finite shape D ⊆ Z2, a non-negative integer matrix B and an integer t0 +be given such that the polynomial fD(x, y) − t ∈ Z[x±1, y±1] has no line polynomial factors +whenever t ̸= t0 and det(B − t0I) ̸= 0. Then there are only finitely many (D, B)-colorings +and one can effectively construct them. In particular, there is an algorithm to determine +whether there exist any (D, B)-colorings. +Proof. Let S = S(D, B) be the SFT of the vector presentations of all (D, B)-colorings. By +Theorem 17 all elements of S are two-periodic. Hence, S is finite, and the claim follows by +Lemma 28. +Corollaries 18, 19, 20, 21 and 22 together with above theorem yield the following corollary. +Corollary 30. The following decision problems are decidable for a given matrix B satisfying +the given conditions. +• The existence of (D, B)-colorings where D is the relative 1-neighborhood of the square +grid and det(B − I) ̸= 0. +• The existence of (D, B)-colorings where D is the relative 1-neighborhood of the trian- +gular grid and det(B + I) ̸= 0. +• The existence of (D, B)-colorings where D is the relative r-neighborhood of the square +grid and B is arbitrary. +• The existence of (D, B)-colorings where D is the relative r-neighborhood of the trian- +gular grid and B is arbitrary. +• The existence of (D, B)-colorings where D is the relative r-neighborhood of the king +grid and det(B) ̸= 0. +25 + +Theorem 31. Given a polynomial f in two variables with line polynomial factors in at +most one parallel direction there is an algorithm to determine whether there exist any two- +dimensional configurations over an alphabet of size n that have low abelian complexity with +respect to f. In fact, there are only finitely many such configurations and one can effectively +construct all of them. +Proof. The set {c ∈ {e1, . . . , en}Z2 | Ifc = 0} of the vector presentations of all configurations +over an alphabet of size n with low abelian complexity with respect to f is an SFT. By +Theorem 23 it contains only two-periodic configurations and hence it is finite. Thus, by +Lemma 28 we have the claim. +9 +Conclusions +We studied two-dimensional perfect colorings and proved a general condition (Theorem 17) +for their forced periodicity using an algebraic approach to multidimensional symbolic dy- +namics. As corollaries of this theorem we obtained new proofs for known results of forced +periodicity in the square and the triangular grid and a new result in the king grid. More- +over, we generalized a statement of forced periodicity of two-dimensional configurations of +low abelian complexity. Also, some observations of algorithmic decidability were made in +the context of forced periodicity. +All our results of forced periodicity of perfect colorings used Theorem 2 and hence con- +cerned only two-dimensional configurations. However, a d-dimensional version of Theorem 2 +exists [15], and so we wonder whether an analogous result to Theorem 17 exists that would +give a sufficient condition for forced periodicity of d-dimensional perfect colorings for arbi- +trary dimension d. Note that clearly every one-dimensional perfect coloring is necessarily +periodic. +References +[1] M. A. Axenovich. On multiple coverings of the infinite rectangular grid with balls of +constant radius. Discrete Mathematics, 268(1):31 – 48, 2003. +[2] A. Ballier, B. Durand, and E. Jeandal. Structural aspects of tilings. In Susanne Albers +and Pascal Weil, editors, 25th International Symposium on Theoretical Aspects of Com- +puter Science, volume 1 of Leibniz International Proceedings in Informatics (LIPIcs), +pages 61–72, Dagstuhl, Germany, 2008. Schloss Dagstuhl–Leibniz-Zentrum fuer Infor- +matik. +[3] S. Bhattacharya. Periodicity and decidability of tilings of Z2. American Journal of +Mathematics, 142, 02 2016. +[4] T. Ceccherini-Silberstein and M. Coornaert. Cellular Automata and Groups. Springer +Monographs in Mathematics. Springer Berlin Heidelberg, 2010. +26 + +[5] G. Cohen, I. Honkala, S. Litsyn, and A. Lobstein. Covering Codes. Elsevier, 1997. +[6] D. A. Cox, J. Little, and D. O’Shea. Ideals, Varieties, and Algorithms: An Introduction +to Computational Algebraic Geometry and Commutative Algebra. Springer, 2015. +[7] N. Geravker and S. A. Puzynina. Abelian Nivat’s conjecture for non-rectangular pat- +terns. arXiv:2111.04690, December 2021. +[8] C. Godsil. Equitable partitions. Paul Erd¨os is Eighty Vol. 1, pages 173–192, 1993. +[9] R. Greenfeld and T. Tao. The structure of translational tilings in Zd. Discrete Analysis, +2021. +[10] R. Greenfeld and T. Tao. A counterexample to the periodic tiling conjecture, 2022. +[11] T. W. Haynes, S. Hedetniemi, and P. Slater. Fundamentals of Domination in Graphs. +CRC Press, 1 edition, 1997. +[12] E. Heikkil¨a, P. Herva, and J. Kari. On perfect coverings of two-dimensional grids. In +Volker Diekert and Mikhail Volkov, editors, Developments in Language Theory, pages +152–163, Cham, 2022. Springer International Publishing. +[13] J. Kari. Theory of cellular automata: A survey. Theoretical Computer Science, 334(1):3– +33, 2005. +[14] J. Kari. Low-complexity tilings of the plane. In Descriptional Complexity of Formal +Systems - 21st IFIP WG 1.02 International Conference, DCFS 2019, volume 11612 of +Lecture Notes in Computer Science, pages 35–45. Springer, 2019. +[15] J. Kari. Expansivity and periodicity in algebraic subshifts. Submitted for publication, +2022. +[16] J. Kari and E. Moutot. Nivat’s conjecture and pattern complexity in algebraic subshifts. +Theoretical Computer Science, 777:379 – 386, 2019. +[17] J. Kari and M. Szabados. An algebraic geometric approach to Nivat’s conjecture. In +Proceedings of ICALP 2015, part II, volume 9135 of Lecture Notes in Computer Science, +pages 273–285, 2015. +[18] J. Kari and M. Szabados. An algebraic geometric approach to Nivat’s conjecture. In- +formation and Computation, 271:104481, 2020. +[19] P. Kurka. Topological and Symbolic Dynamics. Collection SMF. Soci´et´e math´ematique +de France, 2003. +[20] J. C. Lagarias and Y. Wang. Tiling the line with translates of one tile. Inventiones +Mathematicae, 124:341–365, 1996. +27 + +[21] D. Lind and B. Marcus. An Introduction to Symbolic Dynamics and Coding. Cambridge +University Press, 1995. +[22] M. Lothaire. Combinatorics on Words. Cambridge Mathematical Library. Cambridge +University Press, 2 edition, 1997. +[23] E. F. Moore. Machine models of self-reproduction. 1962. +[24] M. Morse and G. A. Hedlund. Symbolic dynamics. American Journal of Mathematics, +60(4):815–866, 1938. +[25] J. R. Myhill. The converse of Moore’s Garden-of-Eden theorem. 1963. +[26] D. Newman. Tesselation of integers. J. Number Theory, 9(1):107–111, 1977. +[27] M. Nivat. Invited talk at the 24th International Colloquium on Automata, Languages, +and Programming (ICALP 1997), 1997. +[28] S. A. Puzynina. +Perfect colorings of radius r > 1 of the infinite rectangular grid. +`Elektron. Mat. Izv., 5:283–292, 2008. +[29] S. A. Puzynina. On periodicity of generalized two-dimensional infinite words. Informa- +tion and Computation, 207(11):1315–1328, 2009. +[30] S. A. Puzynina. Aperiodic two-dimensional words of small abelian complexity. The +Electronic Journal of Combinatorics, 26(4), 2019. +[31] M. Szegedy. Algorithms to tile the infinite grid with finite clusters. Proceedings 39th +Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280), pages +137–145, 1998. +[32] H. Wang. Proving theorems by pattern recognition – II. The Bell System Technical +Journal, 40(1):1–41, 1961. +28 + diff --git a/ndE4T4oBgHgl3EQfug0k/content/tmp_files/load_file.txt b/ndE4T4oBgHgl3EQfug0k/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..399dbd06f0eac8f0f96060dfa397f3c8cdb5e563 --- /dev/null +++ b/ndE4T4oBgHgl3EQfug0k/content/tmp_files/load_file.txt @@ -0,0 +1,1077 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf,len=1076 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content='05232v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content='CO] 12 Jan 2023 On forced periodicity of perfect colorings Pyry Herva and Jarkko Kari Abstract We study forced periodicity of two-dimensional configurations under certain constraints and use an algebraic approach to multidimensional symbolic dynamics in which d- dimensional configurations and finite patterns are presented as formal power series and Laurent polynomials, respectively, in d variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We consider perfect colorings that are configurations such that the number of points of a given color in the neigh- borhood of any point depends only on the color of the point for some fixed relative neighborhood, and we show that by choosing the alphabet suitably any perfect coloring has a non-trivial annihilator, that is, there exists a Laurent polynomial whose formal product with the power series presenting the perfect coloring is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Using known results we obtain a sufficient condition for forced periodicity of two-dimensional per- fect colorings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' As corollaries of this result we get simple new proofs for known results of forced periodicity on the square and the triangular grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Moreover, we obtain a new result concerning forced periodicity of perfect colorings in the king grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We also consider perfect colorings of a particularly simple type: configurations that have low abelian complexity with respect to some shape, and we generalize a result that gives a sufficient condition for such configurations to be necessarily periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Also, some algorithmic aspects are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 1 Introduction We say that a d-dimensional configuration c ∈ AZd, that is, a coloring of the d-dimensional integer grid Zd using colors from a finite set A is a perfect coloring with respect to some finite relative neighborhood D ⊆ Zd if the number of any given color of A in the pattern c|u+D depends only on the color c(u) for any u ∈ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' There is a similar version of this definition for general graphs: a vertex coloring ϕ: V → A of a graph G = (V, E) with a finite set A of colors is a perfect coloring of radius r if the number of any given color in the r-neighborhood of a vertex u ∈ V depends only on the color ϕ(u) of u [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' More generally, the definition of perfect colorings is a special case of the definition of equitable partitions [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If ϕ: V → {0, 1} is a binary vertex coloring of a graph G = (V, E) then we can define a subset C ⊆ V of the vertex set – a code – such that it contains all the vertices with color 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If ϕ is a perfect coloring of radius r, then the code C has the property that the number of codewords of C in the r-neighborhood of a vertex u ∈ V is a if u ̸∈ C and b if u ∈ C for some fixed non-negative integers a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' This kind of code is called a perfect (r, b, a)-covering 1 or simply just a perfect multiple covering [1, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' This definition is related to domination in graphs and covering codes [11, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let D ⊆ Zd be a finite set and A a finite set of colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Two finite patterns p, q ∈ AD are abelian equivalent if the number of occurrences of each symbol in A is the same in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The abelian complexity of a configuration c ∈ AZd with respect to a finite shape D is the number of abelian equivalence classes of patterns of shape D in c [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We note that if c ∈ AZd is a perfect coloring with respect to D and |A| = n, then the abelian complexity of c with respect to D is at most n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Abelian complexity is a widely studied concept in one-dimensional symbolic dynamics and combinatorics on words [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In this paper we study forced periodicity of two-dimensional perfect colorings, that is, we study conditions under which all the colorings are necessarily periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We give a general condition for forced periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' As corollaries of this result we get new proofs for known results [1, 28, 29] concerning forced periodicity of perfect colorings in the square and the triangular grid and a new result for forced periodicity of perfect colorings in the king grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Moreover, we study two-dimensional configurations of low abelian complexity, that is, con- figurations that have abelian complexity 1 with respect to some shape: we generalize a statement of forced periodicity concerning this type of configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We use an algebraic approach [17] to multidimensional symbolic dynamics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=', we present configurations as for- mal power series and finite patterns as Laurent polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' This approach was developed to make progress in a famous open problem in symbolic dynamics – Nivat’s conjecture [27] – concerning forced periodicity of two-dimensional configurations that have a sufficiently low number of m × n rectangular patterns for some m, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The Nivat’s conjecture thus claims a two-dimensional generalization of the Morse-Hedlund theorem [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' This article is an extended version of the conference paper [12] where we considered forced periodicity of perfect coverings, that is, perfect colorings with only two colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The structure of the paper We begin in Section 2 by introducing the basic concepts of symbolic dynamics, cellular automata and graphs, and defining perfect colorings formally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In Section 3 we present the relevant algebraic concepts and the algebraic approach to multidimensional symbolic dynam- ics, and in Section 4 we describe an algorithm to find the line polynomial factors of a given two-dimensional Laurent polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In Section 5 we consider forced periodicity of perfect coverings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=', perfect colorings with only two colors and then in Section 6 we extend the re- sults from the previous section to concern perfect colorings using arbitrarily large alphabets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' After this we prove a statement concerning forced periodicity of two-dimensional configu- rations of low abelian complexity in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In Section 8 we consider some algorithmic questions concerning perfect colorings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 2 2 Preliminaries Basics on symbolic dynamics Let us review briefly some basic concepts of symbolic dynamics relevant to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' For a reference see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [4, 19, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Although our results concern mostly two-dimensional configurations, we state our definitions in an arbitrary dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let A be a finite set (the alphabet) and let d be a positive integer (the dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A d-dimensional configuration over A is a coloring of the infinite grid Zd using colors from A, that is, an element of AZd – the d-dimensional configuration space over the alphabet A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We denote by cu = c(u) the symbol or color that a configuration c ∈ AZd has in cell u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The translation τ t by a vector t ∈ Zd shifts a configuration c such that τ t(c)u = cu−t for all u ∈ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A configuration c is t-periodic if τ t(c) = c, and it is periodic if it is t-periodic for some non-zero t ∈ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Moreover, we say that a configuration is periodic in direction v ∈ Qd \\ {0} if it is kv-periodic for some k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A d-dimensional configuration c is strongly periodic if it has d linearly independent vectors of periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A strongly periodic configuration is periodic in every rational direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Two-dimensional strongly periodic configurations are called two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A finite pattern is an assignment of symbols on some finite shape D ⊆ Zd, that is, an element of AD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In particular, the finite patterns in AD are called D-patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let us denote by A∗ the set of all finite patterns over A where the dimension d is known from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We say that a finite pattern p ∈ AD appears in a configuration c ∈ AZd or that c contains p if τ t(c)|D = p for some t ∈ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' For a fixed shape D, the set of all D-patterns of c is the set LD(c) = {τ t(c)|D | t ∈ Zd} and the set of all finite patterns of c is denoted by L(c) which is called the language of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' For a set S ⊆ AZd of configurations we define LD(S) and L(S) as the unions of LD(c) and L(c) over all c ∈ S, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The pattern complexity P(c, D) of a configuration c ∈ AZd with respect to a shape D is the number of distinct D-patterns that c contains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' For any a ∈ A we denote by |p|a the number of occurrences of the color a in a finite pattern p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Two finite patterns p, q ∈ AD are called abelian equivalent if |p|a = |q|a for all a ∈ A, that is, if the number of occurrences of each color is the same in both p and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The abelian complexity A(c, D) of a configuration c ∈ AZ2 with respect to a finite shape D is the number of different D-patterns in c up to abelian equivalence [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Clearly A(c, D) ≤ P(c, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We say that c has low complexity with respect to D if P(c, D) ≤ |D| and that c has low abelian complexity with respect to D if A(c, D) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The configuration space AZd can be made a compact topological space by endowing A with the discrete topology and considering the product topology it induces on AZd – the prodiscrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' This topology is induced by a metric where two configurations are close if they agree on a large area around the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' So, AZd is a compact metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 3 A subset S ⊆ AZd of the configuration space is a subshift if it is topologically closed and translation-invariant meaning that if c ∈ S, then for all t ∈ Zd also τ t(c) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Equivalently, subshifts can be defined using forbidden patterns: Given a set F ⊆ A∗ of forbidden finite patterns, the set XF = {c ∈ AZd | L(c) ∩ F = ∅} of configurations that avoid all forbidden patterns is a subshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Moreover, every subshift is obtained by forbidding some set of finite patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If F ⊆ A∗ is finite, then we say that XF is a subshift of finite type (SFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The orbit of a configuration c is the set O(c) = {τ t(c) | t ∈ Zd} of its every translate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The orbit closure O(c) is the topological closure of its orbit under the prodiscrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The orbit closure of a configuration c is the smallest subshift that contains c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' It consists of all configurations c′ such that L(c′) ⊆ L(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Cellular automata Let us describe briefly an old result of cellular automata theory that we use in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' See [13] for a more thorough survey on the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A d-dimensional cellular automaton or a CA for short over a finite alphabet A is a map F : AZd −→ AZd determined by a neighborhood vector N = (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , tn) and a local rule f : An −→ A such that F(c)(u) = f(c(u + t1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , c(u + tn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A CA is additive or linear if its local rule is of the form f(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , xn) = a1x1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + anxn where a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , an ∈ R are elements of some finite ring R and A is an R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In Section 6 we consider the surjectivity of cellular automata and use a classic result called the Garden-of-Eden theorem proved by Moore and Myhil that gives a characterization for surjectivity in terms of injectivity on “finite” configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Two configurations c1 and c2 are called asymptotic if the set diff(c1, c2) = {u | c1(u) ̸= c2(u)} of cells where they differ is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A cellular automaton F is pre-injective if F(c1) ̸= F(c2) for any distinct asymptotic configurations c1 and c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Clearly injective CA are pre-injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The Garden-of-Eden theorem states that pre-injectivity of a CA is equivalent to surjectivity: Theorem (Garden-of-Eden theorem, [23, 25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A CA is surjective if and only if it is pre- injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In the one-dimensional setting the Garden-of-Eden theorem yields the following corollary: Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' For a one-dimensional surjective CA every configuration has only a finite number of pre-images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 4 Figure 1: The relative 2-neighborhoods of the square grid, the triangular grid and the king grid, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Graphs In this paper we consider graphs that are simple, undirected and connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A graph G that has vertex set V and edge set E is denoted by G = (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The distance d(u, v) of two vertices u ∈ V and v ∈ V of a graph G = (V, E) is the length of a shortest path between them in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The r-neighborhood of u ∈ V in a graph G = (V, E) is the set Nr(u) = {v ∈ V | d(v, u) ≤ r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The graphs we consider has vertex set V = Z2 and a translation invariant edge set E ⊆ {{u, v} | u, v ∈ Z2, u ̸= v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' This implies that for all r and for any two points u ∈ Z2 and v ∈ Z2 their r-neighborhoods are the same up to translation, that is, Nr(u) = Nr(v) + u − v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Moreover, we assume that all the vertices of G have only finitely many neighbors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=', we assume that the degree of G is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We call these graphs two-dimensional (infinite) grid graphs or just (infinite) grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In a grid graph G, let us call the r-neighborhood of 0 the relative r-neighborhood of G since it determines the r-neighborhood of any vertex in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Indeed, for all u ∈ Z2 we have Nr(u) = Nr + u where Nr is the relative r-neighborhood of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Given the edge set of a grid graph, the relative r-neighborhood is determined for every r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We specify three 2-dimensional infinite grid graphs: The square grid is the infinite grid graph (Z2, ES) with ES = {{u, v} | u − v ∈ {(±1, 0), (0, ±1)}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The triangular grid is the infinite grid graph (Z2, ET ) with ET = {{u, v} | u − v ∈ {(±1, 0), (0, ±1), (1, 1), (−1, −1)}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The king grid is the infinite grid graph (Z2, EK) with EK = {{u, v} | u − v ∈ {(±1, 0), (0, ±1), (±1, ±1)}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The relative 2-neighborhoods of these grid graphs are pictured in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 5 Perfect colorings Let A = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , an} be a finite alphabet of n colors and let D ⊆ Zd be a finite shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A configuration c ∈ AZd is a perfect coloring with respect to D ⊆ Zd or a D-perfect coloring if for all i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , n} there exist numbers bij such that for all u ∈ Zd with cu = aj the number of occurrences of color ai in the D-neighborhood of u, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=', in the pattern c|u+D is exactly bij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The matrix of a D-perfect coloring c is the matrix B = (bij)n×n where the numbers bij are as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A D-perfect coloring with matrix B is called a (perfect) (D, B)-coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Any D-perfect coloring is called simply a perfect coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In other words, a configuration is a perfect coloring if the number of cells of a given color in the given neighborhood of a vertex u depends only on the color of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Perfect colorings are defined also for arbitrary graphs G = (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Again, let A = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , an} be a finite set of n colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A vertex coloring ϕ: V → A of G is an r-perfect coloring with matrix B = (bij)n×n if the number of vertices of color ai in the r-neighborhood of a vertex of color aj is exactly bij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Clearly if G is a translation invariant graph with vertex set Zd, then the r-perfect colorings of G are exactly the D-perfect colorings in AZd where D is the relative r-neighborhood of the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 3 Algebraic concepts We review the basic concepts and some results relevant to us concerning an algebraic ap- proach to multidimensional symbolic dynamics introduced and studied in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' See also [14] for a short survey of the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let c ∈ AZd be a d-dimensional configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The power series presenting c is the formal power series c(X) = c(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , xd) = � u=(u1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=',ud)∈Zd cuxu1 1 · · ·xud d = � u∈Zd cuXu in d variables X = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , xd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We denote the set of all formal power series in d variables X = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , xd) over a domain M by M[[X±1]] = M[[x±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , x±1 d ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If d = 1 or d = 2, we denote x = x1 and y = x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A power series is finitary if it has only finitely many distinct coefficients and integral if its coefficients are all integers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=', if it belongs to the set Z[[X±1]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A configuration is always presented by a finitary power series and a finitary power series always presents a configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' So, from now on we may call any finitary power series a configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We consider also Laurent polynomials which we may call simply just polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We denote the set of Laurent polynomials in d variables X = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , xd) over a ring R by R[X±1] = R[x±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , x±1 d ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The term “proper” is used when we talk about proper (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=', non-Laurent) polynomials and denote the proper polynomial ring over R by R[X] as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We say that two Laurent polynomials have no common factors if all their common factors are units in the polynomial ring under consideration and that they have a common factor if they have a non–unit common factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' For example, in C[X±1] two polynomials have no 6 common factors if all their common factors are constants or monomials, and two proper polynomials in C[X] have no common factors if all their common factors are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The support of a power series c = c(X) = � u∈Zd cuXu is the set supp(c) = {u ∈ Zd | cu ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Clearly a polynomial is a power series with a finite support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The kth dilation of a polynomial f(X) is the polynomial f(Xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' See Figure 2 for an illustration of dilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Figure 2: The supports of the polynomial f(X) = 1 + x−1y−1 + x−1y1 + x1y−1 + x1y1 and its dilations f(X2) and f(X3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The xi-resultant Resxi(f, g) of two proper polynomials f, g ∈ R[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , xd] is the deter- minant of the Sylvester matrix of f and g with respect to variable xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We omit the details which the reader can check from [6], and instead we consider the resultant Resxi(f, g) ∈ R[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , xi−1, xi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , xd] for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , d} as a certain proper polynomial that has the following two properties: Resxi(f, g) is in the ideal generated by f and g, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=', there exist proper polynomials h and l such that hf + lg = Resxi(f, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If two proper polynomials f and g have no common factors in R[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , xd], then Resxi(f, g) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let R be a ring and M a (left) R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The formal product of a polynomial f = f(X) = �m i=1 aiXui ∈ R[X±1] and a power series c = c(X) = � u∈Zd cuXu ∈ M[X±1] is well-defined as the formal power series fc = f(X)c(X) = � u∈Zd (fc)uXu ∈ M[X±1] where (fc)u = m � i=1 aicu−ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We say that a polynomial f = f(X) annihilates (or is an annihilator of ) a power series c = c(X) if fc = 0, that is, if their product is the zero power series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In a typical setting, we assume that A ⊆ Z and hence consider any configuration c ∈ AZd as a finitary and integral power series c(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Since multiplying c(X) by the monomial Xu 7 produces the power series presenting the translation τ u(c) of c by u, we have that c is u-periodic if and only if c(X) is annihilated by the difference polynomial Xu − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' (By a difference polynomial we mean a polynomial Xu − 1 for any u ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=') This means that it is natural to consider multiplication of c by polynomials in C[X±1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' However, note that the product of c and a polynomial f ∈ C[X±1] may not be integral, but it is still finitary, hence a configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We say that a polynomial f periodizes (or is a periodizer of ) a configuration c if fc is strongly periodic, that is, periodic in d linearly independent directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We denote the set of all periodizers with complex coefficients of a configuration c by Per(c) which is an ideal of C[X±1] and hence we call it the periodizer ideal of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Note that annihilators are periodizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Note also that if c has a periodizer f, then (Xu − 1)f is an annihilator of c for some u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, c has a non-trivial (= non-zero) annihilator if and only if it has a non-trivial periodizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The following theorem states that if a configuration has a non-trivial periodizer, then it has in fact an annihilator of a particular simple form – a product of difference polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Theorem 1 ([17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let c ∈ Z[[X±1]] be a configuration in any dimension and assume that it has a non-trivial periodizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Then there exist m ≥ 1 and pairwise linearly independent vectors t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , tm such that (Xt1 − 1) · · ·(Xtm − 1) annihilates c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A line polynomial is a polynomial whose support contains at least two points and the points of the support lie on a unique line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In other words, a polynomial f is a line polynomial if it is not a monomial and there exist vectors u, v ∈ Zd such that supp(f) ⊆ u + Qv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In this case we say that f is a line polynomial in direction v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We say that non-zero vectors v, v′ ∈ Zd are parallel if v′ ∈ Qv, and clearly then a line polynomial in direction v is also a line polynomial in any parallel direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A vector v ∈ Zd is primitive if its components are pairwise relatively prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If v is primitive, then Qv ∩ Zd = Zv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' For any non-zero v ∈ Zd there exists a parallel primitive vector v′ ∈ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, we may assume the vector v in the definition of a line polynomial f to be primitive so that supp(f) ⊆ u + Zv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In the following our preferred presentations of directions are in terms of primitive vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Any line polynomial φ in a (primitive) direction v can be written uniquely in the form φ = Xu(a0 + a1Xv + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + anXnv) = Xu(a0 + a1t + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + antn) where u ∈ Zd, n ≥ 1, a0 ̸= 0, an ̸= 0 and t = Xv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let us call the single variable proper polynomial a0 + a1t + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + antn ∈ C[t] the normal form of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Moreover, for a monomial aXu we define its normal form to be a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' So, two line polynomials in the direction v have the same normal form if and only if they are the same polynomial up to multiplication by Xu, for some u ∈ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Difference polynomials are line polynomials and hence the annihilator provided by The- orem 1 is a product of line polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Annihilation by a difference polynomial means periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' More generally, annihilation of a configuration c by a line polynomial in a primitive direction v can be understood as the annihilation of the one-dimensional v-fibers 8 � k∈Z cu+kvXu+kv of c in direction v, and since annihilation in the one-dimensional setting implies periodicity with a bounded period, we conclude that a configuration is periodic if and only if it is annihilated by a line polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' It is known that if c has a periodizer with line polynomial factors in at most one primitive direction, then c is periodic: Theorem 2 ([18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let c ∈ Z[[x±1, y±1]] be a two-dimensional configuration and let f be a periodizer of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Then the following conditions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If f does not have any line polynomial factors, then c is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If all line polynomial factors of f are in the same primitive direction, then c is periodic in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The periodizer ideal Per(c) = {g ∈ C[x±1, y±1] | gc is two-periodic} of c is a principal ideal generated by a polynomial g = φ1 · · · φm where φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , φm are line polynomials in pairwise non-parallel directions [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Because f ∈ Per(c), we know that g divides f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If f does not have any line polynomial factors, then g = 1 and hence c = gc is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If f has line polynomial factors, and they are in the same primitive direction v, then g is a line polynomial in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Since gc is two-periodic, it is annihilated by (Xkv − 1) for some k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' This implies that the configuration c is annihilated by the line polynomial (Xkv − 1)g in direction v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We conclude that c is periodic in direction v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The proof of the previous theorem sketched above relies heavily on the structure of the ideal Per(c) developed in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We give an alternative proof sketch that mimics the usage of resultants in [16]: Second proof sketch of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The existence of a non-trivial periodizer f implies by Theorem 1 that c has a special annihilator g = φ1 · · ·φm that is a product of (difference) line polynomials φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , φm in pairwise non-parallel directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' All irreducible factors of g are line polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If f does not have any line polynomial factors, then the periodizers f and g do not have common factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We can assume that both are proper polynomials as they can be multiplied by a suitable monomial if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Because f, g ∈ Per(c), also their resultant Resx(f, g) ∈ Per(c), implying that c has a non-trivial annihilator containing only variable y since Resx(f, g) ̸= 0 because f and g have no common factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' This means that c is periodic in the vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Analogously, the y-resultant Resy(f, g) shows that c is horizontally periodic, and hence two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The proof for the case that f has line polynomial factors only in one direction v goes analogously by considering φc instead of c, where φ is the greatest common line polynomial factor of f and g in the direction v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We get that φc is two-periodic, implying that c is periodic in direction v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In this paper we also consider configurations over alphabets A that are finite subsets of Zn, that is, the set of length n integer vectors, and hence study finitary formal power series from the set Zn[[X±1]] for n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In particular, we call this kind of configurations integral vector configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Also in this setting we consider multiplication of power series by polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The coefficients of the polynomials are n×n integer matrices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=', elements 9 of the ring Zn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Since Zn is a (left) Zn×n-module where we consider the vectors of Zn as column vectors, the product of a polynomial f = f(X) ∈ Zn×n[X±1] and a power series c = c(X) ∈ Zn[[X±1]] is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Consequently, we say that c(X) ∈ Zn[[X±1]] is t- periodic if it is annihilated by the polynomial IXt−I and that it is periodic if it is t-periodic for some non-zero t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' There is a natural way to present configurations over arbitrary alphabets as integral vector configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let A = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , an} be a finite alphabet with n elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The vector presentation of a configuration c ∈ AZd is the configuration c′ ∈ {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , en}Zd (or the power series c′(X) ∈ Zn[[X±1]] presenting c′) defined such that c′ u = ei if and only if cu = ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Here by ei ∈ Zn we denote the ith natural base vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=', the vector whose ith component is 1 while all the other components are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Clearly c is t-periodic if and only if its vector presentation is t-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, to study the periodicity of a configuration it is sufficient to study the periodicity of its vector presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The ith layer of c = � cuXu ∈ Zn[[X±1]] is the power series layeri(c) = � c(i) u Xu ∈ Z[[X±1]] where c(i) u is the ith component of cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Clearly c ∈ Zn[[X±1]] is periodic in direction v if and only if for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , n} the ith layer of c is periodic in direction v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Finally, let R be a finite ring and A a finite R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A polynomial f(X) = �n i=1 aiX−ui ∈ R[x±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , x±1 d ] defines an additive CA that has neighborhood vector (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , un) and local rule f ′(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , yn) = a1y1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + anyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' More precisely, the image of a configuration c under the CA determined by f is the configuration fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 4 Finding the line polynomial factors of a given two- variate Laurent polynomial In this section we have d = 2 and hence all our polynomials are in two variables x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The open and closed discrete half planes determined by a non-zero vector v ∈ Z2 are the sets Hv = {u ∈ Z2 | ⟨u, v⊥⟩ > 0} and Hv = {u ∈ Z2 | ⟨u, v⊥⟩ ≥ 0}, respectively, where v⊥ = (v2, −v1) is orthogonal to v = (v1, v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let us also denote by lv = Hv \\ Hv the discrete line parallel to v that goes through the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In other words, the half plane determined by v is the half plane “to the right” of the line lv when moving along the line in the direction of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We say that a finite set D ⊆ Z2 has an outer edge in direction v if there exists a vector t ∈ Z2 such that D ⊆ Hv + t and |D ∩ (lv + t)| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We call D ∩ (lv + t) the outer edge of D in direction v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' An outer edge corresponding to v means that the convex hull of D has an edge in direction v in the clockwise orientation around D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If a finite non-empty set D does not have an outer edge in direction v, then there exists a vector t ∈ Z2 such that D ⊆ Hv + t and |D ∩ (lv + t)| = 1, and then we say that D has a vertex in direction v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We call D ∩ (lv + t) the vertex of D in direction v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We say that a polynomial f has an outer edge or a vertex in direction v if its support has an outer edge or a vertex in direction v, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Note that every non-empty finite shape D has either 10 an edge or a vertex in any non-zero direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Note also that in this context directions v and −v are not the same: a shape may have an outer edge in direction v but no outer edge in direction −v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The following lemma shows that a polynomial can have line polynomial factors only in the directions of its outer edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Lemma 3 ([16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let f be a non-zero polynomial with a line polynomial factor in direction v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Then f has outer edges in directions v and −v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let v ∈ Z2 \\ {0} be a non-zero primitive vector and let f = � fuXu be a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Recall that a v-fiber of f is a polynomial of the form � k∈Z fu+kvXu+kv for some u ∈ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, a non-zero v-fiber of a polynomial is either a line polynomial or a monomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let us denote by Fv(f) the set of different normal forms of all non-zero v-fibers of a polynomial f, which is hence a finite set of one-variate proper polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The following simple example illustrates the concept of fibers and their normal forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let us determine the set Fv(f) for f = f(X) = f(x, y) = 3x + y + xy2 + xy + x3y3 + x4y4 and v = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' By grouping the terms we can write f = 3x + y(1 + xy) + xy(1 + x2y2 + x3y3) = X(1,0) · 3 + X(0,1)(1 + t) + X(1,1)(1 + t2 + t3) where t = X(1,1) = xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Hence, Fv(f) = {3, 1 + t, 1 + t2 + t3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' See Figure 3 for a pictorial illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 3x y xy2 xy x3y3 x4y4 Figure 3: The support of f = 3x + y + xy2 + xy + x3y3 + x4y4 and its different (1, 1)-fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' As noticed in the example above, polynomials are linear combinations of their fibers: for any polynomial f and any non-zero primitive vector v we can write f = Xu1ψ1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + Xunψn for some u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , un ∈ Z2 where ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , ψn ∈ Fv(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We use this in the proof of the next theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A polynomial f has a line polynomial factor in direction v if and only if the polynomials in Fv(f) have a common factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' For any line polynomial φ in direction v, and for any polynomial g, the v-fibers of the product φg have a common factor φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In other words, if a polynomial f has a line polynomial factor φ in direction v, then the polynomials in Fv(f) have the normal form of φ as a common factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' For the converse direction, assume that the polynomials in Fv(f) have a common factor φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Then there exist vectors u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , un ∈ Z2 and polynomials φψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , φψn ∈ Fv(f) such that f = Xu1φψ1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + Xunφψn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Hence, φ is a line polynomial factor of f in direction v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Note that Lemma 3 actually follows immediately from Theorem 5: A vertex instead of an outer edge in direction v or −v provides a non-zero monomial v-fiber, which implies that the polynomials in Fv(f) have no common factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' So, to find out the line polynomial factors of f we first need to find out the possible directions of the line polynomials, that is, the directions of the (finitely many) outer edges of f, and then we need to check for which of these possible directions v the polynomials in Fv(f) have a common factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' There are clearly algorithms to find the outer edges of a given polynomial and to determine whether finitely many line polynomials have a common factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If such a factor exists, then by Theorem 5 the polynomial f has a line polynomial factor in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We have proved the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' There is an algorithm to find the line polynomial factors of a given (Laurent) polynomial in two variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 5 Forced periodicity of perfect colorings with two col- ors In this section we consider forced periodicity of two-dimensional perfect colorings with only two colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Without loss of generality we may assume that A = {a1, a2} = {0, 1} (a1 = 0, a2 = 1) and consider perfect colorings c ∈ AZ2 since the names of the colors do not matter in our considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' So, let c ∈ {0, 1}Z2 be a perfect coloring with respect to D ⊆ Z2 and let B = (bij)2×2 be the matrix of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let us define a set C = {u ∈ Z2 | cu = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' This set has the property that the neighborhood u+D of a point u contains exactly a = b21 points of color 1 if u ̸∈ C and exactly b = b22 points of color 1 if u ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In fact, C is a perfect (multiple) covering of the infinite grid G determined by the relative neighborhood D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' More precisely, the set C is a (perfect) (D, b, a)-covering of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' This is a variant of the following definition: in any graph a subset C of its vertex set is an (r, b, a)-covering if the number of vertices of C in the r-neighborhood of a vertex u is a if u ̸∈ C and b if u ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' See [1] for a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Clearly in translation invariant graphs the (r, b, a)-coverings correspond to (D, b, a)-coverings where D is the relative r-neighborhood of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, it is natural to call any perfect coloring with only two colors a perfect covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Note that a (D, b, a)-covering is a D-perfect 12 coloring with the matrix B = � |D| − a |D| − b a b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The following theorem by Axenovich states that “almost every” (1, b, a)-covering in the square grid is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Theorem 7 ([1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If b−a ̸= 1, then every (1, b, a)-covering in the square grid is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' For a finite set D ⊆ Z2 we define its characteristic polynomial to be the polynomial fD(X) = � u∈D X−u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We denote by 1(X) the constant power series � u∈Z2 Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If c ∈ {0, 1}Z2 is a (D, b, a)-covering, then from the definition we get that fD(X)c(X) = (b − a)c(X) + a1(X) which is equivalent to (fD(X) − (b − a)) c(X) = a1(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, if c is a (D, b, a)-covering, then fD(X) − (b − a) is a periodizer of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Hence, by Theorem 2 the condition that the polynomial fD(X) − (b − a) has no line polynomial factors is a sufficient condition for forced periodicity of a (D, b, a)-covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Hence, we have the following corollary of Theorem 2: Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let D ⊆ Z2 be a finite shape and let b and b be non-negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If g = fD −(b−a) has no line polynomial factors, then every (D, b, a)-covering is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Using our formulation and the algebraic approach we get a simple proof for Theorem 7: Reformulation of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let D be the relative 1-neighborhood of the square grid and assume that b − a ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Then every (D, b, a)-covering is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let c be an arbitrary (D, b, a)-covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The outer edges of g = fD − (b − a) = x−1 + y−1 + 1 − (b − a) + x + y are in directions (1, 1), (−1, −1), (1, −1) and (−1, 1) and hence by Lemma 3 any line polynomial factor of g is either in direction (1, 1) or (1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' For v ∈ {(1, 1), (1, −1)} we have Fv(g) = {1 + t, 1 − (b − a)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' See Figure 4 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Since 1 − (b − a) is a non-trivial monomial, by Theorem 5 the periodizer g ∈ Per(c) has no line polynomial factors and hence the claim follows by corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We also get a similar proof for the following known result concerning the forced periodicity perfect coverings in the square grid with radius r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Theorem 9 ([29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let r ≥ 2 and let D be the relative r-neighborhood of the square grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Then every (D, b, a)-covering is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In other words, all (r, b, a)-coverings in the square grid are two-periodic for all r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let c be an arbitrary (D, b, a)-covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' By Lemma 3 any line polynomial factor of g = fD − (b − a) has direction (1, 1) or (1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' So, assume that v ∈ {(1, 1), (1, −1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We have φ1 = 1 + t + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + tr ∈ Fv(g) and φ2 = 1 + t + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + tr−1 ∈ Fv(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' See Figure 4 for an illustration in the case r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Since φ1 − φ2 = tr, the polynomials φ1 and φ2 have no common factors, and hence by Theorem 5 the periodizer g has no line polynomial factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Corollary 8 gives the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' There are analogous results in the triangular grid, and we can prove them similarly using Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 13 1 + t 1 − (b − a) 1 + t + t2 1 + t 1 + t + t2 + t3 + t4 1 + t + (1 − (b − a))t2 + t3 + t4 1 + t 1 + (1 − (b − a))t + t2 1 + t + t2 1 + t + t2 + t3 Figure 4: Pictorial illustrations for the proofs of Theorems 7, 9, 10, 11 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The constel- lation on the left of the upper row illustrates the proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The constellation in the center of the upper row illustrates the proof of Theorem 9 with r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The constellation on the right of the upper row illustrates the proof of Theorem 12 with r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The constel- lation on the left of the lower row illustrates the proof of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The constellation on the right of the lower row illustrates the proof of Theorem 11 with r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In each of the constellations we have pointed out two normal forms with no common factors in Fv(g) from the points of supp(g) for one of the outer edges v of supp(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Theorem 10 ([29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let D be the relative 1-neighborhood of the triangular grid and assume that b−a ̸= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Then every (D, b, a)-covering in the triangular grid is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In other words, all (1, b, a)-coverings in the triangular grid are two-periodic whenever b − a ̸= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let c be an arbitrary (D, b, a)-covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The outer edges of g = fD − (b − a) = x−1y−1 + x−1 + y−1 + 1 − (b − a) + x + y + xy have directions (1, 1), (−1, −1), (1, 0), (−1, 0), (0, 1) and (0, −1) and hence by Lemma 3 any line polynomial factor of g has direction (1, 1), (1, 0) or (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' So, let v ∈ {(1, 1), (1, 0), (0, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We have Fv(g) = {1+t, 1+(1−(b−a))t+t2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' See Figure 4 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Polynomials φ1 = 1 + t and φ2 = 1 + (1 − (b − a))t + t2 satisfy φ2 1 − φ2 = (1 + b − a)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, they do not have any common factors if b − a ̸= −1 and hence by Theorem 5 the polynomial g has no line polynomial factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The claim follows by Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Theorem 11 ([29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let r ≥ 2 and let D be the relative r-neighborhood of the triangular grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Then every (D, b, a)-covering is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In other words, every (r, b, a)-covering in the triangular grid is two-periodic for all r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let c be an arbitrary (D, b, a)-covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The outer edges of g = fD − (b − a) have directions (1, 1), (−1, −1), (1, 0), (−1, 0), (0, 1) and (0, −1), and hence by Lemma 3 any line polynomial factor of g has direction (1, 1), (1, 0) or (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' So, let v ∈ {(1, 1), (1, 0), (0, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' There exists n ≥ 1 such that 1+t+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content='+tn ∈ Fv(g) and 1+t+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content='+tn+1 ∈ Fv(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' See Figure 4 for an illustration with r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Since these two polynomials have no common factors, by 14 Theorem 5 the polynomial g has no line polynomial factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Again, Corollary 8 yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If a ̸= b, then for all r ≥ 1 any (r, b, a)-covering in the king grid is two-periodic: Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let r ≥ 1 be arbitrary and let D be the relative r-neighborhood of the king grid and assume that a ̸= b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Then any (D, b, a)-covering is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In other words, all (r, b, a)-coverings in the king grid are two-periodic whenever a ̸= b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let c be an arbitrary (D, b, a)-covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The outer edges of g = fD − (b − a) are in directions (1, 0), (−1, 0), (0, 1) and (0, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Hence, by Lemma 3 any line polynomial factor of g has direction (1, 0) or (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let v ∈ {(1, 0), (0, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We have φ1 = 1 + t + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + tr−1 + (1 − (b − a))tr + tr+1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + t2r ∈ Fv(g) and φ2 = 1 + t + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + t2r ∈ Fv(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' See Figure 4 for an illustration in the case r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Since φ2 − φ1 = (b − a)tr is a non-trivial monomial, φ1 and φ2 have no common factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, by Theorem 5 the polynomial g has no line polynomial factors and the claim follows by Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In the above proofs we used the fact that two Laurent polynomials in one variable have no common factors if and only if they generate the entire ideal C[t±1], and they do this if and only if they generate a non-zero monomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' This is known as the weak Nullstellensatz [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A shape D ⊆ Z2 is convex if it is the intersection D = conv(D)∩Z2 where conv(D) ⊆ R2 is the real convex hull of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Above all our shapes were convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Next we generalize the above theorems and give a sufficient condition for forced periodicity of (D, b, a)-coverings for convex D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' So, let D ⊆ Z2 be a finite convex shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Any (D, b, a)-covering has a periodizer g = fD − (b − a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' As earlier, we study whether g has any line polynomial factors since if it does not, then Corollary 8 guarantees forced periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' For any v ̸= 0 the set Fv(fD) contains only polynomials φn = 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + tn−1 for different n ≥ 1 since D is convex: if D contains two points, then D contains every point between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, Fv(g) contains only polynomials φn for different n ≥ 1 and, if b − a ̸= 0, it may also contain a polynomial φn0 − (b − a)tm0 for some n0 ≥ 1 such that φn0 ∈ Fv(fD) and for some m0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If b − a = 0, then g = fD and thus Fv(g) = Fv(fD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Two polynomials φm and φn have a common factor if and only if gcd(m, n) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' More gen- erally, the polynomials φn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , φnr have a common factor if and only if d = gcd(n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , nr) > 1 and, in fact, their greatest common factor is the dth cyclotomic polynomial � 1≤k≤d gcd(k,d)=1 (t − ei· 2πk d ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let us introduce the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' For any polynomial f, we denote by F ′ v(f) the set of normal forms of the non-zero fibers � k∈Z fu+kvXu+kv for all u ̸∈ Zv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In other words, we exclude the fiber through the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let us also denote fibv(f) for the normal form of the fiber � k∈Z fkvXkv through the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We have Fv(f) = F ′ v(f)∪{fibv(f)} if fibv(f) ̸= 0 and Fv(f) = F ′ v(f) if fibv(f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 15 Applying Theorems 2 and 5 we have the following theorem that gives sufficient conditions for every (D, b, a)-covering to be periodic for a finite and convex D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' This theorem generalizes the results proved above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In fact, they are corollaries of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The first part of the theorem was also mentioned in [7] in a slightly different context and in a more general form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let D be a finite convex shape, g = fD − (b − a) and let E be the set of the outer edge directions of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Assume that b − a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' For any v ∈ E denote dv = gcd(n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , nr) where Fv(g) = {φn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , φnr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If dv = 1 holds for all v ∈ E, then every (D, b, a)-covering is two- periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If dv = 1 holds for all but some parallel v ∈ E, then every (D, b, a)-covering is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Assume that b − a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' For any v ∈ E denote dv = gcd(n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , nr) where F ′ v(g) = {φn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , φnr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If the dv’th cyclotomic polynomial and fibv(g) have no common factors for any v ∈ E, then every (D, b, a)-covering is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If the condition holds for all but some parallel v ∈ E, then every (D, b, a)-covering is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' (Note that the condition is satisfied, in particular, if dv = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Assume first that b − a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If dv = 1 for all v ∈ E, then the v-fibers of g have no common factors and hence by Theorem 5 g has no line polynomial factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If dv = 1 holds for all but some parallel v ∈ E, then all the line polynomial factors of g are in parallel directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, the claim follows by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Assume then that b − a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If the dv’th cyclotomic polynomial and fibv(g) have no common factors for all v ∈ E, then by Theorem 5 g has no line polynomial factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If the condition holds for all but some parallel v ∈ E, then all the line polynomial factors of g are in parallel directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, by Theorem 2 the claim holds also in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 6 Forced periodicity of perfect colorings over arbitrar- ily large alphabets In this section we prove a theorem that gives a sufficient condition for forced periodicity of two-dimensional perfect colorings over an arbitrarily large alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' As corollaries of the theorem and theorems from the previous section we obtain conditions for forced periodicity of perfect colorings in two-dimensional infinite grid graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We start by proving some lemmas that work in any dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We consider the vector presentations of perfect colorings because this way we get a non-trivial annihilator for any such vector presentation: Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let c be the vector presentation of a D-perfect coloring over an alphabet of size n with matrix B = (bij)n×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Then c is annihilated by the polynomial f(X) = � u∈D IX−u − B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 16 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Note the similarity of the above annihilator to the periodizer � u∈D X−u − (b − a) of a (D, b, a)-covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let v ∈ Zd be arbitrary and assume that cv = ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Then (Bc)v = Bej is the jth column of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' On the other hand, from the definition of B we have ((� u∈D IX−u)c)v = � u∈D cv+u = �n i=1 bijei which is also the jth column of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, (fc)v = 0 and hence fc = 0 since v was arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The following lemma shows that as in the case of integral configurations with non-trivial annihilators, also the vector presentation of a perfect coloring has a special annihilator which is a product of difference polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' By congruence of two polynomials with integer matrices as coefficients (mod p) we mean that their corresponding coefficients are congruent (mod p) and by congruence of two integer matrices (mod p) we mean that their corresponding components are congruent (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let c be the vector presentation of a D-perfect coloring over an alphabet of size n with matrix B = (bij)n×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Then c is annihilated by the polynomial g(X) = (IXv1 − I) · · ·(IXvm − I) for some vectors v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , vm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' By Lemma 14 the power series c is annihilated by f(X) = � u∈D IX−u − B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let p be a prime larger than ncmax where cmax is the maximum absolute value of the components of the coefficients of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Since the coefficients of f commute with each other, we have for any positive integer k using the binomial theorem that f pk = f pk(X) ≡ � u∈D IX−pku − Bpk (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We have f pk(X)c(X) ≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' There are only finitely many distinct matrices Bpk (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' So, let k and k′ be distinct and such that Bpk ≡ Bpk′ (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Then the coefficients of f ′ = f pk−f pk′ (mod p) are among I and −I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Since f pkc ≡ 0 (mod p) and f pk′ c ≡ 0 (mod p), also f ′c ≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The components of the configuration f ′c are bounded in absolute value by ncmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Since we chose p larger than ncmax, this implies that f ′c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Because f ′ = � u∈P1 IXu − � u∈P2 IXu for some finite subsets P1 and P2 of Zd, the annihilation of c by f ′ is equivalent to the annihilation of every layer of c by f ′′ = � u∈P1 Xu− � u∈P2 Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, every layer of c has a non-trivial annihilator and hence by Theorem 1 every layer of c has a special annihilator which is a product of difference polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let g′ = (Xv1 − 1) · · ·(Xvm − 1) 17 be the product of all these special annihilators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Since g′ annihilates every layer of c, the polynomial g = (IXv1 − I) · · ·(IXvm − I) annihilates c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let p be a prime and let H be an additive CA over Zn p determined by a polyno- mial h = �k i=0 AiXui ∈ Zn×n p [X±1] whose coefficients Ai commute with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Assume that there exist M ∈ Zp \\ {0} and matrices C0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , Ck that commute with each other and with every Ai such that C0A0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + CkAk = M · I holds in Zk×k p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Then H is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Assume the contrary that H is not surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' By the Garden-of-Eden theorem H is not pre-injective and hence there exist two distinct asymptotic configurations c1 and c2 such that H(c1) = H(c2), that is, h(X)c1(X) = h(X)c2(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, h is an annihilator of e = c1 − c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Without loss of generality we may assume that c1(0) ̸= c2(0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=', that e(0) = v ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let l be such that the support supp(e) = {u ∈ Zd | e(u) ̸= 0} of e is contained in a d-dimensional pl × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' × pl hypercube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Note that in Zk×k p we have f pl = k � i=0 Apl i Xplui which is also an annihilator of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Hence, by the choice of l we have Apl i v = 0 for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' By raising the identity C0A0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + CkAk = M · I to power pl and multiplying the result by the vector v from the right we get Mpl · v = Cpl 0 Apl 0 v + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + Cpl k Apl k v = 0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' However, this is a contradiction because Mplv ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, H must be surjective as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Theorem 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let D ⊆ Z2 be a finite shape and assume that there exists an integer t0 such that the polynomial fD − t = � u∈D X−u − t has no line polynomial factors whenever t ̸= t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Then any D-perfect coloring with matrix B is two-periodic whenever det(B − t0I) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If fD −t has no line polynomial factors for any t, then every D-perfect coloring is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let c be the vector presentation of a D-perfect coloring with matrix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' By Lemmas 14 and 15 it has two distinct annihilators: f = � u∈D IX−u −B and g = (IXv1 −I) · · · (IXvm − I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let us replace I by 1 and B by a variable t and consider the corresponding integral polynomials f ′ = � u∈D X−u −t = fD −t and g′ = (Xv1 −1) · · ·(Xvm −1) in C[x, y, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Here X = (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 18 Without loss of generality we may assume that f ′ and g′ are proper polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Indeed, we can multiply f ′ and g′ by monomials such that the obtained polynomials f ′′ and g′′ are proper polynomials and that they have a common factor if and only if f ′ and g′ have a common factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' So, we may consider f ′′ and g′′ instead of f ′ and g′ if they are not proper polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We consider the y-resultant Resy(f ′, g′) of f ′ and g′, and write Resy(f ′, g′) = f0(t) + f1(t)x + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + fk(t)xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' By the properties of resultants Resy(f ′, g′) is in the ideal generated by f ′ and g′, and it can be the zero polynomial only if f ′ and g′ have a common factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Since g′ is a product of line polynomials, any common factor of f ′ and g′ is also a product of line polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In particular, if f ′ and g′ have a common factor, then they have a common line polynomial factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' However, by the assumption f ′ has no line polynomial factors if t ̸= t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, f ′ and g′ may have a common factor only if t = t0 and hence Resy(f ′, g′) can be zero only if t = t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' On the other hand, Resy(f ′, g′) = 0 if and only if f0(t) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' = fk(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We conclude that gcd(f0(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , fk(t)) = (t − t0)m for some m ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, Resy(f ′, g′) = (t − t0)m(f ′ 0(t) + f ′ 1(t)x + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + f ′ k(t)xk) where the polynomials f ′ 0(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , f ′ k(t) have no common factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' By the Euclidean algorithm there are polynomials a0(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , ak(t) such that a0(t)f ′ 0(t) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + ak(t)f ′ k(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' (1) Moreover, the coefficients of the polynomials a0(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , ak(t) are rational numbers because the polynomials f ′ 0(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , f ′ k(t) are integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Note that if f ′ has no line polynomial factors for any t, then m = 0 and hence f ′ i(t) = fi(t) for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let us now consider the polynomial (B − t0I)m(f ′ 0(B) + f ′ 1(B)x + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + f ′ k(B)xk) which is obtained from Resy(f ′, g′) by plugging back I and B in the place of 1 and t, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Since Resy(f ′, g′) is in the ideal generated by f ′ and g′, the above polynomial is in the ideal generated by f and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, it is an annihilator of c because both f and g are annihilators of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Assume that det(B − t0I) ̸= 0 or that m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Now also h = f ′ 0(B) + f ′ 1(B)x + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + f ′ k(B)xk is an annihilator of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Since f ′ 0(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , f ′ k(t) have no common factors, h is non-zero, because otherwise it would be f ′ 0(B) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' = f ′ k(B) = 0 and the minimal polynomial of B would be a common factor of f ′ 0(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , f ′ k(t), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Plugging t = B to Equation 1 we get a0(B)f ′ 0(B) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + ak(B)f ′ k(B) = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 19 Let us multiply the above equation by a common multiple M of all the denominators of the rational numbers appearing in the equation and let us consider it (mod p) where p is a prime that does not divide M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We obtain the following identity a′ 0(B)f ′ 0(B) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' + a′ k(B)f ′ k(B) = M · I ̸≡ 0 (mod p) where all the coefficients in the equation are integer matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' By Lemma 16 the additive CA determined by h = �k i=0 f ′ i(B)xi is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Since h is a polynomial in variable x only, it defines a 1-dimensional CA H which is surjective and which maps every horizontal fiber of c to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Hence, every horizontal fiber of c is a pre-image of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let c′ be a horizontal fiber of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The Garden-of-Eden theorem implies that 0 has finitely many, say N, pre-images under H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Since also every translation of c′ is a pre-image of 0, we conclude that c′ = τ i(c′) for some i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , N − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, (N − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' is a common period of all the horizontal fibers of c and hence c is horizontally periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Repeating the same argumentation for the x-resultant of f ′ and g′ we can show that c is also vertically periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, c is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' As corollaries of the above theorem and theorems from the previous section, we obtain new proofs for forced periodicity of perfect colorings in the square and the triangular grids, and a new result for forced periodicity of perfect colorings in the king grid: Corollary 18 ([29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let D be the relative 1-neighborhood of the square grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Then any D- perfect coloring with matrix B is two-periodic whenever det(B − I) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In other words, any 1-perfect coloring with matrix B in the square grid is two-periodic whenever det(B − I) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In our proof of Theorem 7 it was shown that the polynomial fD − t has no line polynomial factors if t ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, by Theorem 17 any (D, B)-coloring is two-periodic whenever det(B − I) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Corollary 19 ([29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let D be the relative 1-neighborhood of the triangular grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Then any D-perfect coloring with matrix B is two-periodic whenever det(B + I) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In other words, any 1-perfect coloring with matrix B in the triangular grid is two-periodic whenever det(B + I) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In the proof of Theorem 10 it was shown that the polynomial fD − t has no line polynomial factors if t ̸= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, by Theorem 17 any (D, B)-coloring is two-periodic whenever det(B + I) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Corollary 20 ([29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let r ≥ 2 and let D be the relative r-neighborhood of the square grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Then every D-perfect coloring is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In other words, any r-perfect coloring in the square grid is two-periodic for all r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In the proof of Theorem 9 it was shown that the polynomial fD − t has no line polynomial factors for any t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, by Theorem 17 every D-perfect coloring is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 20 Corollary 21 ([29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let r ≥ 2 and let D be the relative r-neighborhood of the triangular grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Then every D-perfect coloring is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In other words, any r-perfect coloring in the triangular grid is two-periodic for all r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In the proof of Theorem 11 it was shown that the polynomial fD − t has no line polynomial factors for any t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, by Theorem 17 every D-perfect coloring is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Corollary 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let r ≥ 1 and let D be the relative r-neighborhood of the king grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Then every D-perfect coloring with matrix B is two-periodic whenever det(B) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In other words, every r-perfect coloring with matrix B in the king grid is two-periodic whenever det(B) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In the proof of Theorem 12 we showed that the polynomial fD − t has no line poly- nomial factors if t ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, by Theorem 17 any (D, B)-coloring is two-periodic whenever det(B) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Note that the results in Corollaries 18, 19, 20 and 21 were stated and proved in [29] in a slightly more general form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Indeed, in [29] it was proved that if a configuration c ∈ AZ2 is annihilated by � u∈D IX−u − B where B ∈ Zn×n is an arbitrary integer matrix whose determinant satisfies the conditions in the four corollaries and D is as in the corollaries, then c is necessarily periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' This kind of configuration was called a generalized centered function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' However, in Lemma 14 we proved that the vector presentation of any D-perfect coloring with matrix B is annihilated by this polynomial, that is, we proved that the vector presentation of a perfect coloring is a generalized centered function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' By analyzing the proof of Theorem 17 we see that the theorem holds also for generalized centered functions and hence the corollaries following it hold also for generalized centered functions, and thus we have the same results as in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 7 Forced periodicity of configurations of low abelian complexity In this section we prove a statement concerning forced periodicity of two-dimensional con- figurations of low abelian complexity which generalizes a result in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In fact, as in [7] we generalize the definition of abelian complexity from finite patterns to polynomials and prove a statement of forced periodicity under this more general definition of abelian complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let c ∈ {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , en}Zd and let D ⊆ Zd be a finite shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Consider the polynomial f = I · fD(X) = � u∈D IX−u ∈ Zn×n[X±1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The ith coefficient of (fc)v = � u∈D I · cv+u tells the number of cells of color ei in the D-neighborhood of v in c and hence the abelian complexity of c with respect to D is exactly the number of distinct coefficients of fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 21 More generally, we define the abelian complexity A(c, f) of an integral vector con- figuration c ∈ AZd where A is finite set of integer vectors with respect to a polynomial f ∈ Zn×n[X±1] as A(c, f) = |{(fc)v | v ∈ Zd}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' This definition can be extended to integral configurations and polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Indeed, we define the abelian complexity A(c, f) of a configuration c ∈ AZd where A ⊆ Z with respect to a polynomial f = � fiXui ∈ Z[X±1] to be the abelian complexity A(c′, f ′) of the vector presentation c′ of c with respect to the polynomial f ′ = I·f = � fi·I·Xui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Consequently, we say that c has low abelian complexity with respect to a polynomial f if A(c, f) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Clearly this definition is consistent with the definition of low abelian complexity of a configuration with respect to a finite shape since if c is an integral configuration, then A(c, D) = 1 if and only if A(c, fD) = 1, and if c is an integral vector configuration, then A(c, D) = 1 if and only if A(c, I · fD) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We study forced periodicity of two-dimensional configurations of low abelian complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Note that a configuration of low abelian complexity is not necessarily periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Indeed, in [30] it was shown that there exist non-periodic two-dimensional configurations that have abelian complexity A(c, D) = 1 for some finite shape D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' However, in [7] it was shown that if A(c, f) = 1 and if the polynomial f has no line polynomial factors, then c is two-periodic assuming that the support of f is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The following theorem strengthens this result and shows that the convexity assumption of the support of the polynomial is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' We obtain this result as a corollary of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Theorem 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let c be a two-dimensional integral configuration over an alphabet of size n and assume that it has low abelian complexity with respect to a polynomial f ∈ Z[x±1, y±1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If f has no line polynomial factors, then c is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If f has line polynomial factors in a unique primitive direction v, then c is v-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, if fD has no line polynomial factors or its line polynomial factors are in a unique primitive direction, then any configuration that has low abelian complexity with respect to D is two-periodic or periodic, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' By the assumption that A(c, f) = 1 we have f ′c′ = c0 1 for some c0 ∈ Zn where c′ is the vector presentation of c and f ′ = I · f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, f periodizes every layer of c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If f has no line polynomial factors, then by Theorem 2 every layer of c′ is two-periodic and hence c′ is two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If f has line polynomial factors in a unique primitive direction v, then by Theorem 2 every layer of c′ is v-periodic and hence also c′ is v-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Since c is periodic if and only if its vector presentation c′ is periodic, the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In [7] a polynomial f ∈ Z[X±1] is called abelian rigid if an integral configuration c having low abelian complexity with respect to f implies that c is strongly periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In the above theorem we proved that if a polynomial f ∈ Z[x±1, y±1] has no line polynomial factors then it is abelian rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Also, the converse holds as proved in [7], that is, if a polynomial f ∈ Z[x±1, y±1] has a line polynomial factor then it is not abelian rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' This means that if f has a line polynomial factor then there exists a configuration which is not two-periodic but has low abelian complexity with respect to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In fact this direction holds for all d, not just for d = 2 as reported in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 22 In the following example we introduce an open problem related to configurations of low abelian complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Example 24 (Periodic tiling problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' This example concerns translational tilings by a single tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In this context by a tile we mean any finite subset F ⊆ Zd and by a tiling by the tile F we mean such subset C ⊆ Zd that every point of the grid Zd has a unique presentation as a sum of an element of F and an element of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Presenting the tiling C as its indicator function we obtain a d-dimensional binary configuration c ∈ {0, 1}Zd defined by cu = � 1, if u ∈ C 0, if u ̸∈ C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The configuration c has exactly |F| different patterns of shape −F, namely the patterns with exactly one symbol 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In other words, it has low complexity with respect to −F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let f = fF = � u∈F X−u be the characteristic polynomial of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Since C is a tiling by F, we have fc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In fact, c has low abelian complexity with respect to f and −F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, by Theorem 23 any tiling by F ⊂ Z2 is two-periodic if fF has no line polynomial factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The periodic tiling problem claims that if there exists a tiling by a tile F ⊆ Zd, then there exists also a periodic tiling by F [20, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' By a simple pigeonholing argument it can be seen that in dimension d = 1 all translational tilings by a single tile are periodic and hence the periodic tiling problem holds in dimension 1 [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' For d ≥ 2 the conjecture is much trickier and only recently it was proved by Bhattacharya that it holds for d = 2 [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In [9] it was presented a slightly different proof in the case d = 2 with some generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' For d ≥ 3 the conjecture is still partly open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' However, very recently it has been proved that for some sufficiently large d the periodic tiling conjecture is false [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 8 Algorithmic aspects All configurations in a subshift are periodic, in particular, if there are no configurations in the subshift at all!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' It is useful to be able to detect such trivial cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The set S(D, b, a) = {c ∈ {0, 1}Z2 | (fD − (b − a))c = a1(X)} of all (D, b, a)-coverings is an SFT for any given finite shape D and non-negative integers b and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Hence, the question whether there exist any (D, b, a)-coverings for a given neigh- borhood D and covering constants b and a is equivalent to the question whether the SFT S(D, b, a) is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The question of emptiness of a given SFT is undecidable in general, but if the SFT is known to be not aperiodic, then the problem becomes decidable as a classic argumentation by Hao Wang shows: Lemma 25 ([32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If an SFT is either the empty set or it contains a strongly periodic config- uration, then its emptiness problem is decidable, that is, there is an algorithm to determine whether there exist any configurations in the SFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 23 In particular, if g = fD − (b − a) has line polynomial factors in at most one direction, then the question whether there exist any (D, b, a)-coverings is decidable: Theorem 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let a finite D ⊆ Z2 and non-negative integers b and a be given such that the polynomial g = fD − (b − a) ∈ Z[x±1, y±1] has line polynomial factors in at most one primitive direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Then there exists an algorithm to determine whether there exist any (D, b, a)-coverings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let S = S(D, b, a) be the SFT of all (D, b, a)-coverings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Since g has line polynomial factors in at most one primitive direction, by Theorem 2 every element of S is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Any two-dimensional SFT that contains periodic configurations contains also two-periodic configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, S is either empty or contains a two-periodic configuration and hence by Lemma 25 there is an algorithm to determine whether S is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' One may also want to design a perfect (D, b, a)-covering for given D, b and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' This can be effectively done under the assumptions of Theorem 26: As we have seen, if S = S(D, b, a) is non-empty, it contains a two-periodic configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' For any two-periodic configuration c it is easy to check if c contains a forbidden pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' By enumerating two-periodic configurations one-by-one one is guaranteed to find eventually one that is in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If the polynomial g has no line polynomial factors, then the following stronger result holds: Theorem 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If the polynomial g = fD − (b − a) has no line polynomial factors for given finite shape D ⊆ Z2 and non-negative integers b and a, then the SFT S = S(D, b, a) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' One can then effectively construct all the finitely many elements of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The proof of the first part of above theorem relies on the fact that a two-dimensional subshift is finite if and only if it contains only two-periodic configurations [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' If g has no line polynomial factors, then every configuration it periodizes (including every configuration in S) is two-periodic by Theorem 2, and hence S is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The second part of the theorem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=', the fact that one can effectively produce all the finitely many elements of S holds generally for finite SFTs in any dimension: Lemma 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Given a finite F ⊆ A∗ such that XF is finite, one can effectively construct the elements of XF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Given a finite F ⊆ A∗ and a pattern p ∈ AD, assuming that strongly periodic configurations are dense in XF, one can effectively check whether p ∈ L(XF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Indeed, we have a semi-algorithm for the positive instances that guesses a strongly periodic configuration c and verifies that c ∈ XF and p ∈ L(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A semi-algorithm for the negative instances exists for any SFT XF and is a standard compactness argument: guess a finite E ⊆ Zd such that D ⊆ E and verify that every q ∈ AE such that q|D = p contains a forbidden subpattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Consequently, given finite F, G ⊆ A∗, assuming that strongly periodic configurations are dense in XF and XG, one can effectively determine whether XF = XG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Indeed, XF ⊆ XG if and only if no p ∈ G is in L(XF), a condition that we have shown above to be decidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Analogously we can test XG ⊆ XF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 24 Finally, let a finite F ⊆ A∗ be given such that XF is known to be finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' All elements of XF are strongly periodic so that strongly periodic configurations are certainly dense in XF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' One can effectively enumerate all finite sets P of strongly periodic configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' For each P that is translation invariant (and hence a finite SFT) one can construct a finite set G ⊆ A∗ of forbidden patterns such that XG = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' As shown above, there is an algorithm to test whether XF = XG = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Since XF is finite, a set P is eventually found such that XF = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let us now turn to the more general question of existence of perfect colorings over alpha- bets of arbitrary size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let D ⊆ Z2 be a finite shape and let B be an n × n integer matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' To determine whether there exist any (D, B)-colorings is equivalent to asking whether the SFT S(D, B) = {c ∈ {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , en}Z2 | gc = 0} is non-empty where g = � u∈D IX−u−B since it is exactly the set of the vector presentations of all (D, B)-colorings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Theorem 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let a finite shape D ⊆ Z2, a non-negative integer matrix B and an integer t0 be given such that the polynomial fD(x, y) − t ∈ Z[x±1, y±1] has no line polynomial factors whenever t ̸= t0 and det(B − t0I) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Then there are only finitely many (D, B)-colorings and one can effectively construct them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In particular, there is an algorithm to determine whether there exist any (D, B)-colorings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Let S = S(D, B) be the SFT of the vector presentations of all (D, B)-colorings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' By Theorem 17 all elements of S are two-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Hence, S is finite, and the claim follows by Lemma 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Corollaries 18, 19, 20, 21 and 22 together with above theorem yield the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Corollary 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The following decision problems are decidable for a given matrix B satisfying the given conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The existence of (D, B)-colorings where D is the relative 1-neighborhood of the square grid and det(B − I) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The existence of (D, B)-colorings where D is the relative 1-neighborhood of the trian- gular grid and det(B + I) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The existence of (D, B)-colorings where D is the relative r-neighborhood of the square grid and B is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The existence of (D, B)-colorings where D is the relative r-neighborhood of the trian- gular grid and B is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The existence of (D, B)-colorings where D is the relative r-neighborhood of the king grid and det(B) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 25 Theorem 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Given a polynomial f in two variables with line polynomial factors in at most one parallel direction there is an algorithm to determine whether there exist any two- dimensional configurations over an alphabet of size n that have low abelian complexity with respect to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In fact, there are only finitely many such configurations and one can effectively construct all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The set {c ∈ {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' , en}Z2 | Ifc = 0} of the vector presentations of all configurations over an alphabet of size n with low abelian complexity with respect to f is an SFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' By Theorem 23 it contains only two-periodic configurations and hence it is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Thus, by Lemma 28 we have the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 9 Conclusions We studied two-dimensional perfect colorings and proved a general condition (Theorem 17) for their forced periodicity using an algebraic approach to multidimensional symbolic dy- namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' As corollaries of this theorem we obtained new proofs for known results of forced periodicity in the square and the triangular grid and a new result in the king grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' More- over, we generalized a statement of forced periodicity of two-dimensional configurations of low abelian complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Also, some observations of algorithmic decidability were made in the context of forced periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' All our results of forced periodicity of perfect colorings used Theorem 2 and hence con- cerned only two-dimensional configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' However, a d-dimensional version of Theorem 2 exists [15], and so we wonder whether an analogous result to Theorem 17 exists that would give a sufficient condition for forced periodicity of d-dimensional perfect colorings for arbi- trary dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Note that clearly every one-dimensional perfect coloring is necessarily periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Axenovich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' On multiple coverings of the infinite rectangular grid with balls of constant radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Discrete Mathematics, 268(1):31 – 48, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Ballier, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Durand, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Jeandal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Structural aspects of tilings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In Susanne Albers and Pascal Weil, editors, 25th International Symposium on Theoretical Aspects of Com- puter Science, volume 1 of Leibniz International Proceedings in Informatics (LIPIcs), pages 61–72, Dagstuhl, Germany, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Schloss Dagstuhl–Leibniz-Zentrum fuer Infor- matik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Bhattacharya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Periodicity and decidability of tilings of Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' American Journal of Mathematics, 142, 02 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [4] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Ceccherini-Silberstein and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Coornaert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Cellular Automata and Groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Springer Monographs in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Springer Berlin Heidelberg, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 26 [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Cohen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Honkala, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Litsyn, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Lobstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Covering Codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Elsevier, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Cox, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Little, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' O’Shea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Springer, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [7] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Geravker and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Puzynina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Abelian Nivat’s conjecture for non-rectangular pat- terns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content='04690, December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [8] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Godsil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Equitable partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Paul Erd¨os is Eighty Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 1, pages 173–192, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Greenfeld and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Tao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The structure of translational tilings in Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Discrete Analysis, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [10] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Greenfeld and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Tao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A counterexample to the periodic tiling conjecture, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [11] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Haynes, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Hedetniemi, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Slater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Fundamentals of Domination in Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' CRC Press, 1 edition, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [12] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Heikkil¨a, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Herva, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Kari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' On perfect coverings of two-dimensional grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In Volker Diekert and Mikhail Volkov, editors, Developments in Language Theory, pages 152–163, Cham, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Springer International Publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Kari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Theory of cellular automata: A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Theoretical Computer Science, 334(1):3– 33, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Kari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Low-complexity tilings of the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In Descriptional Complexity of Formal Systems - 21st IFIP WG 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content='02 International Conference, DCFS 2019, volume 11612 of Lecture Notes in Computer Science, pages 35–45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Springer, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Kari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Expansivity and periodicity in algebraic subshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Submitted for publication, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [16] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Kari and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Moutot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Nivat’s conjecture and pattern complexity in algebraic subshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Theoretical Computer Science, 777:379 – 386, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Kari and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Szabados.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' An algebraic geometric approach to Nivat’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In Proceedings of ICALP 2015, part II, volume 9135 of Lecture Notes in Computer Science, pages 273–285, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Kari and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Szabados.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' An algebraic geometric approach to Nivat’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' In- formation and Computation, 271:104481, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [19] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Kurka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Topological and Symbolic Dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Collection SMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Soci´et´e math´ematique de France, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Lagarias and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Tiling the line with translates of one tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Inventiones Mathematicae, 124:341–365, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 27 [21] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Lind and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Marcus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' An Introduction to Symbolic Dynamics and Coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Cambridge University Press, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Lothaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Combinatorics on Words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Cambridge Mathematical Library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Cambridge University Press, 2 edition, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [23] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Moore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Machine models of self-reproduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 1962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Morse and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Hedlund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Symbolic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' American Journal of Mathematics, 60(4):815–866, 1938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [25] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Myhill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The converse of Moore’s Garden-of-Eden theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 1963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [26] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Newman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Tesselation of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Number Theory, 9(1):107–111, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Nivat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Invited talk at the 24th International Colloquium on Automata, Languages, and Programming (ICALP 1997), 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [28] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Puzynina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Perfect colorings of radius r > 1 of the infinite rectangular grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' `Elektron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Izv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=', 5:283–292, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [29] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Puzynina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' On periodicity of generalized two-dimensional infinite words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Informa- tion and Computation, 207(11):1315–1328, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [30] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Puzynina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Aperiodic two-dimensional words of small abelian complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The Electronic Journal of Combinatorics, 26(4), 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [31] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Szegedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Algorithms to tile the infinite grid with finite clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content='98CB36280), pages 137–145, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' [32] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' Proving theorems by pattern recognition – II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' The Bell System Technical Journal, 40(1):1–41, 1961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} +page_content=' 28' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE4T4oBgHgl3EQfug0k/content/2301.05232v1.pdf'} diff --git a/ntE0T4oBgHgl3EQf8gLK/content/2301.02790v1.pdf b/ntE0T4oBgHgl3EQf8gLK/content/2301.02790v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..249bffc72b2157074dcef6bd1d5e93d436928030 --- /dev/null +++ b/ntE0T4oBgHgl3EQf8gLK/content/2301.02790v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c3d5d0be321475867913f9ac3dc9e8a53d9b359b5ab8b9ec3ffe49cf224b095f +size 1620133 diff --git a/ntE0T4oBgHgl3EQf8gLK/vector_store/index.faiss b/ntE0T4oBgHgl3EQf8gLK/vector_store/index.faiss new file mode 100644 index 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a/qdE4T4oBgHgl3EQfUgzF/content/tmp_files/2301.05017v1.pdf.txt b/qdE4T4oBgHgl3EQfUgzF/content/tmp_files/2301.05017v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..503c39a3aff6f0ee1c5a8dbc0d118188d21f6bb7 --- /dev/null +++ b/qdE4T4oBgHgl3EQfUgzF/content/tmp_files/2301.05017v1.pdf.txt @@ -0,0 +1,1754 @@ +arXiv:2301.05017v1 [eess.SP] 11 Jan 2023 +1 +Low PAPR MIMO-OFDM Design Based on +Convolutional Autoencoder +Yara Huleihel, and Haim H. Permuter, Senior Member, IEEE +Abstract +An enhanced framework for peak-to-average power ratio (PAPR) reduction and waveform design for +Multiple-Input-Multiple-Output (MIMO) orthogonal frequency-division multiplexing (OFDM) systems, +based on a convolutional-autoencoder (CAE) architecture, is presented. The end-to-end learning-based +autoencoder (AE) for communication networks represents the network by an encoder and decoder, +where in between, the learned latent representation goes through a physical communication channel. +We introduce a joint learning scheme based on projected gradient descent iteration to optimize the +spectral mask behavior and MIMO detection under the influence of a non-linear high power amplifier +(HPA) and a multipath fading channel. The offered efficient implementation novel waveform design +technique utilizes only a single PAPR reduction block for all antennas. It is throughput-lossless, as +no side information is required at the decoder. Performance is analyzed by examining the bit error +rate (BER), the PAPR, and the spectral response and compared with classical PAPR reduction MIMO +detector methods on 5G simulated data. The suggested system exhibits competitive performance when +considering all optimization criteria simultaneously. We apply gradual loss learning for multi-objective +optimization and show empirically that a single trained model covers the tasks of PAPR reduction, +spectrum design, and MIMO detection together over a wide range of SNR levels. +Index Terms +Deep learning, Autoencoder, Multiple-Input-Multiple-Output, Orthogonal frequency-division mul- +tiplexing, Peak-to-average power ratio, Wireless signal processing. +January 13, 2023 +DRAFT + +2 +I. INTRODUCTION +The Multiple-input-multiple-output (MIMO) scheme is a widely used technique for enhancing +channel capacity and transmission reliability, thanks to the diversity and multiplexing gains. +Orthogonal frequency division multiplexing (OFDM) is a waveform design method known for +providing high bandwidth efficiency, high throughput, simple equalization in wireless transmis- +sion, and efficient hardware implementation. For these reasons, it has been adopted as a standard +technology in various wireless communication systems, such as WiFi, 4G, and 5G standards for +wireless communications. Nonetheless, significant drawbacks of the OFDM multi-carrier system +appear in the form of adjacent channel power ratio (ACPR) limitations and the tendency to +produce signals with a high peak-to-average power ratio (PAPR) in the time-domain, since many +subcarrier components are added via a fast Fourier transform (FFT) operation. The contribution +of each subcarrier to the total power is dynamic, which makes the total power highly variable. +In particular, the high PAPR in MIMO-OFDM systems is exacerbated as the number of antennas +increases [1]. The demand for higher energy efficiency as well as reduced power consumption is +expected to increase for future radio systems [2]. Moreover, future communication system design +and equipment are expected to be more compatible with machine learning (ML) implementation +requirements, for example, allowing learning in the field to make some design choices [2]. +Accordingly, waveform designs using ML techniques are becoming significantly attractive. +A high power amplifier (HPA) is required to provide enough output power for reliable com- +munication. In practical systems, the HPA is not linear and distorts the transmitted signal. As a +result, severe non-linear signal distortions are found when these high PAPR signals pass through +the non-linear HPA. The resulting signal exhibits spectral regrowth in the form of in-band +signal distortions and out-of-band radiation [3], and the bit error rate (BER) increases. Hence, +it is crucial to develop PAPR reduction techniques for MIMO-OFDM systems to increase their +efficiency in handling large data streams and to reduce their error rates. Training and applying +DRAFT +January 13, 2023 + +3 +the PAPR reduction block to each antenna individually exacerbates the complexity, cost, and +redundancy in proportion to the number of transmit antennas in the system. Instead, in this +work, a single PAPR reduction block jointly operates on all antenna OFDM sequences, and it +is designed according to the maximum PAPR value of all antenna sequences. +A central difficulty of the multiple transmitter (TX) and receiver (RX) antenna structure is +posed by the need for joint detection of the data symbols sent by each transmitter. Unfortunately, +the optimal MIMO detection solution imposes an NP-hard problem on the receiver. Consequently, +various sub-optimal yet feasible detection algorithms have been proposed. Other than classical +model-driven solutions, an increasing effort has been dedicated to ML, and, specifically, deep +learning (DL) based techniques to solve the MIMO detection problem, and more generally, various +wireless communication tasks. +The design of OFDM waveform signals aims to simultaneously achieve a high data rate, high +spectral efficiency (measured by the ACPR), and low computational complexity [4], [5]. This +design is highly affected by the non-linear effects of the HPA. While keeping the PAPR level +low is favorable, it is of particular importance to have acceptable signal spectral behavior and +BER, which are often referred to as waveform design. In order to fulfill that, this work suggests +an overall communication network multi-objective optimization, such that the transmitter, HPA, +channel, and the receiver, are represented as a single optimization block. Instead of separately +optimizing different components of the transmitter and the receiver, an end-to-end convolutional- +autoencoder (CAE) learning model is proposed. This end-to-end optimization block is presented +as a constrained optimization problem where the transmitted signal estimation is the objective, +and the PAPR and ACPR requirements are the constraints. MIMO detection over multiple channel +realizations is performed as a part of the end-to-end joint optimization model, utilizing an iterative +approach based on convolutional layers, and a gradual loss learning approach. We evaluate +the performance of our algorithms over both additive white Gaussian noise (AWGN) and 3rd +Generation Partnership Project (3GPP) fading channels [6]. By analyzing the BER, PAPR, and +January 13, 2023 +DRAFT + +4 +spectrum performance, we show that the proposed end-to-end learning approach can integrate +different communication network blocks to balance those performance objectives successfully. +We show that the suggested scheme is able to achieve better spectral performance for higher +HPA efficiency operation. Various OFDM PAPR reduction techniques have been proposed in +the literature, as well as for MIMO detection. Generally, these techniques can be categorized +into model-driven and data-driven techniques. The first category refers to standard approaches +in classical communications theory, while the second relies on recently developed approaches +based on ML techniques. The following subsections review different earlier solutions for the +above-mentioned problems. +A. Classical Approaches (Model Driven) for MIMO Detection +Many MIMO detection algorithms have been developed over the years. The maximum like- +lihood estimation (MLE) solution is optimal for the joint detection of transmitted symbols in a +MIMO system. However, its exponential computational and time complexity (due to exhaustive +searches over all possible transmitted signals) render it infeasible when the number of transmitters +and the modulation order are high. An example of suboptimal high accuracy non-linear detection +algorithms are those based on sphere decoding (SD) [7], but they become computationally +expensive as the number of antennas grows. The general idea is based on a lattice search for a +solution in an iterative manner, and the accuracy/complexity ratio strongly depends on the value +chosen for the radius parameter. More advanced detectors include the successive interference +cancellation (SIC) based detectors [8] and the semi-definite relaxation detectors [9]. +B. Machine-Learning-Based Schemes (Data Driven) for MIMO Detection +The motivation for DL-based detectors is to enhance the performance of classical model- +driven detection algorithms by learning, from the training data set, an optimized mapping of +the received signals onto the transmitted symbols. In [10], [11], an iteration-based algorithm +for implementing a receiver for MIMO detection was suggested. One of the highlights of the +DRAFT +January 13, 2023 + +5 +presented model’s framework is that it enables training through different random communication +channel realizations. In [12], a model-based algorithm was suggested, where a classical SD +algorithm was integrated with a neural network (NN) that was trained to optimize the selection of +the initial radius. In [13], a neural detector-based transformer architecture implements a recurrent +estimation scheme by learning an iterative decoding algorithm. +In [14]–[16] an AE was offered to design a physical layer in which DL-based CSI encoding +was suggested for different scenarios to achieve lower BER together with better robustness to +the wireless channel characteristics. Thorough surveys and analysis are presented in [17]. +C. Classical Approaches (Model Driven) for PAPR Reduction +PAPR reduction schemes are roughly classified into three categories. The signal distortion +category consists of techniques such as clipping and filtering (CF) [18], [19], which limit the +peak envelope of the input signal in the time domain to a predetermined value. The multiple +signaling probabilistic category includes methods such as selective mapping (SLM) [19], [20], +partial transmit sequence (PTS) [20], ton reservation and ton injection [21], and constellation +shaping [22]. The main principle of SLM is to generate different candidates for each OFDM block +by multiplying the symbols vector with a set of different pseudo-random sequences and choosing +the candidate with the lowest PAPR. The third category is the coding technique category [21], +[23], attempting to reduce the occurrence probability of the same phase signals. +Earlier schemes were mainly developed for single-antenna systems. Extended works which +applied the single-antenna PAPR reduction schemes on each antenna of the MIMO configuration +separately are found in, e.g., [24], but those required considerable computations, cost, and +complexity. Model-driven approaches to simultaneously reduce PAPR over all antennas were +also proposed. In [25], instead of applying SLM to each antenna, the sequence with the highest +PAPR over all transmit antennas was selected. Usually, SLM and PTS methods demand side +information (SI) to be sent to the receiver along with each transmitted data block for retrieving +the original data. The need for SI requires extra bandwidth overhead, and the incorrect detection +January 13, 2023 +DRAFT + +6 +of the SI bits over the channel will lead to significant degradation in the BER performance of +the receiver in the MIMO-OFDM system. +D. Deep-Learning-Based Schemes (Data Driven) for PAPR Reduction +In recent years much research has been dedicated to applying DL techniques in the design +and optimization of wireless communication networks, e.g., [4], [11], [26]. Several papers +have proposed DL methods to handle PAPR reduction. For example, the authors of [27], [28], +added a NN to reduce the complexity of the active constellation scheme, followed by CF. In +[29], [30] the authors present an AE solution for PAPR reduction, while minimizing the BER +degradation. In [31] a CAE was suggested for the implementation of an end-to-end SISO − OFDM +communication network that simultaneously reduces the PAPR and reconstructs the transmitted +symbols, while keeping acceptable spectral requirements. Another learning-based approach, +which considers the reduction of the PAPR and ACPR together with the maximization of the +achievable information rate for a single-carrier waveform above multipath channels, was proposed +in [32]. The authors in [33] proposed a deep NN combined with SLM to mitigate the high PAPR +issue of OFDM signal types. +All of the above papers consider a SISO network. A PAPR reduction scheme assisted by DL +for a MIMO-OFDM system was suggested in [34]. The authors apply selective tone reservation +[35] on each antenna separately and then apply unused beam reservation [36] on all antennas +together. An ML-based method for approximating the optimal tabular hyperparameters required +for using selective tone reservation and unused beam reservation was suggested. +E. Main Contributions +Some of the aforementioned PAPR reduction approaches suffer from in-band interference, out- +of-band distortions, and high computational complexity. Moreover, published ML-based solutions +mostly handle single antenna scenarios. Those who deal with PAPR reduction for MIMO systems +use ML only for the PAPR reduction block and not for the end-to-end network implementation. +This paper aims to handle the PAPR problem in MIMO systems as an integral part of a waveform +DRAFT +January 13, 2023 + +7 +design objective. In particular, we design a communication system that simultaneously achieves +PAPR reduction, acceptable spectral behavior of the PA’s output, and good BER performance. +The suggested end-to-end network aims to resolve the MIMO detection problem as a part of the +other mentioned objectives. To the best of our knowledge, this approach is new. Novelties we +introduce include using a CAE combined with a gradual loss learning technique to handle the +multi-objective optimization of the network, and adding the effect of the HPA on an integrated +end-to-end MIMO communication system. We present an iterative MIMO detection algorithm +integrated into transmitter-receiver end-to-end communication system joint optimization. We +demonstrate our algorithm’s results on 5G MIMO-OFDM Matlab toolbox simulated data, and +we compare our method with classical methods for PAPR reduction and waveform design, and +show competitive results for all the objectives mentioned above. The proposed algorithm offers +performance improvement for future wireless communication systems. We show that our model +provides competitive PAPR reduction, waveform design, and detection results. +The rest of this paper is structured as follows. In Section II, the problem is defined and +formulated separately for MIMO detection, and for PAPR reduction as a part of the MIMO-OFDM +system. We then present the proposed DL-based system architecture for the multi-objective +optimization and explain the training procedure in Section III. Section IV provides detailed +numerical simulation results and insights. Finally, Section V gives concluding remarks. +II. NOTATION AND PROBLEM DEFINITION +In this section, we introduce the notation and the problem definition. +A. Notation +Throughout this paper, we use the following notations. The set of real numbers is denoted +by R, while the set of complex numbers is denoted by C. Random variables will be denoted +by capital letters, and their realizations will be denoted by lower-case letters, e.g., X and x, +respectively. Calligraphic letters denote sets, e.g., X . We use the notation Xn to denote the +random vector (X1, X2, . . . , Xn) and xn to denote the realization of such a random vector. +January 13, 2023 +DRAFT + +8 +The expectation operator is denoted by E [·]. (·)∗, (·)† denote the conjugate, and pseudo-inverse +operators, respectively. +B. Problem Definition +In this section, we describe mathematically each part of the integrated problem of MIMO de- +tection together with PAPR reduction and spectrum constraints. First, we give a brief introduction +to the end-to-end setup used in our system. +1) MIMO detection model: Let us assume a standard MIMO-OFDM system with Nt transmit +antennas and Nr receive antennas. Transmission is considered over a memoryless complex- +valued channel model, while assuming frequency flatness and slow fading. A MIMO system can +be modeled by the following complex baseband model: +y = Hx + n, +(1) +where x ∈ CNt is the transmitted complex symbol vector drawn from a finite discrete constel- +lation of size |M|, H ∈ CNr×Nt is the complex baseband channel matrix that is related to a +specific subcarrier, n ∈ CNr is complex background AWGN seen at the receiver, and y ∈ CNr is +the received complex vector resulting from the propagation of the transmitted symbols through +the channel contaminated by AWGN. +As the proposed implementation is based on a real-valued NN model determined by the DL +Pytorch library, (1) is expressed with real values by splitting and concatenating each signal into +its real and imaginary parts: +x = + + +Re{x} +Im{x} + + , y = + + +Re{y} +Im{y} + + , n = + + +Re{n} +Im{n} + + , H = + + +Re{H} +−Im{H} +Im{H} +Re{H} + + . +(2) +In the MIMO detection problem, the objective is to detect the transmitted symbols, x, given +the received data y. The optimal solution for the MIMO detection of the transmitted symbols +problem defined above is given by the MLE algorithm, that is, +ˆxmle = arg min +x∈X Nt +||y − Hx||2, +(3) +DRAFT +January 13, 2023 + +9 +where X denotes the set of possible transmitted symbols (i.e., signal constellation). The solution +of (3) requires an exhaustive search over all |M|Nt possible transmitted vectors. Therefore, +it is infeasible for an actual implementation where large-scale MIMO setups and/or a large +constellation are in use. +2) PAPR problem in MIMO-OFDM: In an OFDM system with N complex orthogonal sub- +carriers, the discrete-time transmitted OFDM signal at the nt antenna, is given by the inverse +discrete Fourier transform (IDFT): +xnt,n = +1 +√ +N +N−1 +� +k=0 +Xnt,kej 2π +LN kn, 0 ≤ n ≤ LN − 1, 1 ≤ nt ≤ Nt, +(4) +where {Xnt,k}N−1 +k=0 are random input symbols per antenna, modulated by a finite constellation, +and L ≥ 1 is the over-sampling factor (L = 1 is the Nyquist sampling rate). As shown in +[21], oversampling by a factor of four results in a good approximation of the continuous-time +PAPR of complex OFDM signals. The discussed problem considers non-linear HPAs at each +of the Nt TX branches. We assume that the HPAs in all branches have the same non-linear +characteristic, which is a reasonable assumption, considering current wireless MIMO systems. +Also, in a discrete implementation, the same HPAs are usually used. +The PAPR of the transmitted signal in (4) is defined as the ratio between the maximum peak +power and the average power of the OFDM signal. Specifically, the PAPR at the nt-th transmit +antenna is defined by: +PAPRnt ≜ max0≤n≤LN−1 |xnt,n|2 +E|xnt,n|2 +. +(5) +For the entire MIMO-OFDM system, the PAPR reduction method we use will consider the +maximum PAPR among all Nt transmit antennas, as the same PA model is used in all branches: +PAPRMIMO−OFDM = +max +1≤nt≤Nt PAPRnt. +(6) +As HPA non-linearity causes spectral regrowth, an important assessment for the spectral purity +of the system is the ACPR criterion, which is the ratio between the power of the adjacent channel +January 13, 2023 +DRAFT + +10 +and the power of the main channel. Following [6], we define it as +ACPR ≜ +max +�� 3BW/2 +BW/2 Pss(f) df, +� BW/2 +−3BW/2 Pss(f) df +� +� BW/2 +−BW/2 Pss(f) df +, +(7) +where Pss(·) is the power spectral density (PSD) of the signal at the HPA’s output, and BW is +the primary channel bandwidth, which is assumed to be equal to the data signal bandwidth. +A +block +diagram +of +the +communication +system +model +is +shown +in +Fig. +1. + +Q + +PSfrag replacements +Xk +xn +PAPR & Spectral +Optimization +Enc +Filter +xF +n +Power +Amplifier +G [·] +xP +n +Channel ++ +yn +DFT +Yk +Reconstruct +ˆXk +IDFT +& Detect +wn +Fig. 1: General system model diagram. +Specifically, the encoder and fil- +ter blocks mitigate the PAPR effect +and design the waveform to com- +ply with predefined spectral mask +requirements. For example, the en- +coder block can model a clipping +operation, while the filter can be a standard band-pass filter (BPF). The filtered signal xF +n is +amplified by a non-linear HPA. The amplified signal, xP +n = G(xF +n), is transmitted through a +fading channel with AWGN. The channel decoder receives the noisy signal and attempts to +reconstruct and detect the transmitted signal. For model-driven approaches, a classical detection +algorithm, e.g., MLE, is applied for detecting the estimated symbol denoted by ˆXk. +The role of the HPA is to convert the low-level transmission signal to a high power signal, +capable of driving the antenna at the desired power level. The HPA has to operate close to its +saturation region for maximal power efficiency. If the HPA exceeds the saturation point and +enters the non-linear area of operation, the output signal becomes non-linear. Accordingly, to +operate the amplifier only in the linear region, we need to make sure that the amplifier operates +at a power level that is lower than the saturation point. This is achieved by down-scaling the +input signal by an input back-off (IBO) factor. The drawback of adding the IBO attenuation is +that the output power decreases, which makes the HPA power-inefficient. +DRAFT +January 13, 2023 + +11 +Saturation +point +Actual +operating +point +Input +Back-Off +(0.707) +Output Back-Off +PSfrag replacements +Ain +Aout +3dB +Fig. 2: RAPP HPA output versus input +signal for different smoothness p values. +There are several commonly used models for the +non-linearity of an HPA. Here, we will focus on +the RAPP behavioral amplifier model [37], which +is very accurate for solid-state-power amplifiers. +The model’s AM/AM conversion is given by +G(Ain) = v · Ain · +� +1 + +�vAin +A0 +�2p�− 1 +2p +, +(8) +where Ain is the input amplitude, A0 is the limiting +output amplitude, v is the small signal gain, p is a +smoothness parameter controlling the transition from the linear region to the saturation region, +and G(A) is the output amplitude. Figure 2 shows RAPP HPA outputs versus input for several +smoothing factor values. +III. PROPOSED WAVEFORM DESIGN STRUCTURE +In this section, we describe our multi-objective optimization CAE model architecture. Mo- +tivated by research evidence of powerful learning ability, under the same conditions of the +MIMO-OFDM examined structure, it is expected that the proposed CAE model will achieve +good enough performance to be compared with classical PAPR reduction methods combined +with the MLE detector. We first briefly discuss the general CAE concept. Then, we describe our +algorithm building blocks, and the joint PAPR reduction, spectral design, and detection operation +in detail. The motivation and structure of the iteration-based decoder with regard to handling +the MIMO detection problem as a part of the CAE network will be explained. The proposed +architecture in Fig. 3 is then elaborated, including the Bussgang’s non-linearity compensation +theorem, followed by a description of the gradual learning process. Last, the training procedure +of the CAE multi-objective optimization network operation will be described. +Fig. 3a shows the overall end-to-end communication network structure implemented by a +CAE model, where the encoder and the decoder are the trainable blocks. Each mentioned loss +January 13, 2023 +DRAFT + +12 + + +/ + + +PSfrag replacements +Xk +Zero +Zero +pad +IFFTxn +2 +calc +xB +n +Filter +BO +PA +xF +n +PAPR +PAPR +Calculation +Calculation +Calculation +L2b +xP +n +MIMO +Channel +Xk +yn +ACPR +L3 +Loss +L1 +Reconstruction +Encoder f(x) +α +α +Decoder g(y) +Nt × Nsc +Nt × Nsc +Nt +Nt +Nt +Nt +Nt +Nt +Nt +Nt +Nt +Nt +Nt +Nt +Nr +Yk +xE +n +unpad +Nr × Nsc +FFT +Transmitter +Receiver +ˆXk ++wn +L2a +(a) Conv-AE overall scheme. ++ +PSfrag replacements +Input Tensor +Conv+BN+GELU +Fully Connected +Predicted Tensor +Kernel +Nsc +Nsc +Nsc +Nsc +Power Norm Layer +Nt +Nt +Nt +Nt +ch2 +L2 +loss +L3 +fc1 +conv1 +conv2 +PAPR Reduction Block - Encoder f(x) +conv3 +3 +3 +3 +ch1 +ch1 +1 +1 +1 +(b) PAPR reduction block - Encoder scheme f(x). +. . . +PSfrag replacements +Xi−1 +H∗HXk+1 +Xk+1 +H∗H() +ith iteration +Nsc +Nsc +Nsc +Nsc +Nsc +Nsc +Nr +Nr +Nr +Nr +ch2 +ch2 +H∗Y +H∗Y +H∗Y +H∗HXk +Xk +H∗HXi−1 +Concat +Concat +conv4 +conv4 +conv5 +conv5 +Detection Block - Decoder g(x) +fc2 + Softmax +Layer +Layer +kth iteration +3 +3 +3 +3 +3 +3 +3 +3 +ch1 +ch1 +fc2 +Xi +(c) Detection block - Decoder scheme g(x). +Fig. 3: Structure of the proposed conv-AE. +component is calculated using the operations appearing in the red blocks in the figure. It can +be noticed that there are two PAPR calculation-based loss components, as will be detailed later, +that helped achieve improved spectral behavior and BER results. Fig. 3b describes the encoder +structure. It is constructed of 2D convolutional layers to be compatible with the input example +dimensions, a fully connected layer, and a power normalization layer at the output. We also added +a skip connection to improve stability and performance. Fig. 3c shows the decoder structure. It +is also implemented with 2D convolutional layers. To be able to optimize the network under +DRAFT +January 13, 2023 + +13 +different communication channel realizations, we used an iterative-based solution to the MIMO +detection problem. We emphasize that the presented CAE model, once trained on a single training +set, presents competitive results for the discussed multi-objective optimization problem in the +presence of a wide range of noise power levels, without any knowledge of the SNR level. +A. Convolutional Autoencoder (CAE) +The proposed implementation uses an AE learning system based on a convolutional neural +network (CNN). The general structure of an AE consists of two main blocks: the encoder f(x) +and the decoder g(x), where x is the input data. The AE is trained to minimize a certain joint +loss function, which we denote by L(x, g(f(x))). An end-to-end communication system can be +interpreted as an AE in which the encoder and the decoder are part of the transmitter and the +receiver, respectively, and can jointly optimize them through an end-to-end learning procedure. +CNNs are widely used for feature extraction and pattern recognition in ML models. Compared +with a fully connected (FC) network, a CNN has significantly fewer connections between adjacent +layers, and thus fewer parameters and weights to train, resulting in lower complexity and much +faster training. +B. Iteration-Based Model Motivation +An iterative approach based on convolutional layers, was adopted to implement the decoder. +The MIM0 detector was designed to solve the MLE optimization (3) using the projected gradient +descent approach, where the input to the detector is a linear combination of features instead of +the channel output directly. Utilizing such detectors significantly improved the detection results +when various channel realizations were integrated. The detector input feature selection was based +on the following projected gradient decent approach used to optimize (9): +||y − Hx||2. +(9) +As shown in [10], [11], such iterations are updated by +ˆxk+1 = Pc +� +ˆxk − δk +∂||y − Hx||2 +∂x +���� +x=ˆxk +� += Pc +� +ˆxk − δk +� +HHy − HHH ˆxk +�� +, +(10) +January 13, 2023 +DRAFT + +14 +where ˆxk is the objective estimation in the k’th iteration, HH indicates the conjugate transpose +operation over the channel matrix H, δk is the step size, and Pc is the non-linear projection +operator. The above can be adapted to a deep NN iterative solution by the following linear +combination form: +ˆxk+1 = Pc(ˆxk + δ1kHHy + δ2kHHH ˆxk), +(11) +where δ1k and δ2k are learned hyper-parameters to be optimized. One of the motivations for using +the described iterative decoder approach was a former work published in [10], [11]. In our work, it +is designed as part of the joint encoder-decoder multi-task optimization. Also, it was implemented +with convolutional layers, that enabled better computational and performance capabilities for +the joint optimization end-to-end system, and the per subcarrier/antenna alternating analysis. A +softmax layer is added at the output of the CAE to generate probabilistic outputs. In the following +section, we provide the complete detection procedure. +C. Proposed CAE Architecture +This section introduces the suggested CAE learning network implementation of the MIMO- +OFDM system for the multi-objective optimization task. We consider a MIMO-OFDM scheme +with Nt transmit antennas and Nr receive antennas, where the OFDM is of order K. The input +is represented by a matrix in the frequency domain, i.e. +X = + + + + + + + + + + +X(1)(1) +X(1)(2) +· · · +X(1)(K) +X(2)(1) +X(2)(2) +· · · +X(2)(K) +... +... +... +... +X(Nt)(1) +X(Nt)(2) +· · · +X(Nt)(K) + + + + + + + + + + +, +(12) +where, for any nt ∈ [1, Nt] and k ∈ [1, K], X(nt)(k) is a M-QAM constellation complex-valued +symbol. +In Fig. 3a, we illustrate the general structure of the end-to-end communication network +implemented by the CAE configuration. Specifically, we consider a transmitter that takes the +DRAFT +January 13, 2023 + +15 +two-dimensional matrix X as an input. The transmitter’s output goes through a MIMO channel, +together with AWGN. Finally, the noisy channel outputs are fed into a receiver to estimate X. +The operations within the transmitter and the receiver are described below. +• Transmitter: the input signal X is zero-padded on the subcarriers’ dimension and converted +to the time domain via an IFFT applied on each of the transmitter branches, outputting +{xn}LN−1 +n=0 . These symbols serve as the input to the encoder, which acts as a PAPR reduction +block, followed by a BPF filter for optimizing the spectral behavior by reducing the out- +of-band radiation. Its frequency response is a rectangular window with the same bandwidth +as X(nt) +k +. Then, a predefined IBO is applied just before the signal is amplified by the HPA. +• Receiver: the distorted OFDM symbols are divided by an α factor to compensate for the +non-linear distortions, as will be detailed in the following. Finally, the proposed CAE decoder +reconstructs and detects the estimated MIMO-OFDM transmitted signals. +The encoder comprises three convolutional layers, and the decoder is composed of iterative +construction of convolutional layers. Each convolutional layer is followed by a non-linear ac- +tivation function and batch normalization [38], and then a fully connected layer. In addition, a +residual connection is added to the encoder block, which sums (element-wise) the input to +the second convolutional layer and the output of the third convolutional layer. It turns out +that this modification improves the overall performance of the suggested scheme significantly. +The intuition is that adding another path for data to reach the latter parts of the NN makes it +easier to optimize the mapping [39]. Furthermore, the encoder has a power normalization layer, +which ensures that the transmitted signal meets the power constraints of unit average energy +per OFDM symbol. This way, the intended SNR is maintained. We tested several activation +functions, including sigmoid, rectified linear unit (RELU), Gaussian error linear unit (GELU), +and scaled exponential linear unit (SELU) [40]. Empirically, it was found that SELU activation +provides the best results for our CAE scheme. +As illustrated in Fig. 3a and Fig. 3b, since the encoder is responsible for the PAPR reduction, +January 13, 2023 +DRAFT + +16 +which is calculated per OFDM symbol, we start with per antenna analysis, where each antenna is +treated separately. A one-dimensional kernel per TX branch handles this. The encoder architecture +can be described by the following: +f(x) = ρLf +�����W f +Lf +� +ρLf −1 +� +... +� +ρ1 +�����W f +1x + bf +1 +���� +bnorm +��� +... +� ++bf +Lf +���� +bnorm ++ +�����W f +1x + bf +1 +���� +bnorm +�� +, +(13) +where Lf is the number of the encoder’s convolutional layers, W f +i , and bf +i are the encoder’s +weight matrix and bias vector, respectively, for the i’th layer, with size determined as a part of +the network design. ρi(·) is the activation function of the i’th layer, and bnorm means the layer +passes through a batch normalization. +The next part of this process applies the non-linear HPAs on each TX branch of the transmitter +time domain signals, each composed of all subcarriers. The signal is then converted via FFT to the +frequency domain, and the zero-unpadding block removes the out-of-band samples. Afterward, +frequency domain analysis is performed on each subcarrier transmitted through all Nt antennas. +Each subcarrier is transmitted through its related complex baseband channel described by a +(Nr × Nt) matrix, and the AWGN is added as well. +To continue with the per subcarrier analysis on the receiver side, we need to overcome the +non-linearity of the HPA. To that end, we compensate the receiver input signal by applying +an attenuation factor represented by α. Bussgang’s decomposition theorem [41] states that if a +zero-mean Gaussian signal passes through a memoryless non-linear device, then the output-input +cross-correlation function is proportional to the input autocovariance. Accordingly, the value of +α is chosen to minimize the variance of the non-linear signal distortions, such that it is attempted +that the transmitted signal in each transmitter branch is linearly separated, and thus represented +as a sum of the signal and distortion. It can be shown that +α = E +� +xF +nxP +n +� +E (|xFn|2) , +(14) +DRAFT +January 13, 2023 + +17 +where xP +n is the complex output signal of the PA, and xP +n is its complex conjugate. By assuming +that the PSD of the in-band distortion is approximately flat [42] in the frequency domain, the +output signal of the HPA on the k-th subcarrier can then be expressed as +XP(k) = α(k)XF(k) + D(k), +(15) +where D(k) is the non-linear distortion on the k-th subcarrier. The same model is assumed for +all PAs; therefore, it can be concluded that αnt = α. +At the MIMO decoder, Fig. 3c, the per subcarrier analysis is continued, meaning that different +subcarriers of the same RX branch will not be mixed. To generalize our end-to-end structure, +we modified the 1D kernel to a 2D kernel at the decoder part. It also helped better reconstruct +the signal after the encoder layers. As explained in the previous sub-section, we use an iterative +procedure to implement the decoder, which is designed for signal reconstruction and detection. +A general mathematical description of one iteration, k, of the presented decoder is given by +dk = +� +ˆxk−1, δ1kHHy, δ2kHHH ˆxk−1 +� +ˆxk = g(dk) += ρLg,k +�����W g +Lg,k +� +ρLg−1,k +� +... +� +ρ1,k +�����W g +1,kdT +k + bg +1,k +���� +bnorm +��� +... +� ++ bg +Lg,k +���� +bnorm +� +, +where Lg, W g +i , and bg +i , have the same definitions as described for the encoder’s block, only +that these apply for the decoder. The decoder’s input features vector, dk, was initialized by +randomizing a prediction ˆx0. Initialization by zeros resulted in performance degradation. +D. Training of the CAE Network +We train a single CAE model for all tested SNR values. We use the AdamW optimizer [43] +that runs back-propagation to optimize the model during training. This optimizer is designed to +improve gradients when L2 regularization is used. Our loss function is set to solve the constrained +optimization problem by handling three objectives: accurate signal reconstruction (minimal BER), +minimal PAPR, and acceptable ACPR. +We solve this constrained optimization problem by recasting it as an unconstrained problem +by constructing the Lagrangian function and augmenting the objective function with a quadratic +January 13, 2023 +DRAFT + +18 +penalty term [44]. The augmented Lagrangian (AL) combines the Lagrangian formulation with a +weighted quadratic penalty function. The general AL for an inequality-constrained problem can +be described by +Fρk(x, λk) = f(x) + λk +1c1(x) + 1 +2ρk +1 ∥c1(x)∥2 +2 + 1 +2ρk +2 +�� +max{0, λk +2 + ρk +2c2(x)} +�2 − (λk +2)2� +, +(16) +where f denotes the objective function, ρk ≜ (ρk +1, ρk +2) are positive penalty parameters, λk ≜ +(λk +1, λk +2) are the Lagrangian multipliers, the c1-involved expressions handle the equality constraint, +and c2 is for the inequality constraint. Equation (16) considers the elimination of a slack variable +s ≥ 0 that was introduced in the representation of the inequality constraint to transform it into +a relaxed equality constraint. As suggested in [45], the minimizer s = max{0, c2(x) − λ2 +1 +ρ2} +was used. k is the iteration number for updating the Lagrangian multipliers and penalty term, +according to the following rule derived by the dual ascent method [44], +xk+1 := arg min +x +Fρk(x, λk) +(17) +λk+1 +1 +:= λk +1 + ρk +1c1(xk+1) +(18) +λk+1 +2 +:= max{0, λk +2 + ρk +2c2(xk+1)}. +(19) +We saw better convergence and more stable results for different BO values by adding the +quadratic penalty function and adaptively updating the multipliers instead of keeping them +constant. Since adaptive penalty parameter update was not beneficial for the examined cases, it +was added as a fixed hyperparameter. +Following the above-described general inequality constraint optimization problem, the formu- +lation of our loss function based on the appropriate objective and constraints, represented by +DRAFT +January 13, 2023 + +19 +four loss components L1, L2a, L2b, and L3, is +L(x, ˆx, λk +2a, λk +2b, λk +3) = L1(x, ˆx) + λk +2aL2a(x) + ρ2a +2 ∥L2a(x)∥2 +2 ++ λk +2bL2b(x) + ρ2b +2 ∥L2b(x)∥2 +2 + 1 +2ρ3 +�� +max{0, λk +3 + ρ3L3(x)} +�2 − (λk +3)2� +, +(20) +where λ2a, λ2b, λ3, ρ2a, ρ2b, and ρ3 are the appropriate Lagrange multipliers and penalty pa- +rameters, accordingly. These are considered hyper-parameters, which balance the contribution +of each loss component to the joint loss function. We start with a moderate value of λ2a, λ2b +and λ3 and then iterate for a better value according to the resulting PAPR loss of each iteration +and some predetermined PAPR threshold value. Better performance was observed for relatively +small λ2b values, with very low ρ2b, meaning that λ2b was kept almost constant during training. +The loss function we use for optimizing the signal reconstruction and detection is the sum +of negative log loss function of the predicted output probability of the real and imaginary parts +of each symbol, with L2 regularization to reduce over-fitting. Denoting by x the input sample +(which is also the output target), ˆx as the estimated signal, Θ as the model’s weights, and λ1 +as a hyperparameter for tuning the L2 regularization, the loss function for each OFDM MIMO +sample is given by, +L1(x, ˆx) = − +� Nt +� +j=1 +Nsc +� +s=1 +Nc +� +q=1 +1 +� +Re{xj} = lq +� +log Pθ +� +Re +� +ˆxj� += lq +� ++ +Nt +� +j=1 +Nsc +� +s=1 +Nc +� +q=1 +1 +� +Im{xj} = lq +� +log Pθ +� +Im +� +ˆxj� += lq +�� ++ λ1 ∥Θ∥2 +2 , +(21) +where Nc = +� +|M| denotes the number of the real value possibilities, lq, of each of the real +and imaginary parts of the transmitted modulated symbol. +The PAPR minimization part is handled with two loss components, where one, L2a, is +calculated according to the BPF input, xE +n, and the other one, L2b, according to the BPF output, +January 13, 2023 +DRAFT + +20 +xF +n (cf. Fig. 3a). These are our equality constraints, defined by +L2a(x) = PAPR{xE +n}, +(22) +L2b(x) = PAPR{xF +n}. +(23) +Other than the role of PAPR minimization handled by either of the components, L2a(x), signif- +icantly improved the BER result together with the ACPR, while L2b(x), enabled us to control +and obtain better spectral performance, meaning lower ACPR with lower output back-off (OBO) +values. The ACPR loss component is given by +L3(x) = ACPR{xP +n} − ACPRreq, +(24) +where xP +n is the PA’s output, and ACPRreq is the required ACPR value, which is usually dictated +by a standard. ACPRreq was set according to the 5G standard requirements for high spectral +purity: ACPRreq ≤ −45dB [6]; thus, L3(x) defines our inequality constraint. +We have applied a gradual loss learning technique. In the first stage, the loss function consisted +only of L1 and optimized only the reconstruction loss. Then, after a predetermined number of +epochs, the loss function defined in (20) was used to reduce the PAPR and improve the spectral +behavior. The gradual loss learning enables better control and stability in tuning the different +criteria’ trade-offs. +IV. RESULTS AND INSIGHTS +A. Data Generation and Experimental Setup +To train and test the proposed data-driven model, the MATLAB® 5G Toolbox™ [46] was +used. This toolbox provides 5G radio-standard-compliant functions to generate accurate data +for MIMO-OFDM transmission, according to specified constellation sizes and examined MIMO +setups. MIMO-OFDM transmissions over fading channels were simulated, where TDL-D type +channels - a 13 delay tap channel with a 30ns delay spread, as described in the 3gpp specification +document [6], were used for our implemented algorithm. +DRAFT +January 13, 2023 + +21 +We consider a MIMO-OFDM system with K =72 subcarriers over 14 OFDM symbols per +frame. 4375 batches of 32 MIMO samples each were used for a single training set, where +the input and output of the CAE sample shape is [2/4 − antennas, (72 − subcarriers) × +(2 − complex − parts)×(4 − oversampling)]. An oversampling factor L = 4, and smoothness +factor p = 2 were considered. We trained three identical CAE models on the following setups: +1) QPSK modulation scheme with a 2 × 2 MIMO setting, with 3GPP multipath channel. +2) 16-QAM modulation scheme with a 4 × 4 MIMO setting, with 3GPP multipath channel. +3) 16-QAM modulation scheme with a 4 × 4 MIMO setting, with AWGN channel. +To provide an unbiased performance evaluation of the final training model, the training and +test data sets were generated independently, i.e. OFDM symbols, channel realization, and noise +were randomized independently. In the following, we give numerical performance results of our +multi-objective CAE model compared to a classical CF algorithm with a clipping ratio of 4.08 +dB, and to SLM with U = 64 phase sequences, with MLE added for MIMO detection. The +inference part was performed on 7000 MIMO samples for each SNR point. +B. Training Setup +As a part of the experimental analysis, we performed an extensive exploration of different +model structures and hyper-parameters, including the number of layers, kernel sizes, number +of convolutional layer channels, regularization, dropout, number of decoder iterations, batch- +normalization, learning rate, AL parameters, trained SNR value, and epoch number. We found +that the best performance versus complexity on both examined MIMO setups was achieved for +the same model structure, only with different training data sets. As the constellation, the number +of subcarriers per OFDM symbols, and the number of antennas are higher, the training is longer, +and it is harder to achieve the desired results. The proposed CAE structure for the above 4 × 4 +MIMO system is described in Table I, where λ(0) +2a , λ(0) +2b , and λ(0) +3 +are the values of the first iteration +when the AL epochs start, and ’Grad start’ indicates the number of initial epochs where only the +reconstruction loss is counted, optimizing the unconstrained problem. ’LR’ indicates the learning +January 13, 2023 +DRAFT + +22 +TABLE I: CAE Proposed Structure +Transmitter +Receiver +Parameter +Value +Kernel +Ch-in +Ch-out +Value +Kernel +Ch-in +Ch-out +Input size +4 × 720 +- +- +- +12 × 144 +- +- +Conv (SELU) +- +1 × 3 +1 +21 +- +3 × 3 +1 +15 +Conv (SELU) +- +1 × 3 +21 +15 +- +3 × 3 +15 +21 +Conv (SELU) +- +1 × 3 +15 +21 +- +- +- +- +FC (Linear) output size +4 × 720 +- +- +- +12 × 144 +- +- +- +Decoder iterations +- +- +- +- +10 +- +- +- +Conv padding +LR +Epochs num +Grad start +SNR train +λ(0) +2a , λ(0) +2b , λ(0) +3 +ρ2a, ρ2b, ρ3 +2 +0.001 +140 +45 +40 dB +0.015, 0.001, 0.005 +0.0015, 0.00001, 0.001 +rate. Training on any of the data sets with the same best SNR value used for noise generation, +’SNR train’, showed the top overall inference performance for any tested SNRs. Adding dropout +had no benefit in all examined setups. +C. BER Analysis +The calculation of BER versus Peak Signal to Noise Ratio (P SNR) is used here as a key +parameter to measure the reconstruction and detection of the transmitted signal. Considering a +normalized channel, i.e. E ∥H∥2 = 1, the P SNR is defined as the ratio between the MIMO +system maximal emitted energy, PT, and the noise power, σ2 +w, such that +P SNR = PT +σ2w +. +(25) +As shown in Fig. 4, the CAE has competitive BER vs. P SNR performance compared to the +other standard examined methods in most of the P SNR range, where a visible gain is achieved at +the higher part. As the NN does not assume any specific physical model, it has better robustness +to distortions. That is, the MIMO-OFDM signal reconstruction and detection of HPA-distorted +data as a part of the multi-objective optimization proposed by our end-to-end DL scheme has +the benefit over the common algorithms. +DRAFT +January 13, 2023 + +23 +0 +5 +10 +15 +20 +25 +30 +35 +40 +P-SNR[dB] +10 +−3 +10 +−2 +10 +−1 +BER +No_PAPR_reduction +CAE +CF +SLM +(a) 16-QAM, 4x4 MIMO with 3GPP multipath channel +0 +5 +10 +15 +20 +25 +30 +35 +40 +P-SNR[dB] +10 +−5 +10 +−4 +10 +−3 +10 +−2 +10 +−1 +BER +No_PAPR_reduction +CAE +CF +SLM +(b) 16-QAM, 4x4 MIMO with AWGN channel +0 +5 +10 +15 +20 +25 +30 +35 +40 +P-SNR[dB] +10 +−4 +10 +−3 +10 +−2 +10 +−1 +BER +No_PAPR_reduction +CAE +CF +SLM +(c) 4-QAM, 2x2 MIMO with 3GPP multipath channel +Fig. 4: BER vs. P SNR of the considered methods and setups. +D. CCDF for PAPR Comparison +To demonstrate the PAPR performance of different methods, complementary cumulative dis- +tribution function (CCDF) curves are presented in Fig. 5. The CCDF of the PAPR denotes the +probability that the PAPR exceeds a certain threshold, i.e. P(PAPRMIMO−OFDM > PAPR0). The +PAPR is calculated according to the BPF output, xF +n. As can be observed in Fig. 5, the proposed +CAE achieves the better performance of PAPR reduction compared to the CF and SLM methods. +However, still the BER and spectral behavior are more important for performance evaluation. +January 13, 2023 +DRAFT + +24 +5 +6 +7 +8 +9 +10 +11 +PAPR0[dB] +10 +−3 +10 +−2 +10 +−1 +10 +0 +CCDF=Pr(PAPR>PAPR0) +No_PAPR_reduction +CAE +CF +SLM +(a) 16-QAM, 4x4 MIMO with 3GPP multipath channel +5 +6 +7 +8 +9 +10 +11 +12 +PAPR0[dB] +10 +−3 +10 +−2 +10 +−1 +10 +0 +CCDF=Pr(PAPR>PAPR0) +No_PAPR_reduction +CAE +CF +SLM +(b) 16-QAM, 4x4 MIMO with AWGN channel +4 +5 +6 +7 +8 +9 +10 +11 +PAPR0[dB] +10 +−3 +10 +−2 +10 +−1 +10 +0 +CCDF=Pr(PAPR>PAPR0) +No_PAPR_reduction +CAE +CF +SLM +(c) 4-QAM, 2x2 MIMO with 3GPP multipath channel +Fig. 5: CCDF of PAPR of the considered methods. +E. Spectrum Analysis +Figure 6 compare the spectral performance in terms of the PSD of the transmitted signals for +all examined methods. The dashed rectangle shows perfect spectral behavior for a linear HPA +with no non-linear components. +The proposed CAE decreases the out-of-band distortions at the expense of lower transmitted +power efficiency. Observing the spectral behavior as a part of the experimental analysis showed +us that there is a trade-off between increasing the IBO and increasing the Lagrange multiplier +associated with the PAPR loss component, λ2b. As shown in Fig. 7, while increasing the IBO +mostly shifts the CAE curve downwards, increasing λ2b causes the curve to bend more. +DRAFT +January 13, 2023 + +25 +0 +50 +100 +150 +200 +250 +300 +350 +Subcarrier umber +−70 +−60 +−50 +−40 +−30 +−20 +−10 +0 +PSD[dB] +No_PAPR_reductio _ oPA +No_PAPR_reductio +CAE +CF +SLM +(a) 16-QAM, 4x4 MIMO with 3GPP multipath channel +0 +50 +100 +150 +200 +250 +300 +350 +Subcarrier umber +−70 +−60 +−50 +−40 +−30 +−20 +−10 +0 +PSD[dB] +No_PAPR_reductio _ oPA +No_PAPR_reductio +CAE +CF +SLM +(b) 16-QAM, 4x4 MIMO with AWGN channel +0 +50 +100 +150 +200 +250 +300 +350 +Subcarrier umber +−70 +−60 +−50 +−40 +−30 +−20 +−10 +0 +PSD[dB] +No_PAPR_reductio _ oPA +No_PAPR_reductio +CAE +CF +SLM +(c) 4-QAM, 2x2 MIMO with 3GPP multipath channel +Fig. 6: PSD for the considered methods. +(a) moderate λ2b +(b) high λ2b +(c) moderate IBO +(d) high IBO +Fig. 7: Spectral mask behavior trade-off between IBO and λ2b. +The transmitter’s OBO, which evaluates the power efficiency of the system, is defined as the +ratio between the maximal radiated power that is the maximal power transmitted by all the HPAs +in the network, PT, and the mean transmitted power at the HPAs’ input, i.e. +OBO = +PT +�Nt +m=1 E +� +|xBn,m|2�. +(26) +January 13, 2023 +DRAFT + +26 +TABLE II: ACPR and OBO +4QAM 2X2 MIMO +16QAM 4X4 MIMO +Parameter +CAE +FC-AE +CF +SLM +No-reduction +CAE +FC-AE +CF +SLM +No-reduction +ACPR[dB] +-39.87 +-37.26 +-39.08 +-37.73 +-34.99 +-37.88 +-36.53 +-39.004 +-37.67 +-35.01 +OBO[dB] +5.92 +6.62 +6.74 +6.78 +6.86 +6.09 +6.77 +6.74 +6.78 +6.84 +−55 +−50 +−45 +−40 +−35 +−30 +−25 +ACPR[dB] +2 +3 +4 +5 +6 +7 +8 +9 +10 +OBO[dB] +No_PAPR_reduction +CAE +CF +SLM +Fig. 8: OBO vs. ACPR for the considered meth- +ods for 16-QAM with 4x4 MIMO setup and +3GPP multipath channel. +The maximum radiated power is defined to +be PT = 1. PT is divided equally between +the HPAs. In other words, the maximal trans- +mitted power of each HPA is PT/Nt. As we +defined all HPAs to have the same charac- +teristics, the saturation level of each HPA is +A0 = +� +PT/NT. Table II compares the ACPR +and the OBO of the proposed CAE to the other +methods. As shown, the ACPR of the CAE is +comparable with the considered methods. +In Fig. 8 we further compare the OBO performance for different ACPR values. It can be +seen that the CAE system requires lower OBOs, which is better overall power efficiency, while +maintaining better BER compared to the other methods. +F. Autoencoder - FC vs. CNN +We investigated various NN types for the AE, in particular, FC and CNN. Figure 9a compares +the BER performance of two AE architectures: the proposed CAE, which contains convolutional +layers, and a fully connected autoencoder (FC-AE), which contains only FC layers. It can be +observed that the CAE network has better BER performance compared to the FC-AE. As shown +in Table II, the ACPR of the CAE is better than that of the FC-AE. Moreover, the CAE has +lower complexity and thus faster training. The three convolutional layers have a total of 1953 +parameters, while for three FC layers of sizes 3500, 2500, and 3500, as were used for the FC-AE +DRAFT +January 13, 2023 + +27 +0 +5 +10 +15 +20 +25 +30 +35 +40 +P-SNR[dB] +10 +−3 +10 +−2 +10 +−1 +BER +FC_AE +CAE +(a) FC-AE and CAE +0 +5 +10 +15 +20 +25 +30 +35 +40 +P-SNR[dB] +10 +−3 +10 +−2 +10 +−1 +BER +CAE_Non_Gradual +CAE_Gradual +(b) Fixed and gradual loss learning +Fig. 9: BER vs. P SNR learning approaches comparison of 16-QAM, 4x4 MIMO with 3GPP +multipath channel. +in Fig. 9a and Table II, the number of parameters is around 107. +G. Fixed vs. Gradual Loss Learning +To show the benefits of using a gradual loss learning procedure, Fig. 9b compares its BER +performance to that of a fixed-loss training procedure, where the loss function’s weights are fixed +for the entire training. It can be observed that the gradual loss learning procedure significantly +improves the BER. In addition, improving the BER while keeping the PAPR and spectral +performance at the desired levels is easier to control when applying the gradual loss learning +method than manipulating loss function weights in fixed-loss training. Also, spectral performance +and PAPR reduction were harder to control and provide similar performance. +V. CONCLUSIONS AND FUTURE WORK +In this study, we have presented a CAE model for PAPR reduction and waveform design in +a MIMO-OFDM communication system. We have applied a gradual loss learning method to +optimize the performance in terms of three objectives: low BER, low PAPR, and adherence to +ACPR spectral requirements, on top of the AL multipliers optimization technique. The presented +CAE structure trainable parts included a neural PAPR reduction block, followed by a BPF filter to +optimize the spectral behavior at the transmitter, and a neural iterative MIMO detection block at +January 13, 2023 +DRAFT + +28 +the receiver, both were simultaneously optimized as a part of the end-to-end network design. The +proposed CAE was shown to outperform the CF and the SLM algorithms at the examined cases. +Future work can extend the MIMO scenario to higher modulation schemes and larger MIMO +setups, aiming to achieve a functional utility for future wireless communication networks. +REFERENCES +[1] T. Jiang and G. Zhu, “Complement block coding for reduction in peak-to-average power ratio of ofdm signals,” IEEE +Communications Magazine, vol. 43, no. 9, p. S17–S22, 2005. +[2] M. Z. Chowdhury, M. Shahjalal, S. Ahmed, and Y. M. Jang, “6g wireless communication systems: Applications, +requirements, technologies, challenges, and research directions,” IEEE Open Journal of the Communications Society, +vol. 1, pp. 957–975, 2020. +[3] A. Bo, Y. Zhi-Xing, P. C.-Y. Z. Tao-Tao, and G. Jian-Hua, “Effects of papr reduction on hpa predistortion,” IEEE +Transactions on Consumer Electronics, vol. 51, no. 4, pp. 1143–1147, 2005. +[4] C.-X. Wang, M. D. Renzo, S. Stanczak, S. Wang, and E. G. Larsson, “Artificial intelligence enabled wireless networking +for 5g and beyond: Recent advances and future challenges,” IEEE Wireless Communications, vol. 27, no. 1, pp. 16–23, +2020. +[5] Y. S. Cho, J. Kim, W. Y. Yang, and C. G. Kang, MIMO-OFDM wireless communications with MATLAB. +John Wiley & +Sons, 2010. +[6] 3GPP, +“3rd +generation +partnership +project +(3gpp). +study +on +channel +model +for +frequencies +from +0.5 +to +100 +ghz,” +3rd +Generation +Partnership +Project +(3GPP), +Technical +Specification +(TS). +[Online]. +Available: +https://www.3gpp.org/ftp/Specs/archive/38 series/38.901 +[7] B. Hassibi and H. Vikalo, “On sphere decoding algorithm 1: expected complexity,” IEEE Transaction on Signal Processing, +vol. 53, no. 8, pp. 2806–2818, 2005. +[8] P. W. Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Valenzuela, “V-blast: An architecture for realizing very high data +rates over the rich-scattering wireless channel,” in 1998 URSI international symposium on signals, systems, and electronics. +Conference proceedings (Cat. No. 98EX167). +IEEE, 1998, pp. 295–300. +[9] B. Steingrimsson, Z.-Q. Luo, and K. M. Wong, “Soft quasi-maximum-likelihood detection for multiple-antenna wireless +channels,” IEEE Transactions on Signal Processing, vol. 51, no. 11, pp. 2710–2719, 2003. +[10] N. Samuel, T. Diskin, and A. Wiesel, “Learning to detect,” IEEE Transactions on Signal Processing, vol. 67, no. 10, pp. +2554–2564, 2019. +[11] O. Sholev, H. H. Permuter, E. Ben-Dror, and W. Liang, “Neural network MIMO detection for coded wireless communication +with impairments,” in 2020 IEEE Wireless Communications and Networking Conference (WCNC), 2020, pp. 1–8. +[12] M. Mohammadkarimi, M. Mehrabi, M. Ardakani, and Y. Jing, “Deep learning-based sphere decoding,” IEEE Transactions +on Wireless Communications, vol. 18, no. 9, pp. 4368–4378, 2019. +DRAFT +January 13, 2023 + +29 +[13] K. Pratik, B. D. Rao, and M. Welling, “Re-mimo: Recurrent and permutation equivariant neural mimo detection,” IEEE +Transactions on Signal Processing, vol. 69, pp. 459–473, 2020. +[14] T. J. O’Shea, T. Erpek, and T. C. Clancy, “Deep learning based mimo communications,” arXiv preprint arXiv:1707.07980, +2017. +[15] J. Song, C. H¨ager, J. Schr¨oder, T. J. O’Shea, E. Agrell, and H. Wymeersch, “Benchmarking and interpreting end-to-end +learning of mimo and multi-user communication,” IEEE Transactions on Wireless Communications, 2022. +[16] T. O’shea and J. Hoydis, “An introduction to deep learning for the physical layer,” IEEE Transactions on Cognitive +Communications and Networking, vol. 3, no. 4, pp. 563–575, 2017. +[17] M. A. Albreem, M. Juntti, and S. Shahabuddin, “Massive mimo detection techniques: A survey,” IEEE Communications +Surveys & Tutorials, vol. 21, no. 4, pp. 3109–3132, 2019. +[18] I. Gutman, I. Iofedov, and D. Wulich, “Iterative decoding of iterative clipped and filtered OFDM signal,” IEEE Transactions +on Communications, vol. 61, no. 10, pp. 4284–4293, 2013. +[19] A. P. More and S. B. Somani, “The reduction of PAPR in OFDM systems using clipping and SLM method,” in 2013 +International Conference on Information Communication and Embedded Systems (ICICES), 2013, pp. 593–597. +[20] Di-xiao Wu, “Selected mapping and partial transmit sequence schemes to reduce PAPR in OFDM systems,” in 2011 +International Conference on Image Analysis and Signal Processing, 2011, pp. 1–5. +[21] T. Jiang and Y. Wu, “An overview: Peak-to-average power ratio reduction techniques for OFDM signals,” IEEE Transactions +on Broadcasting, vol. 54, no. 2, pp. 257–268, 2008. +[22] A. Mobasher and A. K. Khandani, “Integer-based constellation-shaping method for PAPR reduction in OFDM systems,” +IEEE Transactions on Communications, vol. 54, no. 1, pp. 119–127, 2006. +[23] Z. Q. Taha and X. Liu, “An adaptive coding technique for PAPR reduction,” in IEEE GLOBECOM 2007 - IEEE Global +Telecommunications Conference, 2007, pp. 376–380. +[24] Y. Lee, Y. You, W. Jeon, J. Paik, and H. Song, “Peak-to-average power ratio in mimo-ofdm systems using selective +mapping,” IEEE Communications Letters, vol. 7, no. 12, pp. 575–577, 2003. +[25] A. Namitha and S. Sameer, “A bandwidth efficient selective mapping technique for the papr reduction in spatial multiplexing +mimo-ofdm wireless communication system,” Physical Communication, vol. 25, pp. 128–138, 2017. +[26] A. Zappone, M. Di Renzo, and M. Debbah, “Wireless networks design in the era of deep learning: Model-based, AI-based, +or both?” IEEE Transactions on Communications, vol. 67, no. 10, pp. 7331–7376, 2019. +[27] I. Sohn, “A low complexity PAPR reduction scheme for OFDM systems via neural networks,” IEEE Communications +Letters, vol. 18, no. 2, pp. 225–228, 2014. +[28] I. Sohn and S. C. Kim, “Neural network based simplified clipping and filtering technique for PAPR reduction of OFDM +signals,” IEEE Communications Letters, vol. 19, no. 8, pp. 1438–1441, 2015. +[29] M. Kim, W. Lee, and D. Cho, “A novel papr reduction scheme for OFDM system based on deep learning,” IEEE +Communications Letters, vol. 22, no. 3, pp. 510–513, 2018. +January 13, 2023 +DRAFT + +30 +[30] L. Shi, X. Zhang, W. Wang, Z. Wang, A. Vladimirescu, Y. Zhang, and J. Wang, “PAPR reduction based on deep +autoencoder for VLC DCO-OFDM system,” in 2019 IEEE International Symposium on Broadband Multimedia Systems +and Broadcasting (BMSB), 2019, pp. 1–4. +[31] Y. Huleihel, E. Ben-Dror, and H. H. Permuter, “Low papr waveform design for ofdm systems based on convolutional +autoencoder,” in 2020 IEEE International Conference on Advanced Networks and Telecommunications Systems (ANTS). +IEEE, 2020, pp. 1–6. +[32] F. A. Aoudia and J. Hoydis, “Waveform learning for next-generation wireless communication systems,” IEEE Transactions +on Communications, 2022. +[33] L. Hao, D. Wang, W. Cheng, J. Li, and A. Ma, “Performance enhancement of ACO-OFDM-based VLC systems using a +hybrid autoencoder scheme,” Optics Communications, vol. 442, pp. 110–116, 2019. +[34] A. Kalinov, R. Bychkov, A. Ivanov, A. Osinsky, and D. Yarotsky, “Machine learning-assisted papr reduction in massive +mimo,” IEEE Wireless Communications Letters, vol. 10, no. 3, pp. 537–541, 2020. +[35] A. Ivanov and D. Lakontsev, “Selective tone reservation for papr reduction in wireless communication systems,” in 2017 +IEEE International Workshop on Signal Processing Systems (SiPS). +IEEE, 2017, pp. 1–6. +[36] A. Ivanov, A. Volokhatyi, D. Lakontsev, and D. Yarotsky, “Unused beam reservation for papr reduction in massive mimo +system,” in 2018 IEEE 87th Vehicular Technology Conference (VTC Spring). +IEEE, 2018, pp. 1–5. +[37] P. Kenington, “Methods linearize RF transmitters and power amps,” Microwaves & RF, vol. 37, no. 13, pp. 102–116, 1998. +[38] S. Santurkar, D. Tsipras, A. Ilyas, and A. Madry, “How does batch normalization help optimization?” in Advances in +Neural Information Processing Systems, 2018, pp. 2483–2493. +[39] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in Proceedings of the IEEE conference +on computer vision and pattern recognition, 2016, pp. 770–778. +[40] G. Klambauer, T. Unterthiner, A. Mayr, and S. Hochreiter, “Self-normalizing neural networks,” in Advances in neural +information processing systems, 2017, pp. 971–980. +[41] H. E. Rowe, “Memoryless nonlinearities with gaussian inputs: Elementary results,” The BELL system technical Journal, +vol. 61, no. 7, pp. 1519–1525, 1982. +[42] G. L. Stuber, J. R. Barry, S. W. Mclaughlin, Y. Li, M. A. Ingram, and T. G. Pratt, “Broadband mimo-ofdm wireless +communications,” Proceedings of the IEEE, vol. 92, no. 2, pp. 271–294, 2004. +[43] I. Loshchilov and F. Hutter, “Decoupled weight decay regularization,” arXiv preprint arXiv:1711.05101, 2017. +[44] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating +direction method of multipliers,” Foundations and Trends® in Machine learning, vol. 3, no. 1, pp. 1–122, 2011. +[45] D. G. Luenberger and Y. Yinyu, Linear and nonlinear programming. +Springer, 1984, vol. 2. +[46] “Matlab +5g +toolbox,” +2020, +the +MathWorks, +Natick, +MA, +USA. +[Online]. +Available: +https://www.mathworks.com/products/5g.html +DRAFT +January 13, 2023 + diff --git a/qdE4T4oBgHgl3EQfUgzF/content/tmp_files/load_file.txt b/qdE4T4oBgHgl3EQfUgzF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9c25f030f9f8b4a556a8a8ead8b5f651a14679bb --- /dev/null +++ b/qdE4T4oBgHgl3EQfUgzF/content/tmp_files/load_file.txt @@ -0,0 +1,826 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf,len=825 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='05017v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='SP] 11 Jan 2023 1 Low PAPR MIMO-OFDM Design Based on Convolutional Autoencoder Yara Huleihel, and Haim H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Permuter, Senior Member, IEEE Abstract An enhanced framework for peak-to-average power ratio (PAPR) reduction and waveform design for Multiple-Input-Multiple-Output (MIMO) orthogonal frequency-division multiplexing (OFDM) systems, based on a convolutional-autoencoder (CAE) architecture, is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The end-to-end learning-based autoencoder (AE) for communication networks represents the network by an encoder and decoder, where in between, the learned latent representation goes through a physical communication channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' We introduce a joint learning scheme based on projected gradient descent iteration to optimize the spectral mask behavior and MIMO detection under the influence of a non-linear high power amplifier (HPA) and a multipath fading channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The offered efficient implementation novel waveform design technique utilizes only a single PAPR reduction block for all antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' It is throughput-lossless, as no side information is required at the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Performance is analyzed by examining the bit error rate (BER), the PAPR, and the spectral response and compared with classical PAPR reduction MIMO detector methods on 5G simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The suggested system exhibits competitive performance when considering all optimization criteria simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' We apply gradual loss learning for multi-objective optimization and show empirically that a single trained model covers the tasks of PAPR reduction, spectrum design, and MIMO detection together over a wide range of SNR levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Index Terms Deep learning, Autoencoder, Multiple-Input-Multiple-Output, Orthogonal frequency-division mul- tiplexing, Peak-to-average power ratio, Wireless signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' January 13, 2023 DRAFT 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' INTRODUCTION The Multiple-input-multiple-output (MIMO) scheme is a widely used technique for enhancing channel capacity and transmission reliability, thanks to the diversity and multiplexing gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Orthogonal frequency division multiplexing (OFDM) is a waveform design method known for providing high bandwidth efficiency, high throughput, simple equalization in wireless transmis- sion, and efficient hardware implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' For these reasons, it has been adopted as a standard technology in various wireless communication systems, such as WiFi, 4G, and 5G standards for wireless communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Nonetheless, significant drawbacks of the OFDM multi-carrier system appear in the form of adjacent channel power ratio (ACPR) limitations and the tendency to produce signals with a high peak-to-average power ratio (PAPR) in the time-domain, since many subcarrier components are added via a fast Fourier transform (FFT) operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The contribution of each subcarrier to the total power is dynamic, which makes the total power highly variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' In particular, the high PAPR in MIMO-OFDM systems is exacerbated as the number of antennas increases [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The demand for higher energy efficiency as well as reduced power consumption is expected to increase for future radio systems [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Moreover, future communication system design and equipment are expected to be more compatible with machine learning (ML) implementation requirements, for example, allowing learning in the field to make some design choices [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Accordingly, waveform designs using ML techniques are becoming significantly attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' A high power amplifier (HPA) is required to provide enough output power for reliable com- munication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' In practical systems, the HPA is not linear and distorts the transmitted signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' As a result, severe non-linear signal distortions are found when these high PAPR signals pass through the non-linear HPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The resulting signal exhibits spectral regrowth in the form of in-band signal distortions and out-of-band radiation [3], and the bit error rate (BER) increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Hence, it is crucial to develop PAPR reduction techniques for MIMO-OFDM systems to increase their efficiency in handling large data streams and to reduce their error rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Training and applying DRAFT January 13, 2023 3 the PAPR reduction block to each antenna individually exacerbates the complexity, cost, and redundancy in proportion to the number of transmit antennas in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Instead, in this work, a single PAPR reduction block jointly operates on all antenna OFDM sequences, and it is designed according to the maximum PAPR value of all antenna sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' A central difficulty of the multiple transmitter (TX) and receiver (RX) antenna structure is posed by the need for joint detection of the data symbols sent by each transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Unfortunately, the optimal MIMO detection solution imposes an NP-hard problem on the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Consequently, various sub-optimal yet feasible detection algorithms have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Other than classical model-driven solutions, an increasing effort has been dedicated to ML, and, specifically, deep learning (DL) based techniques to solve the MIMO detection problem, and more generally, various wireless communication tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The design of OFDM waveform signals aims to simultaneously achieve a high data rate, high spectral efficiency (measured by the ACPR), and low computational complexity [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' This design is highly affected by the non-linear effects of the HPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' While keeping the PAPR level low is favorable, it is of particular importance to have acceptable signal spectral behavior and BER, which are often referred to as waveform design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' In order to fulfill that, this work suggests an overall communication network multi-objective optimization, such that the transmitter, HPA, channel, and the receiver, are represented as a single optimization block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Instead of separately optimizing different components of the transmitter and the receiver, an end-to-end convolutional- autoencoder (CAE) learning model is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' This end-to-end optimization block is presented as a constrained optimization problem where the transmitted signal estimation is the objective, and the PAPR and ACPR requirements are the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' MIMO detection over multiple channel realizations is performed as a part of the end-to-end joint optimization model, utilizing an iterative approach based on convolutional layers, and a gradual loss learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' We evaluate the performance of our algorithms over both additive white Gaussian noise (AWGN) and 3rd Generation Partnership Project (3GPP) fading channels [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' By analyzing the BER, PAPR, and January 13, 2023 DRAFT 4 spectrum performance, we show that the proposed end-to-end learning approach can integrate different communication network blocks to balance those performance objectives successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' We show that the suggested scheme is able to achieve better spectral performance for higher HPA efficiency operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Various OFDM PAPR reduction techniques have been proposed in the literature, as well as for MIMO detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Generally, these techniques can be categorized into model-driven and data-driven techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The first category refers to standard approaches in classical communications theory, while the second relies on recently developed approaches based on ML techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The following subsections review different earlier solutions for the above-mentioned problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Classical Approaches (Model Driven) for MIMO Detection Many MIMO detection algorithms have been developed over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The maximum like- lihood estimation (MLE) solution is optimal for the joint detection of transmitted symbols in a MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' However, its exponential computational and time complexity (due to exhaustive searches over all possible transmitted signals) render it infeasible when the number of transmitters and the modulation order are high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' An example of suboptimal high accuracy non-linear detection algorithms are those based on sphere decoding (SD) [7], but they become computationally expensive as the number of antennas grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The general idea is based on a lattice search for a solution in an iterative manner, and the accuracy/complexity ratio strongly depends on the value chosen for the radius parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' More advanced detectors include the successive interference cancellation (SIC) based detectors [8] and the semi-definite relaxation detectors [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Machine-Learning-Based Schemes (Data Driven) for MIMO Detection The motivation for DL-based detectors is to enhance the performance of classical model- driven detection algorithms by learning, from the training data set, an optimized mapping of the received signals onto the transmitted symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' In [10], [11], an iteration-based algorithm for implementing a receiver for MIMO detection was suggested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' One of the highlights of the DRAFT January 13, 2023 5 presented model’s framework is that it enables training through different random communication channel realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' In [12], a model-based algorithm was suggested, where a classical SD algorithm was integrated with a neural network (NN) that was trained to optimize the selection of the initial radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' In [13], a neural detector-based transformer architecture implements a recurrent estimation scheme by learning an iterative decoding algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' In [14]–[16] an AE was offered to design a physical layer in which DL-based CSI encoding was suggested for different scenarios to achieve lower BER together with better robustness to the wireless channel characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Thorough surveys and analysis are presented in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Classical Approaches (Model Driven) for PAPR Reduction PAPR reduction schemes are roughly classified into three categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The signal distortion category consists of techniques such as clipping and filtering (CF) [18], [19], which limit the peak envelope of the input signal in the time domain to a predetermined value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The multiple signaling probabilistic category includes methods such as selective mapping (SLM) [19], [20], partial transmit sequence (PTS) [20], ton reservation and ton injection [21], and constellation shaping [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The main principle of SLM is to generate different candidates for each OFDM block by multiplying the symbols vector with a set of different pseudo-random sequences and choosing the candidate with the lowest PAPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The third category is the coding technique category [21], [23], attempting to reduce the occurrence probability of the same phase signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Earlier schemes were mainly developed for single-antenna systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Extended works which applied the single-antenna PAPR reduction schemes on each antenna of the MIMO configuration separately are found in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=', [24], but those required considerable computations, cost, and complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Model-driven approaches to simultaneously reduce PAPR over all antennas were also proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' In [25], instead of applying SLM to each antenna, the sequence with the highest PAPR over all transmit antennas was selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Usually, SLM and PTS methods demand side information (SI) to be sent to the receiver along with each transmitted data block for retrieving the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The need for SI requires extra bandwidth overhead, and the incorrect detection January 13, 2023 DRAFT 6 of the SI bits over the channel will lead to significant degradation in the BER performance of the receiver in the MIMO-OFDM system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Deep-Learning-Based Schemes (Data Driven) for PAPR Reduction In recent years much research has been dedicated to applying DL techniques in the design and optimization of wireless communication networks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=', [4], [11], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Several papers have proposed DL methods to handle PAPR reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' For example, the authors of [27], [28], added a NN to reduce the complexity of the active constellation scheme, followed by CF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' In [29], [30] the authors present an AE solution for PAPR reduction, while minimizing the BER degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' In [31] a CAE was suggested for the implementation of an end-to-end SISO − OFDM communication network that simultaneously reduces the PAPR and reconstructs the transmitted symbols, while keeping acceptable spectral requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Another learning-based approach, which considers the reduction of the PAPR and ACPR together with the maximization of the achievable information rate for a single-carrier waveform above multipath channels, was proposed in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The authors in [33] proposed a deep NN combined with SLM to mitigate the high PAPR issue of OFDM signal types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' All of the above papers consider a SISO network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' A PAPR reduction scheme assisted by DL for a MIMO-OFDM system was suggested in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The authors apply selective tone reservation [35] on each antenna separately and then apply unused beam reservation [36] on all antennas together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' An ML-based method for approximating the optimal tabular hyperparameters required for using selective tone reservation and unused beam reservation was suggested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Main Contributions Some of the aforementioned PAPR reduction approaches suffer from in-band interference, out- of-band distortions, and high computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Moreover, published ML-based solutions mostly handle single antenna scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Those who deal with PAPR reduction for MIMO systems use ML only for the PAPR reduction block and not for the end-to-end network implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' This paper aims to handle the PAPR problem in MIMO systems as an integral part of a waveform DRAFT January 13, 2023 7 design objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' In particular, we design a communication system that simultaneously achieves PAPR reduction, acceptable spectral behavior of the PA’s output, and good BER performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The suggested end-to-end network aims to resolve the MIMO detection problem as a part of the other mentioned objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' To the best of our knowledge, this approach is new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Novelties we introduce include using a CAE combined with a gradual loss learning technique to handle the multi-objective optimization of the network, and adding the effect of the HPA on an integrated end-to-end MIMO communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' We present an iterative MIMO detection algorithm integrated into transmitter-receiver end-to-end communication system joint optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' We demonstrate our algorithm’s results on 5G MIMO-OFDM Matlab toolbox simulated data, and we compare our method with classical methods for PAPR reduction and waveform design, and show competitive results for all the objectives mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The proposed algorithm offers performance improvement for future wireless communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' We show that our model provides competitive PAPR reduction, waveform design, and detection results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The rest of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' In Section II, the problem is defined and formulated separately for MIMO detection, and for PAPR reduction as a part of the MIMO-OFDM system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' We then present the proposed DL-based system architecture for the multi-objective optimization and explain the training procedure in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Section IV provides detailed numerical simulation results and insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Finally, Section V gives concluding remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' NOTATION AND PROBLEM DEFINITION In this section, we introduce the notation and the problem definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Notation Throughout this paper, we use the following notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The set of real numbers is denoted by R, while the set of complex numbers is denoted by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Random variables will be denoted by capital letters, and their realizations will be denoted by lower-case letters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=', X and x, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Calligraphic letters denote sets, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=', X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' We use the notation Xn to denote the random vector (X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' , Xn) and xn to denote the realization of such a random vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' January 13, 2023 DRAFT 8 The expectation operator is denoted by E [·].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' (·)∗, (·)† denote the conjugate, and pseudo-inverse operators, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Problem Definition In this section, we describe mathematically each part of the integrated problem of MIMO de- tection together with PAPR reduction and spectrum constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' First, we give a brief introduction to the end-to-end setup used in our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 1) MIMO detection model: Let us assume a standard MIMO-OFDM system with Nt transmit antennas and Nr receive antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Transmission is considered over a memoryless complex- valued channel model, while assuming frequency flatness and slow fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' A MIMO system can be modeled by the following complex baseband model: y = Hx + n, (1) where x ∈ CNt is the transmitted complex symbol vector drawn from a finite discrete constel- lation of size |M|, H ∈ CNr×Nt is the complex baseband channel matrix that is related to a specific subcarrier, n ∈ CNr is complex background AWGN seen at the receiver, and y ∈ CNr is the received complex vector resulting from the propagation of the transmitted symbols through the channel contaminated by AWGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' As the proposed implementation is based on a real-valued NN model determined by the DL Pytorch library, (1) is expressed with real values by splitting and concatenating each signal into its real and imaginary parts: x = \uf8ee \uf8ef\uf8f0 Re{x} Im{x} \uf8f9 \uf8fa\uf8fb , y = \uf8ee \uf8ef\uf8f0 Re{y} Im{y} \uf8f9 \uf8fa\uf8fb , n = \uf8ee \uf8ef\uf8f0 Re{n} Im{n} \uf8f9 \uf8fa\uf8fb , H = \uf8ee \uf8ef\uf8f0 Re{H} −Im{H} Im{H} Re{H} \uf8f9 \uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' (2) In the MIMO detection problem, the objective is to detect the transmitted symbols, x, given the received data y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The optimal solution for the MIMO detection of the transmitted symbols problem defined above is given by the MLE algorithm, that is, ˆxmle = arg min x∈X Nt ||y − Hx||2, (3) DRAFT January 13, 2023 9 where X denotes the set of possible transmitted symbols (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=', signal constellation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The solution of (3) requires an exhaustive search over all |M|Nt possible transmitted vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Therefore, it is infeasible for an actual implementation where large-scale MIMO setups and/or a large constellation are in use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 2) PAPR problem in MIMO-OFDM: In an OFDM system with N complex orthogonal sub- carriers, the discrete-time transmitted OFDM signal at the nt antenna, is given by the inverse discrete Fourier transform (IDFT): xnt,n = 1 √ N N−1 � k=0 Xnt,kej 2π LN kn, 0 ≤ n ≤ LN − 1, 1 ≤ nt ≤ Nt, (4) where {Xnt,k}N−1 k=0 are random input symbols per antenna, modulated by a finite constellation, and L ≥ 1 is the over-sampling factor (L = 1 is the Nyquist sampling rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' As shown in [21], oversampling by a factor of four results in a good approximation of the continuous-time PAPR of complex OFDM signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The discussed problem considers non-linear HPAs at each of the Nt TX branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' We assume that the HPAs in all branches have the same non-linear characteristic, which is a reasonable assumption, considering current wireless MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Also, in a discrete implementation, the same HPAs are usually used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The PAPR of the transmitted signal in (4) is defined as the ratio between the maximum peak power and the average power of the OFDM signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Specifically, the PAPR at the nt-th transmit antenna is defined by: PAPRnt ≜ max0≤n≤LN−1 |xnt,n|2 E|xnt,n|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' (5) For the entire MIMO-OFDM system, the PAPR reduction method we use will consider the maximum PAPR among all Nt transmit antennas, as the same PA model is used in all branches: PAPRMIMO−OFDM = max 1≤nt≤Nt PAPRnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' (6) As HPA non-linearity causes spectral regrowth, an important assessment for the spectral purity of the system is the ACPR criterion, which is the ratio between the power of the adjacent channel January 13, 2023 DRAFT 10 and the power of the main channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Following [6], we define it as ACPR ≜ max �� 3BW/2 BW/2 Pss(f) df, � BW/2 −3BW/2 Pss(f) df � � BW/2 −BW/2 Pss(f) df , (7) where Pss(·) is the power spectral density (PSD) of the signal at the HPA’s output, and BW is the primary channel bandwidth, which is assumed to be equal to the data signal bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' A block diagram of the communication system model is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Q PSfrag replacements Xk xn PAPR & Spectral Optimization Enc Filter xF n Power Amplifier G [·] xP n Channel + yn DFT Yk Reconstruct ˆXk IDFT & Detect wn Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 1: General system model diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Specifically, the encoder and fil- ter blocks mitigate the PAPR effect and design the waveform to com- ply with predefined spectral mask requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' For example, the en- coder block can model a clipping operation, while the filter can be a standard band-pass filter (BPF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The filtered signal xF n is amplified by a non-linear HPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The amplified signal, xP n = G(xF n), is transmitted through a fading channel with AWGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The channel decoder receives the noisy signal and attempts to reconstruct and detect the transmitted signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' For model-driven approaches, a classical detection algorithm, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=', MLE, is applied for detecting the estimated symbol denoted by ˆXk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The role of the HPA is to convert the low-level transmission signal to a high power signal, capable of driving the antenna at the desired power level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The HPA has to operate close to its saturation region for maximal power efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' If the HPA exceeds the saturation point and enters the non-linear area of operation, the output signal becomes non-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Accordingly, to operate the amplifier only in the linear region, we need to make sure that the amplifier operates at a power level that is lower than the saturation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' This is achieved by down-scaling the input signal by an input back-off (IBO) factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The drawback of adding the IBO attenuation is that the output power decreases, which makes the HPA power-inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' DRAFT January 13, 2023 11 Saturation point Actual operating point Input Back-Off (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='707) Output Back-Off PSfrag replacements Ain Aout 3dB Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 2: RAPP HPA output versus input signal for different smoothness p values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' There are several commonly used models for the non-linearity of an HPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Here, we will focus on the RAPP behavioral amplifier model [37], which is very accurate for solid-state-power amplifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The model’s AM/AM conversion is given by G(Ain) = v · Ain · � 1 + �vAin A0 �2p�− 1 2p , (8) where Ain is the input amplitude, A0 is the limiting output amplitude, v is the small signal gain, p is a smoothness parameter controlling the transition from the linear region to the saturation region, and G(A) is the output amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Figure 2 shows RAPP HPA outputs versus input for several smoothing factor values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' PROPOSED WAVEFORM DESIGN STRUCTURE In this section, we describe our multi-objective optimization CAE model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Mo- tivated by research evidence of powerful learning ability, under the same conditions of the MIMO-OFDM examined structure, it is expected that the proposed CAE model will achieve good enough performance to be compared with classical PAPR reduction methods combined with the MLE detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' We first briefly discuss the general CAE concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Then, we describe our algorithm building blocks, and the joint PAPR reduction, spectral design, and detection operation in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The motivation and structure of the iteration-based decoder with regard to handling the MIMO detection problem as a part of the CAE network will be explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The proposed architecture in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 3 is then elaborated, including the Bussgang’s non-linearity compensation theorem, followed by a description of the gradual learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Last, the training procedure of the CAE multi-objective optimization network operation will be described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 3a shows the overall end-to-end communication network structure implemented by a CAE model, where the encoder and the decoder are the trainable blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Each mentioned loss January 13, 2023 DRAFT 12 / PSfrag replacements Xk Zero Zero pad IFFTxn 2 calc xB n Filter BO PA xF n PAPR PAPR Calculation Calculation Calculation L2b xP n MIMO Channel Xk yn ACPR L3 Loss L1 Reconstruction Encoder f(x) α α Decoder g(y) Nt × Nsc Nt × Nsc Nt Nt Nt Nt Nt Nt Nt Nt Nt Nt Nt Nt Nr Yk xE n unpad Nr × Nsc FFT Transmitter Receiver ˆXk +wn L2a (a) Conv-AE overall scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' + PSfrag replacements Input Tensor Conv+BN+GELU Fully Connected Predicted Tensor Kernel Nsc Nsc Nsc Nsc Power Norm Layer Nt Nt Nt Nt ch2 L2 loss L3 fc1 conv1 conv2 PAPR Reduction Block - Encoder f(x) conv3 3 3 3 ch1 ch1 1 1 1 (b) PAPR reduction block - Encoder scheme f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' PSfrag replacements Xi−1 H∗HXk+1 Xk+1 H∗H() ith iteration Nsc Nsc Nsc Nsc Nsc Nsc Nr Nr Nr Nr ch2 ch2 H∗Y H∗Y H∗Y H∗HXk Xk H∗HXi−1 Concat Concat conv4 conv4 conv5 conv5 Detection Block - Decoder g(x) fc2 + Softmax Layer Layer kth iteration 3 3 3 3 3 3 3 3 ch1 ch1 fc2 Xi (c) Detection block - Decoder scheme g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 3: Structure of the proposed conv-AE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' component is calculated using the operations appearing in the red blocks in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' It can be noticed that there are two PAPR calculation-based loss components, as will be detailed later, that helped achieve improved spectral behavior and BER results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 3b describes the encoder structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' It is constructed of 2D convolutional layers to be compatible with the input example dimensions, a fully connected layer, and a power normalization layer at the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' We also added a skip connection to improve stability and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 3c shows the decoder structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' It is also implemented with 2D convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' To be able to optimize the network under DRAFT January 13, 2023 13 different communication channel realizations, we used an iterative-based solution to the MIMO detection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' We emphasize that the presented CAE model, once trained on a single training set, presents competitive results for the discussed multi-objective optimization problem in the presence of a wide range of noise power levels, without any knowledge of the SNR level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Convolutional Autoencoder (CAE) The proposed implementation uses an AE learning system based on a convolutional neural network (CNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The general structure of an AE consists of two main blocks: the encoder f(x) and the decoder g(x), where x is the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The AE is trained to minimize a certain joint loss function, which we denote by L(x, g(f(x))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' An end-to-end communication system can be interpreted as an AE in which the encoder and the decoder are part of the transmitter and the receiver, respectively, and can jointly optimize them through an end-to-end learning procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' CNNs are widely used for feature extraction and pattern recognition in ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Compared with a fully connected (FC) network, a CNN has significantly fewer connections between adjacent layers, and thus fewer parameters and weights to train, resulting in lower complexity and much faster training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Iteration-Based Model Motivation An iterative approach based on convolutional layers, was adopted to implement the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The MIM0 detector was designed to solve the MLE optimization (3) using the projected gradient descent approach, where the input to the detector is a linear combination of features instead of the channel output directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Utilizing such detectors significantly improved the detection results when various channel realizations were integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The detector input feature selection was based on the following projected gradient decent approach used to optimize (9): ||y − Hx||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' (9) As shown in [10], [11], such iterations are updated by ˆxk+1 = Pc � ˆxk − δk ∂||y − Hx||2 ∂x ���� x=ˆxk � = Pc � ˆxk − δk � HHy − HHH ˆxk �� , (10) January 13, 2023 DRAFT 14 where ˆxk is the objective estimation in the k’th iteration, HH indicates the conjugate transpose operation over the channel matrix H, δk is the step size, and Pc is the non-linear projection operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The above can be adapted to a deep NN iterative solution by the following linear combination form: ˆxk+1 = Pc(ˆxk + δ1kHHy + δ2kHHH ˆxk), (11) where δ1k and δ2k are learned hyper-parameters to be optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' One of the motivations for using the described iterative decoder approach was a former work published in [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' In our work, it is designed as part of the joint encoder-decoder multi-task optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Also, it was implemented with convolutional layers, that enabled better computational and performance capabilities for the joint optimization end-to-end system, and the per subcarrier/antenna alternating analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' A softmax layer is added at the output of the CAE to generate probabilistic outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' In the following section, we provide the complete detection procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Proposed CAE Architecture This section introduces the suggested CAE learning network implementation of the MIMO- OFDM system for the multi-objective optimization task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' We consider a MIMO-OFDM scheme with Nt transmit antennas and Nr receive antennas, where the OFDM is of order K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The input is represented by a matrix in the frequency domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' X = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X(1)(1) X(1)(2) · · X(1)(K) X(2)(1) X(2)(2) · · X(2)(K) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' X(Nt)(1) X(Nt)(2) · · X(Nt)(K) \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , (12) where, for any nt ∈ [1, Nt] and k ∈ [1, K], X(nt)(k) is a M-QAM constellation complex-valued symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 3a, we illustrate the general structure of the end-to-end communication network implemented by the CAE configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Specifically, we consider a transmitter that takes the DRAFT January 13, 2023 15 two-dimensional matrix X as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The transmitter’s output goes through a MIMO channel, together with AWGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Finally, the noisy channel outputs are fed into a receiver to estimate X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The operations within the transmitter and the receiver are described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Transmitter: the input signal X is zero-padded on the subcarriers’ dimension and converted to the time domain via an IFFT applied on each of the transmitter branches, outputting {xn}LN−1 n=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' These symbols serve as the input to the encoder, which acts as a PAPR reduction block, followed by a BPF filter for optimizing the spectral behavior by reducing the out- of-band radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Its frequency response is a rectangular window with the same bandwidth as X(nt) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Then, a predefined IBO is applied just before the signal is amplified by the HPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Receiver: the distorted OFDM symbols are divided by an α factor to compensate for the non-linear distortions, as will be detailed in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Finally, the proposed CAE decoder reconstructs and detects the estimated MIMO-OFDM transmitted signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The encoder comprises three convolutional layers, and the decoder is composed of iterative construction of convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Each convolutional layer is followed by a non-linear ac- tivation function and batch normalization [38], and then a fully connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' In addition, a residual connection is added to the encoder block, which sums (element-wise) the input to the second convolutional layer and the output of the third convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' It turns out that this modification improves the overall performance of the suggested scheme significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The intuition is that adding another path for data to reach the latter parts of the NN makes it easier to optimize the mapping [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Furthermore, the encoder has a power normalization layer, which ensures that the transmitted signal meets the power constraints of unit average energy per OFDM symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' This way, the intended SNR is maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' We tested several activation functions, including sigmoid, rectified linear unit (RELU), Gaussian error linear unit (GELU), and scaled exponential linear unit (SELU) [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Empirically, it was found that SELU activation provides the best results for our CAE scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 3a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 3b, since the encoder is responsible for the PAPR reduction, January 13, 2023 DRAFT 16 which is calculated per OFDM symbol, we start with per antenna analysis, where each antenna is treated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' A one-dimensional kernel per TX branch handles this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The encoder architecture can be described by the following: f(x) = ρLf �����W f Lf � ρLf −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' � ρ1 �����W f 1x + bf 1 ���� bnorm ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' � +bf Lf ���� bnorm + �����W f 1x + bf 1 ���� bnorm �� , (13) where Lf is the number of the encoder’s convolutional layers, W f i , and bf i are the encoder’s weight matrix and bias vector, respectively, for the i’th layer, with size determined as a part of the network design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' ρi(·) is the activation function of the i’th layer, and bnorm means the layer passes through a batch normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The next part of this process applies the non-linear HPAs on each TX branch of the transmitter time domain signals, each composed of all subcarriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The signal is then converted via FFT to the frequency domain, and the zero-unpadding block removes the out-of-band samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Afterward, frequency domain analysis is performed on each subcarrier transmitted through all Nt antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Each subcarrier is transmitted through its related complex baseband channel described by a (Nr × Nt) matrix, and the AWGN is added as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' To continue with the per subcarrier analysis on the receiver side, we need to overcome the non-linearity of the HPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' To that end, we compensate the receiver input signal by applying an attenuation factor represented by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Bussgang’s decomposition theorem [41] states that if a zero-mean Gaussian signal passes through a memoryless non-linear device, then the output-input cross-correlation function is proportional to the input autocovariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Accordingly, the value of α is chosen to minimize the variance of the non-linear signal distortions, such that it is attempted that the transmitted signal in each transmitter branch is linearly separated, and thus represented as a sum of the signal and distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' It can be shown that α = E � xF nxP n � E (|xFn|2) , (14) DRAFT January 13, 2023 17 where xP n is the complex output signal of the PA, and xP n is its complex conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' By assuming that the PSD of the in-band distortion is approximately flat [42] in the frequency domain, the output signal of the HPA on the k-th subcarrier can then be expressed as XP(k) = α(k)XF(k) + D(k), (15) where D(k) is the non-linear distortion on the k-th subcarrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The same model is assumed for all PAs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' therefore, it can be concluded that αnt = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' At the MIMO decoder, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 3c, the per subcarrier analysis is continued, meaning that different subcarriers of the same RX branch will not be mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' To generalize our end-to-end structure, we modified the 1D kernel to a 2D kernel at the decoder part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' It also helped better reconstruct the signal after the encoder layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' As explained in the previous sub-section, we use an iterative procedure to implement the decoder, which is designed for signal reconstruction and detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' A general mathematical description of one iteration, k, of the presented decoder is given by dk = � ˆxk−1, δ1kHHy, δ2kHHH ˆxk−1 � ˆxk = g(dk) = ρLg,k �����W g Lg,k � ρLg−1,k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' � ρ1,k �����W g 1,kdT k + bg 1,k ���� bnorm ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' � + bg Lg,k ���� bnorm � , where Lg, W g i , and bg i , have the same definitions as described for the encoder’s block, only that these apply for the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The decoder’s input features vector, dk, was initialized by randomizing a prediction ˆx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Initialization by zeros resulted in performance degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Training of the CAE Network We train a single CAE model for all tested SNR values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' We use the AdamW optimizer [43] that runs back-propagation to optimize the model during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' This optimizer is designed to improve gradients when L2 regularization is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Our loss function is set to solve the constrained optimization problem by handling three objectives: accurate signal reconstruction (minimal BER), minimal PAPR, and acceptable ACPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' We solve this constrained optimization problem by recasting it as an unconstrained problem by constructing the Lagrangian function and augmenting the objective function with a quadratic January 13, 2023 DRAFT 18 penalty term [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The augmented Lagrangian (AL) combines the Lagrangian formulation with a weighted quadratic penalty function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The general AL for an inequality-constrained problem can be described by Fρk(x, λk) = f(x) + λk 1c1(x) + 1 2ρk 1 ∥c1(x)∥2 2 + 1 2ρk 2 �� max{0, λk 2 + ρk 2c2(x)} �2 − (λk 2)2� , (16) where f denotes the objective function, ρk ≜ (ρk 1, ρk 2) are positive penalty parameters, λk ≜ (λk 1, λk 2) are the Lagrangian multipliers, the c1-involved expressions handle the equality constraint, and c2 is for the inequality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Equation (16) considers the elimination of a slack variable s ≥ 0 that was introduced in the representation of the inequality constraint to transform it into a relaxed equality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' As suggested in [45], the minimizer s = max{0, c2(x) − λ2 1 ρ2} was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' k is the iteration number for updating the Lagrangian multipliers and penalty term, according to the following rule derived by the dual ascent method [44], xk+1 := arg min x Fρk(x, λk) (17) λk+1 1 := λk 1 + ρk 1c1(xk+1) (18) λk+1 2 := max{0, λk 2 + ρk 2c2(xk+1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' (19) We saw better convergence and more stable results for different BO values by adding the quadratic penalty function and adaptively updating the multipliers instead of keeping them constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Since adaptive penalty parameter update was not beneficial for the examined cases, it was added as a fixed hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Following the above-described general inequality constraint optimization problem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' the formu- lation of our loss function based on the appropriate objective and constraints,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' represented by DRAFT January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 2023 19 four loss components L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' L2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' L2b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' and L3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' is L(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' ˆx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' λk 2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' λk 2b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' λk 3) = L1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' ˆx) + λk 2aL2a(x) + ρ2a 2 ∥L2a(x)∥2 2 + λk 2bL2b(x) + ρ2b 2 ∥L2b(x)∥2 2 + 1 2ρ3 �� max{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' λk 3 + ρ3L3(x)} �2 − (λk 3)2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' (20) where λ2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' λ2b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' λ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' ρ2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' ρ2b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' and ρ3 are the appropriate Lagrange multipliers and penalty pa- rameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' These are considered hyper-parameters, which balance the contribution of each loss component to the joint loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' We start with a moderate value of λ2a, λ2b and λ3 and then iterate for a better value according to the resulting PAPR loss of each iteration and some predetermined PAPR threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Better performance was observed for relatively small λ2b values, with very low ρ2b, meaning that λ2b was kept almost constant during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The loss function we use for optimizing the signal reconstruction and detection is the sum of negative log loss function of the predicted output probability of the real and imaginary parts of each symbol, with L2 regularization to reduce over-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Denoting by x the input sample (which is also the output target),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' ˆx as the estimated signal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Θ as the model’s weights,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' and λ1 as a hyperparameter for tuning the L2 regularization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' the loss function for each OFDM MIMO sample is given by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' L1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' ˆx) = − � Nt � j=1 Nsc � s=1 Nc � q=1 1 � Re{xj} = lq � log Pθ � Re � ˆxj� = lq � + Nt � j=1 Nsc � s=1 Nc � q=1 1 � Im{xj} = lq � log Pθ � Im � ˆxj� = lq �� + λ1 ∥Θ∥2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' (21) where Nc = � |M| denotes the number of the real value possibilities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' lq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' of each of the real and imaginary parts of the transmitted modulated symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The PAPR minimization part is handled with two loss components, where one, L2a, is calculated according to the BPF input, xE n, and the other one, L2b, according to the BPF output, January 13, 2023 DRAFT 20 xF n (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' These are our equality constraints, defined by L2a(x) = PAPR{xE n}, (22) L2b(x) = PAPR{xF n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' (23) Other than the role of PAPR minimization handled by either of the components, L2a(x), signif- icantly improved the BER result together with the ACPR, while L2b(x), enabled us to control and obtain better spectral performance, meaning lower ACPR with lower output back-off (OBO) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The ACPR loss component is given by L3(x) = ACPR{xP n} − ACPRreq, (24) where xP n is the PA’s output, and ACPRreq is the required ACPR value, which is usually dictated by a standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' ACPRreq was set according to the 5G standard requirements for high spectral purity: ACPRreq ≤ −45dB [6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' thus, L3(x) defines our inequality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' We have applied a gradual loss learning technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' In the first stage, the loss function consisted only of L1 and optimized only the reconstruction loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Then, after a predetermined number of epochs, the loss function defined in (20) was used to reduce the PAPR and improve the spectral behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The gradual loss learning enables better control and stability in tuning the different criteria’ trade-offs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' RESULTS AND INSIGHTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Data Generation and Experimental Setup To train and test the proposed data-driven model, the MATLAB® 5G Toolbox™ [46] was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' This toolbox provides 5G radio-standard-compliant functions to generate accurate data for MIMO-OFDM transmission, according to specified constellation sizes and examined MIMO setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' MIMO-OFDM transmissions over fading channels were simulated, where TDL-D type channels - a 13 delay tap channel with a 30ns delay spread, as described in the 3gpp specification document [6], were used for our implemented algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' DRAFT January 13, 2023 21 We consider a MIMO-OFDM system with K =72 subcarriers over 14 OFDM symbols per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 4375 batches of 32 MIMO samples each were used for a single training set, where the input and output of the CAE sample shape is [2/4 − antennas, (72 − subcarriers) × (2 − complex − parts)×(4 − oversampling)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' An oversampling factor L = 4, and smoothness factor p = 2 were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' We trained three identical CAE models on the following setups: 1) QPSK modulation scheme with a 2 × 2 MIMO setting, with 3GPP multipath channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 2) 16-QAM modulation scheme with a 4 × 4 MIMO setting, with 3GPP multipath channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 3) 16-QAM modulation scheme with a 4 × 4 MIMO setting, with AWGN channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' To provide an unbiased performance evaluation of the final training model, the training and test data sets were generated independently, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' OFDM symbols, channel realization, and noise were randomized independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' In the following, we give numerical performance results of our multi-objective CAE model compared to a classical CF algorithm with a clipping ratio of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='08 dB, and to SLM with U = 64 phase sequences, with MLE added for MIMO detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The inference part was performed on 7000 MIMO samples for each SNR point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Training Setup As a part of the experimental analysis, we performed an extensive exploration of different model structures and hyper-parameters, including the number of layers, kernel sizes, number of convolutional layer channels, regularization, dropout, number of decoder iterations, batch- normalization, learning rate, AL parameters, trained SNR value, and epoch number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' We found that the best performance versus complexity on both examined MIMO setups was achieved for the same model structure, only with different training data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' As the constellation, the number of subcarriers per OFDM symbols, and the number of antennas are higher, the training is longer, and it is harder to achieve the desired results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The proposed CAE structure for the above 4 × 4 MIMO system is described in Table I, where λ(0) 2a , λ(0) 2b , and λ(0) 3 are the values of the first iteration when the AL epochs start, and ’Grad start’ indicates the number of initial epochs where only the reconstruction loss is counted, optimizing the unconstrained problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' ’LR’ indicates the learning January 13, 2023 DRAFT 22 TABLE I: CAE Proposed Structure Transmitter Receiver Parameter Value Kernel Ch-in Ch-out Value Kernel Ch-in Ch-out Input size 4 × 720 12 × 144 Conv (SELU) 1 × 3 1 21 3 × 3 1 15 Conv (SELU) 1 × 3 21 15 3 × 3 15 21 Conv (SELU) 1 × 3 15 21 FC (Linear) output size 4 × 720 12 × 144 Decoder iterations 10 Conv padding LR Epochs num Grad start SNR train λ(0) 2a , λ(0) 2b , λ(0) 3 ρ2a, ρ2b, ρ3 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='001 140 45 40 dB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='015, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='0015, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='00001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='001 rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Training on any of the data sets with the same best SNR value used for noise generation, ’SNR train’, showed the top overall inference performance for any tested SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Adding dropout had no benefit in all examined setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' BER Analysis The calculation of BER versus Peak Signal to Noise Ratio (P SNR) is used here as a key parameter to measure the reconstruction and detection of the transmitted signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Considering a normalized channel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' E ∥H∥2 = 1, the P SNR is defined as the ratio between the MIMO system maximal emitted energy, PT, and the noise power, σ2 w, such that P SNR = PT σ2w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' (25) As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 4, the CAE has competitive BER vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' P SNR performance compared to the other standard examined methods in most of the P SNR range, where a visible gain is achieved at the higher part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' As the NN does not assume any specific physical model, it has better robustness to distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' That is, the MIMO-OFDM signal reconstruction and detection of HPA-distorted data as a part of the multi-objective optimization proposed by our end-to-end DL scheme has the benefit over the common algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' DRAFT January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 2023 23 0 5 10 15 20 25 30 35 40 P-SNR[dB] 10 −3 10 −2 10 −1 BER No_PAPR_reduction CAE CF SLM (a) 16-QAM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 4x4 MIMO with 3GPP multipath channel 0 5 10 15 20 25 30 35 40 P-SNR[dB] 10 −5 10 −4 10 −3 10 −2 10 −1 BER No_PAPR_reduction CAE CF SLM (b) 16-QAM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 4x4 MIMO with AWGN channel 0 5 10 15 20 25 30 35 40 P-SNR[dB] 10 −4 10 −3 10 −2 10 −1 BER No_PAPR_reduction CAE CF SLM (c) 4-QAM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 2x2 MIMO with 3GPP multipath channel Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 4: BER vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' P SNR of the considered methods and setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' CCDF for PAPR Comparison To demonstrate the PAPR performance of different methods, complementary cumulative dis- tribution function (CCDF) curves are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The CCDF of the PAPR denotes the probability that the PAPR exceeds a certain threshold, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' P(PAPRMIMO−OFDM > PAPR0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The PAPR is calculated according to the BPF output, xF n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' As can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 5, the proposed CAE achieves the better performance of PAPR reduction compared to the CF and SLM methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' However, still the BER and spectral behavior are more important for performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 2023 DRAFT 24 5 6 7 8 9 10 11 PAPR0[dB] 10 −3 10 −2 10 −1 10 0 CCDF=Pr(PAPR>PAPR0) No_PAPR_reduction CAE CF SLM (a) 16-QAM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 4x4 MIMO with 3GPP multipath channel 5 6 7 8 9 10 11 12 PAPR0[dB] 10 −3 10 −2 10 −1 10 0 CCDF=Pr(PAPR>PAPR0) No_PAPR_reduction CAE CF SLM (b) 16-QAM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 4x4 MIMO with AWGN channel 4 5 6 7 8 9 10 11 PAPR0[dB] 10 −3 10 −2 10 −1 10 0 CCDF=Pr(PAPR>PAPR0) No_PAPR_reduction CAE CF SLM (c) 4-QAM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 2x2 MIMO with 3GPP multipath channel Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 5: CCDF of PAPR of the considered methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Spectrum Analysis Figure 6 compare the spectral performance in terms of the PSD of the transmitted signals for all examined methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The dashed rectangle shows perfect spectral behavior for a linear HPA with no non-linear components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The proposed CAE decreases the out-of-band distortions at the expense of lower transmitted power efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Observing the spectral behavior as a part of the experimental analysis showed us that there is a trade-off between increasing the IBO and increasing the Lagrange multiplier associated with the PAPR loss component, λ2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 7, while increasing the IBO mostly shifts the CAE curve downwards, increasing λ2b causes the curve to bend more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' DRAFT January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 2023 25 0 50 100 150 200 250 300 350 Subcarrier umber −70 −60 −50 −40 −30 −20 −10 0 PSD[dB] No_PAPR_reductio _ oPA No_PAPR_reductio CAE CF SLM (a) 16-QAM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 4x4 MIMO with 3GPP multipath channel 0 50 100 150 200 250 300 350 Subcarrier umber −70 −60 −50 −40 −30 −20 −10 0 PSD[dB] No_PAPR_reductio _ oPA No_PAPR_reductio CAE CF SLM (b) 16-QAM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 4x4 MIMO with AWGN channel 0 50 100 150 200 250 300 350 Subcarrier umber −70 −60 −50 −40 −30 −20 −10 0 PSD[dB] No_PAPR_reductio _ oPA No_PAPR_reductio CAE CF SLM (c) 4-QAM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 2x2 MIMO with 3GPP multipath channel Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 6: PSD for the considered methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' (a) moderate λ2b (b) high λ2b (c) moderate IBO (d) high IBO Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 7: Spectral mask behavior trade-off between IBO and λ2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The transmitter’s OBO, which evaluates the power efficiency of the system, is defined as the ratio between the maximal radiated power that is the maximal power transmitted by all the HPAs in the network, PT, and the mean transmitted power at the HPAs’ input, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' OBO = PT �Nt m=1 E � |xBn,m|2�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' (26) January 13, 2023 DRAFT 26 TABLE II: ACPR and OBO 4QAM 2X2 MIMO 16QAM 4X4 MIMO Parameter CAE FC-AE CF SLM No-reduction CAE FC-AE CF SLM No-reduction ACPR[dB] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='87 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='26 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='08 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='73 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='99 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='88 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='53 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='004 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='67 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='01 OBO[dB] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='92 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='62 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='74 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='78 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='86 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='09 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='77 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='74 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='78 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='84 −55 −50 −45 −40 −35 −30 −25 ACPR[dB] 2 3 4 5 6 7 8 9 10 OBO[dB] No_PAPR_reduction CAE CF SLM Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 8: OBO vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' ACPR for the considered meth- ods for 16-QAM with 4x4 MIMO setup and 3GPP multipath channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The maximum radiated power is defined to be PT = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' PT is divided equally between the HPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' In other words, the maximal trans- mitted power of each HPA is PT/Nt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' As we defined all HPAs to have the same charac- teristics, the saturation level of each HPA is A0 = � PT/NT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Table II compares the ACPR and the OBO of the proposed CAE to the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' As shown, the ACPR of the CAE is comparable with the considered methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 8 we further compare the OBO performance for different ACPR values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' It can be seen that the CAE system requires lower OBOs, which is better overall power efficiency, while maintaining better BER compared to the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Autoencoder - FC vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' CNN We investigated various NN types for the AE, in particular, FC and CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Figure 9a compares the BER performance of two AE architectures: the proposed CAE, which contains convolutional layers, and a fully connected autoencoder (FC-AE), which contains only FC layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' It can be observed that the CAE network has better BER performance compared to the FC-AE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' As shown in Table II, the ACPR of the CAE is better than that of the FC-AE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Moreover, the CAE has lower complexity and thus faster training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The three convolutional layers have a total of 1953 parameters, while for three FC layers of sizes 3500, 2500, and 3500, as were used for the FC-AE DRAFT January 13, 2023 27 0 5 10 15 20 25 30 35 40 P-SNR[dB] 10 −3 10 −2 10 −1 BER FC_AE CAE (a) FC-AE and CAE 0 5 10 15 20 25 30 35 40 P-SNR[dB] 10 −3 10 −2 10 −1 BER CAE_Non_Gradual CAE_Gradual (b) Fixed and gradual loss learning Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 9: BER vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' P SNR learning approaches comparison of 16-QAM, 4x4 MIMO with 3GPP multipath channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 9a and Table II, the number of parameters is around 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Fixed vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Gradual Loss Learning To show the benefits of using a gradual loss learning procedure, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 9b compares its BER performance to that of a fixed-loss training procedure, where the loss function’s weights are fixed for the entire training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' It can be observed that the gradual loss learning procedure significantly improves the BER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' In addition, improving the BER while keeping the PAPR and spectral performance at the desired levels is easier to control when applying the gradual loss learning method than manipulating loss function weights in fixed-loss training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Also, spectral performance and PAPR reduction were harder to control and provide similar performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' CONCLUSIONS AND FUTURE WORK In this study, we have presented a CAE model for PAPR reduction and waveform design in a MIMO-OFDM communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' We have applied a gradual loss learning method to optimize the performance in terms of three objectives: low BER, low PAPR, and adherence to ACPR spectral requirements, on top of the AL multipliers optimization technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The presented CAE structure trainable parts included a neural PAPR reduction block, followed by a BPF filter to optimize the spectral behavior at the transmitter, and a neural iterative MIMO detection block at January 13, 2023 DRAFT 28 the receiver, both were simultaneously optimized as a part of the end-to-end network design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' The proposed CAE was shown to outperform the CF and the SLM algorithms at the examined cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Future work can extend the MIMO scenario to higher modulation schemes and larger MIMO setups, aiming to achieve a functional utility for future wireless communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' REFERENCES [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Jiang and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Zhu, “Complement block coding for reduction in peak-to-average power ratio of ofdm signals,” IEEE Communications Magazine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 43, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' S17–S22, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Chowdhury, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Shahjalal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Ahmed, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Jang, “6g wireless communication systems: Applications, requirements, technologies, challenges, and research directions,” IEEE Open Journal of the Communications Society, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 957–975, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Bo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Zhi-Xing, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Tao-Tao, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Jian-Hua, “Effects of papr reduction on hpa predistortion,” IEEE Transactions on Consumer Electronics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 1143–1147, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Renzo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Stanczak, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Wang, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Larsson, “Artificial intelligence enabled wireless networking for 5g and beyond: Recent advances and future challenges,” IEEE Wireless Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 16–23, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [5] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Cho, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Kim, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Yang, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Kang, MIMO-OFDM wireless communications with MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' John Wiley & Sons, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [6] 3GPP, “3rd generation partnership project (3gpp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' study on channel model for frequencies from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='5 to 100 ghz,” 3rd Generation Partnership Project (3GPP), Technical Specification (TS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Available: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='3gpp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='org/ftp/Specs/archive/38 series/38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='901 [7] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Hassibi and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Vikalo, “On sphere decoding algorithm 1: expected complexity,” IEEE Transaction on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 53, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 2806–2818, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [8] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Wolniansky, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Foschini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Golden, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Valenzuela, “V-blast: An architecture for realizing very high data rates over the rich-scattering wireless channel,” in 1998 URSI international symposium on signals, systems, and electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Conference proceedings (Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 98EX167).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' IEEE, 1998, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 295–300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [9] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Steingrimsson, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Luo, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Wong, “Soft quasi-maximum-likelihood detection for multiple-antenna wireless channels,” IEEE Transactions on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 2710–2719, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [10] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Samuel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Diskin, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Wiesel, “Learning to detect,” IEEE Transactions on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 67, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 2554–2564, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [11] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Sholev, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Permuter, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Ben-Dror, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Liang, “Neural network MIMO detection for coded wireless communication with impairments,” in 2020 IEEE Wireless Communications and Networking Conference (WCNC), 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Mohammadkarimi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Mehrabi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Ardakani, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Jing, “Deep learning-based sphere decoding,” IEEE Transactions on Wireless Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 18, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 4368–4378, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' DRAFT January 13, 2023 29 [13] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Pratik, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Rao, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Welling, “Re-mimo: Recurrent and permutation equivariant neural mimo detection,” IEEE Transactions on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 69, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 459–473, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [14] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' O’Shea, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Erpek, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Clancy, “Deep learning based mimo communications,” arXiv preprint arXiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='07980, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Song, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' H¨ager, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Schr¨oder, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' O’Shea, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Agrell, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Wymeersch, “Benchmarking and interpreting end-to-end learning of mimo and multi-user communication,” IEEE Transactions on Wireless Communications, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [16] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' O’shea and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Hoydis, “An introduction to deep learning for the physical layer,” IEEE Transactions on Cognitive Communications and Networking, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 563–575, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Albreem, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Juntti, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Shahabuddin, “Massive mimo detection techniques: A survey,” IEEE Communications Surveys & Tutorials, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 3109–3132, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [18] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Gutman, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Iofedov, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Wulich, “Iterative decoding of iterative clipped and filtered OFDM signal,” IEEE Transactions on Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 61, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 4284–4293, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' More and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Somani, “The reduction of PAPR in OFDM systems using clipping and SLM method,” in 2013 International Conference on Information Communication and Embedded Systems (ICICES), 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 593–597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [20] Di-xiao Wu, “Selected mapping and partial transmit sequence schemes to reduce PAPR in OFDM systems,” in 2011 International Conference on Image Analysis and Signal Processing, 2011, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [21] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Jiang and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Wu, “An overview: Peak-to-average power ratio reduction techniques for OFDM signals,” IEEE Transactions on Broadcasting, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 54, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 257–268, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Mobasher and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Khandani, “Integer-based constellation-shaping method for PAPR reduction in OFDM systems,” IEEE Transactions on Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 54, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 119–127, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [23] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Taha and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Liu, “An adaptive coding technique for PAPR reduction,” in IEEE GLOBECOM 2007 - IEEE Global Telecommunications Conference, 2007, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 376–380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [24] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Lee, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' You, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Jeon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Paik, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Song, “Peak-to-average power ratio in mimo-ofdm systems using selective mapping,” IEEE Communications Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 575–577, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [25] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Namitha and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Sameer, “A bandwidth efficient selective mapping technique for the papr reduction in spatial multiplexing mimo-ofdm wireless communication system,” Physical Communication, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 25, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 128–138, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [26] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Zappone, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Di Renzo, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Debbah, “Wireless networks design in the era of deep learning: Model-based, AI-based, or both?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' IEEE Transactions on Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 67, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 7331–7376, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [27] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Sohn, “A low complexity PAPR reduction scheme for OFDM systems via neural networks,” IEEE Communications Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 18, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 225–228, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [28] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Sohn and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Kim, “Neural network based simplified clipping and filtering technique for PAPR reduction of OFDM signals,” IEEE Communications Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 19, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 1438–1441, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [29] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Kim, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Lee, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Cho, “A novel papr reduction scheme for OFDM system based on deep learning,” IEEE Communications Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 22, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 510–513, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' January 13, 2023 DRAFT 30 [30] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Shi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Wang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Vladimirescu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Zhang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Wang, “PAPR reduction based on deep autoencoder for VLC DCO-OFDM system,” in 2019 IEEE International Symposium on Broadband Multimedia Systems and Broadcasting (BMSB), 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [31] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Huleihel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Ben-Dror, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Permuter, “Low papr waveform design for ofdm systems based on convolutional autoencoder,” in 2020 IEEE International Conference on Advanced Networks and Telecommunications Systems (ANTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' IEEE, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [32] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Aoudia and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Hoydis, “Waveform learning for next-generation wireless communication systems,” IEEE Transactions on Communications, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [33] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Hao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Cheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Li, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Ma, “Performance enhancement of ACO-OFDM-based VLC systems using a hybrid autoencoder scheme,” Optics Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 442, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 110–116, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [34] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Kalinov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Bychkov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Ivanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Osinsky, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Yarotsky, “Machine learning-assisted papr reduction in massive mimo,” IEEE Wireless Communications Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 10, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 537–541, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [35] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Ivanov and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Lakontsev, “Selective tone reservation for papr reduction in wireless communication systems,” in 2017 IEEE International Workshop on Signal Processing Systems (SiPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' IEEE, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [36] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Ivanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Volokhatyi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Lakontsev, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Yarotsky, “Unused beam reservation for papr reduction in massive mimo system,” in 2018 IEEE 87th Vehicular Technology Conference (VTC Spring).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' IEEE, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [37] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Kenington, “Methods linearize RF transmitters and power amps,” Microwaves & RF, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 37, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 13, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 102–116, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [38] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Santurkar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Tsipras, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Ilyas, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Madry, “How does batch normalization help optimization?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' in Advances in Neural Information Processing Systems, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 2483–2493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [39] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Ren, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Sun, “Deep residual learning for image recognition,” in Proceedings of the IEEE conference on computer vision and pattern recognition, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 770–778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [40] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Klambauer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Unterthiner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Mayr, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Hochreiter, “Self-normalizing neural networks,” in Advances in neural information processing systems, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 971–980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [41] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Rowe, “Memoryless nonlinearities with gaussian inputs: Elementary results,” The BELL system technical Journal, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 61, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 1519–1525, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [42] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Stuber, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Barry, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Mclaughlin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Ingram, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Pratt, “Broadband mimo-ofdm wireless communications,” Proceedings of the IEEE, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 92, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 271–294, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [43] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Loshchilov and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Hutter, “Decoupled weight decay regularization,” arXiv preprint arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='05101, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [44] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Boyd, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Parikh, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Chu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Peleato, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends® in Machine learning, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 1–122, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [45] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Luenberger and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Yinyu, Linear and nonlinear programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Springer, 1984, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [46] “Matlab 5g toolbox,” 2020, the MathWorks, Natick, MA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content=' Available: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfUgzF/content/2301.05017v1.pdf'} +page_content='mathworks.' metadata={'source': 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0000000000000000000000000000000000000000..bfeccbf18bab1b957cea993e825a0b5f826ee9d8 Binary files /dev/null and b/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf differ diff --git a/rNE5T4oBgHgl3EQfJw5V/content/tmp_files/2301.05460v1.pdf.txt b/rNE5T4oBgHgl3EQfJw5V/content/tmp_files/2301.05460v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2e2d0750bbe0c6883248d8bdd8064f7909b48622 --- /dev/null +++ b/rNE5T4oBgHgl3EQfJw5V/content/tmp_files/2301.05460v1.pdf.txt @@ -0,0 +1,358 @@ +arXiv:2301.05460v1 [cs.DS] 13 Jan 2023 +Quick Minimization of Tardy Processing Time +on a Single Machine +Baruch Schieber * +Pranav Sitaraman † +Abstract +We consider the problem of minimizing the total processing time of tardy jobs on a sin- +gle machine. This is a classical scheduling problem, first considered by [Lawler and Moore +1969], that also generalizes the Subset Sum problem. Recently, it was shown that this prob- +lem can be solved efficiently by computing (max, min)-skewed-convolutions. The running +time of the resulting algorithm is equivalent, up to logarithmic factors, to the time it takes +to compute a (max, min)-skewed-convolution of two vectors of integers whose sum is O(P), +where P is the sum of the jobs’ processing times. We further improve the running time of the +minimum tardy processing time computation by introducing a job “bundling” technique and +achieve a ˜O +� +P2−1/α� +running time, where ˜O(Pα) is the running time of a (max, min)-skewed- +convolution of vectors of size P. This results in a ˜O +� +P7/5� +time algorithm for tardy processing +time minimization, an improvement over the previously known ˜O +� +P5/3� +time algorithm. +1 +Introduction +The input to the Minimum Tardy Processing Time (MTPT) Problem consists of n jobs each of +which is associated with a due date and processing time pi ∈ N. Consider a (nonpreemptive) +schedule of these jobs on a single machine that can execute only one job at a time. A job is tardy if +it terminates after its due date. The MTPT Problem is to find a schedule of the jobs that minimizes +the total processing time of the tardy jobs. In the standard scheduling notation the MTPT problem +is denoted 1|| ∑ pjUj. +Consider an instance of MTPT in which all the jobs have the same due date d. Let P = +∑n +j=1 pj > d. The decision whether the total processing time of the tardy jobs is exactly P − d +(which is optimal in this case) is equivalent to finding whether there exists a subset of the jobs +whose processing time sums to d. This is equivalent to the Subset Sum problem. It follows that +MTPT is NP-hard. MTPT is weakly NP-hard and Lawler and Moore [6] gave an O(P · n) time +algorithm for this problem. +Bringmann et al. [2] introduced a new convolution variant called a (max, min)-skewed-convolution. +They gave an algorithm for MTPT that uses (max, min)-skewed-convolutions, and proved that up +*Dept. of Computer Science, New Jersey Institute of Technology, Newark, NJ 07102, USA. sbar@njit.edu +†Edison Academy Magnet School, Edison, NJ 08837, USA. sitaraman.pranav@gmail.com +1 + +to logarithmic factors, the running time of this algorithm is equivalent to the time it takes to com- +pute a (max, min)-skewed-convolution of integers that sum to O(P). They also gave an ˜O +� +P7/4� +time algorithm1 for computing a (max, min)-skewed-convolution of integers that sum to O(P), +which results in an ˜O +� +P7/4� +time algorithm for the MTPT problem. Klein et al. [3] further im- +proved the algorithm for computing a (max, min)-skewed-convolution and achieved an ˜O +� +P5/3� +running time, and thus an ˜O +� +P5/3� +time algorithm for the MTPT problem. +A natural approach to further improve the MTPT algorithm is by improving the running time +of a (max, min)-skewed computation. However, obtaining an ˜o +� +P3/2� +time algorithm for com- +puting a (max, min)-skewed-convolution seems difficult as this would imply an improvement to +the best known (and decades old) algorithm for computing a (max, min)-convolution [5]. We +were able to “break” the ˜O +� +P3/2� +barrier by introducing a job “bundling” technique. Applying +this technique in conjunction with the best known algorithm for computing a (max, min)-skewed- +convolution yields an ˜O +� +P7/5� +time algorithm for the MTPT problem. This algorithm outper- +forms Lawler and Moore’s algorithm [6] in instances where n = ˜ω +� +P2/5� +. In general, applying +our technique in conjunction with an ˜O(Pα) time algorithm for computing a (max, min)-skewed- +convolution yields an ˜O +� +P2−1/α� +time for the MTPT problem. +The rest of the paper is organized as follows. In Section 2 we introduce our notations and +describe the prior work that we apply in our algorithm. Section 3 specifies our algorithm, and +Section 4 has some concluding remarks. +2 +Preliminaries +Our notations follow the notations in [2]. The input to the MTPT problem is a set of n jobs J = +{J1, J2, . . . , Jn}. Each job Jj ∈ J has due date ej ∈ N and processing time pj ∈ N. Let D# denote +the number of distinct due dates, and denote the monotone sequence of distinct due dates by +d1 < d2 < d3 < · · · < dD#, with d0 = 0. Let Jk ⊆ J be the set of jobs with due date dk. Let +D = ∑D# +i=1 di and P = ∑n +i=1 pi. For any I = {i, . . . , j}, where 1 ≤ i ≤ j ≤ D#, let JI = � +i∈I Ji and +PI = ∑Ji∈JI pi. +Recall that the goal is to schedule the jobs in J so that the total processing time of tardy jobs +is minimized. Since we only consider non-preemptive schedules, any schedule S corresponds to +a permutation σS : {1, . . . , n} → {1, . . . , n} of the job indices. The completion time of job Jj ∈ J in +schedule S is Cj = ∑σS(i)≤σS(j) pi, and j is tardy in S if Cj > ej. Therefore, we can consider that our +algorithm seeks to minimize ∑Jj∈J ,Cj>ej pj. +Next, we recall the definition of convolutions and describe the techniques developed in [2] and +used by our algorithm. Given two vectors A and B of dimension n + 1 and two binary operations ◦ +and •, the (◦, •)-convolution applied on A and B results in a 2n + 1 dimensional vector C, defined +as: +C[k] = ⃝min{k,n} +i=max{0,k−n}A[i] • B[k − i], ∀ k ∈ {0, . . . , 2n} . +A (max, min)-skewed-convolution applied on A and B results in a 2n + 1 dimensional vector +C, defined as: +C[k] = +min{k,n} +max +i=max{0,k−n} min {A[i], B[k − i] + k} , ∀ k ∈ {0, . . . , 2n} . +1The notation ˜O(·) hides all logarithmic factors. +2 + +Bringmann et al. [2] apply an equivalent form of (max, min)-skewed-convolution defined as +C[k] = +min{k,n} +max +i=max{0,k−n} min {A[i], B[k − i] − i} , ∀ k ∈ {0, . . . , 2n} . +Below, we use this equivalent form as well. +Let X and Y be two integral vectors. Define the sumset X ⊕ Y = {x + y : x ∈ X, y ∈ Y}. It is +not difficult to see that the sumset can be inferred from a (+, ·)-convolution of X1 and X2 which +can be computed in ˜O(P) time for X, Y ⊆ {0, . . . , P} as in [1]. +The set of all subset sums of entries of X, denoted S(X), is defined as S(X) = {∑x∈Z x : +Z ⊆ X}. These subset sums can be calculated in ˜O(∑x∈X x) time by successive computations of +sumsets [4]. We note that we always have 0 ∈ S(X). Define the t−prefix and t−suffix of S(X) as +pref(S, t) = {x ∈ S(X) ∧ x ≤ t} and suff(S, t) = {x ∈ S(X) ∧ x > t}. +We say that a subset of jobs J ′ ⊆ J can be scheduled feasibly starting at time t if there exists a +schedule of these jobs starting at time t such that all jobs are executed by their due date. Note that +it is enough to check whether all jobs in J ′ are executed by their due date in the earliest due date +first (EDD) schedule of these jobs starting at t. +For a consecutive subset of indices I = {i0, i0 + 1, . . . , i1}, with 1 ≤ i0 ≤ i1 ≤ D#, define an +integral vector M(I) as follows. The entry M(I)[x] equals −∞ if none of the subsets of jobs in JI +with total processing time exactly x can be scheduled feasibly. Otherwise, M(I)[x] equals the latest +time t starting at which a subset of jobs in JI with total processing time x can be scheduled feasibly. +Applying the algorithm for (max, min)-skewed-convolutions given in [3], we get an ˜O +� +PI5/3� +time algorithm for computing M(I), where PI = ∑Ji∈JI pi. +In addition to the algorithm that uses (max, min)-skewed-convolutions, Bringmann et al. [2] +gave a second algorithm for the MTPT problem. The running time of this algorithm is ˜O(P · D#). +We use a version of this algorithm in our algorithm and for completeness we describe it in Algo- +rithm 1. +Algorithm 1 The ˜O(P · D#) time algorithm +1: Let d1 < . . . < dD# denote the different due dates of jobs in J . +2: for i = 1, . . . , D# do +3: +Compute Xi = {pj : Jj ∈ Ji} +4: +Compute S(Xi) +5: Let S0 = ∅. +6: for i = 1, . . . , D# do +7: +Compute Si = Si−1 ⊕ S(Xi). +8: +Remove any x ∈ Si with x > di. +9: Return P − x, where x is the maximum value in SD#. +3 +The Algorithm +We define job bundles by coloring due dates in red and blue. The blue due dates are the bundled +ones. +3 + +Choose some δ ∈ (0, 1). For each k = 1, 2, . . . D#, color the due date dk red if ∑Ji∈Jk pi > P1−δ. +To determine the bundles we repeat the following procedure until all due dates are colored. +Let m be the largest index for which due date dm is not yet colored. Find the smallest k < m +that satisfies the following conditions. +Condition 1: None of the due dates dk, . . . , dm are colored red. +Condition 2: ∑m +i=k ∑Jj∈Ji pj ≤ P1−δ +Color all due dates dk, dk+1, . . . , dm blue and “bundle” them into one group, denoted B(k, m). We +say that due date dk is the start of the bundle and dm is the end of the bundle. +Lemma 1. The number of red due dates is O +� +Pδ� +and the number of bundles is O +� +Pδ� +. +Proof. Clearly, there can be at most Pδ due dates with ∑Ji∈Jk pi > P1−δ. Consider a bundle B(k, m). +Since k < m is the smallest index that satisfies the two conditions above, it is either true that dk−1 +is red or ∑m +i=k−1 ∑Jj∈Ji pj > P1−δ. +(i) Since there are at most Pδ red due dates, there can be at most only Pδ bundles B(k, m) for +which dk−1 is red. +(ii) Consider the sum ∑m +i=k−1 ∑Jj∈Ji pj, for a bundle B(k, m). Note that pj of a job Jj ∈ Jk−1 ∪ Jm +may appear in at most one more sum that corresponds to a different bundle, while pj of a job +Jj ∈ �m−1 +i=k Ji cannot appear in any other such sum. Thus, the total of all sums cannot exceed +2P. It follows that there are at most 2Pδ bundles B(k, m) for which ∑m +i=k−1 ∑Jj∈Ji pj > P1−δ. +■ +Algorithm 2 called SOLVE(J ), given below, follows the structure of Algorithm 1 with addi- +tional processing of entire bundles that avoids processing each due date in the bundles individ- +ually. We prove later that processing a bundle takes ˜O +� +P(1−δ)·α + P +� +time, where ˜O(Pα) is the +running time of the algorithm needed for computing a (max, min)-skewed-convolution. Process- +ing each red due date takes ˜O(P) time. Substituting δ = 1 − 1 +α yields a total running time of +˜O +� +P · P1−1/α� = ˜O +� +P2−1/α� +. +Theorem 1. Algorithm SOLVE(J ) returns the longest feasible schedule that starts at d0. +Proof. Consider iteration i of Algorithm SOLVE(J ), for an index i such that either di is a red due +date or di is the end of some bundle B(k, i). To prove the theorem it suffices to prove that at the +end of any such iteration i the set T consists of the processing times of all feasible schedules of +jobs in �i +j=1 Jj that start at d0. The proof is by induction. The basis is trivial since T is initialized +to {0}. Consider such an iteration i and suppose that the claim holds for all iterations i′ < i such +that either di′ is a red due date or di′ is the end of some bundle B(k′, i′). We distinguish two cases. +Case 1: di is a red due date. In this case it must be that either di−1 is also a red due date or di−1 is +the end of some bundle B(k′, i − 1). Thus, by our induction hypothesis, at the start of iteration i +the set T consists of the processing times of all feasible schedules of subsets of jobs in �i−1 +j=1 Jj that +start at d0. Since iteration i sets T = T ⊕ S(Xi) (Line 5), the claim follows. +4 + +Algorithm 2 SOLVE(J ) +1: Let T = {0}. +2: For each red due date di, compute Xi = +� +pj : ej = di +� +and S(Xi). +3: for i = 1, . . . , D# do +4: +if di is a red due date then +5: +Compute T = T ⊕ S(Xi). +6: +Remove any x ∈ T with x > di. +7: +else if di is the end of some bundle B(k, i) then +8: +Let I = {k, . . . , i}. +9: +Compute the vector M(I). +10: +Let Si = {x ∈ {0, . . . , PI} : M(I)[x] ̸= −∞}. +11: +if dk − PI ≥ 0 then +12: +Let T = T ∪ (pref(T, dk − PI) ⊕ Si). +13: +Let M′ be an integral vector of dimension dk and initialize M′ = −∞. +14: +For each x ∈ suff(T, dk − PI), let M′[x] = 0. +15: +for y = 0, . . . , dk − 1 + PI do +16: +Let C[y] = maxy +x=0 min{M′[x], M(I)[y − x] − x}. +17: +Let Ti = {x ∈ {0, . . . 2PI − 1} : C[x] = 0}. +18: +Let T = T ∪ Ti. +19: +Remove any x ∈ T with x > di. +20: Return P − x, where x is the maximum value in T. +Case 2: di is the end of some bundle B(k, i). By our induction hypothesis, at the start of iteration i +the set T consists of the processing times of all feasible schedules of subsets of jobs in �k−1 +j=1 Jj that +start at d0. Let I = {k, . . . , i}. The maximum length of any feasible schedule of subsets of jobs in +JI is PI. Since the earliest due date of these jobs is dk we are guaranteed that any such feasible +schedule can start at any time up to (and including) dk − PI (assuming that dk − PI ≥ 0). By the +definition of M(I) the set Si = {x ∈ {0, . . . , PI} : M(I)[x] ̸= −∞} consists of the processing times +of all feasible schedules of subsets of jobs in JI (Line 10). pref(T, dk − PI) consists of the processing +times of all feasible schedules of subsets of jobs in �k−1 +j=1 Jj that start at d0 and end at any time up +to (and including) dk − PI. Since iteration i sets T = T ∪ (pref(T, dk − PI) ⊕ Si) (Line 12), after this +line T consists of all the feasible schedules of subsets of jobs in �i +j=1 Jj that start at d0 and also +satisfy the condition that the sum of the lengths of the jobs in �k−1 +j=1 Jj that are scheduled is at most +dk − PI. +The set T is still missing the lengths of all the feasible schedules of subsets of jobs in �i +j=1 Jj that +start at d0 in which the sum of the lengths of the jobs in �k−1 +j=1 Jj exceeds dk − PI. These schedules +are added to T in Lines 13–18 of SOLVE(J ). Consider such a feasible schedule of length y in which +the length of the jobs in �k−1 +j=1 Jj is some x > dk − PI, which implies that M′[x] = 0. To complement +the prefix of this schedule by a feasible schedule of a subset of jobs in JI that starts at x and is of +length y − x we must have M(I)[y − x] ≥ x or min{M′[x], M(I)[y − x] − x} = 0. Lines 15–16 of +SOLVE(J ) check if such a feasible schedule exists. +■ +Lemma 2. The running time of algorithm SOLVE(J ) is ˜O +� +P(1−δ)·α + P +� +· Pδ. +Proof. By Lemma 1 the number of iterations that are not vacuous is Pδ. It is not difficult to see +that all operations other than the computation of the vectors M(I) and C take ˜O(P) time. The +5 + +vector M(I) is computed as in [2] in ˜O(PIα) time. The vector C is also computed via a (max, min)- +skewed-convolution and thus its computation time is proportional to the sum of lengths of the +vectors M(I) and M′ (up to logarithmic factors). Naively, this sum of lengths is dk + PI. However, +since M′[x] = −∞ for all x ≤ dk − PI, we can ignore these entries and implement the convolution +in ˜O(PIα) time. Recall that by the definition of bundles PI ≤ P1−δ. Thus, the lemma is proved. +■ +4 +Conclusions +We have shown a ˜O +� +P7/5� +time algorithm for tardy processing time minimization, an improve- +ment over the previously known ˜O +� +P5/3� +time algorithm. Improving this bound further is an +interesting open problem. In general, by applying our job “bundling” technique we can achieve a +˜O +� +P2−1/α� +running time, where ˜O(Pα) is the running time of a (max, min)-skewed-convolution of +vectors of size P. Since it is reasonable to assume that computing a (max, min)-skewed-convolution +requires ˜Ω +� +P3/2� +time, our technique is unlikely to yield a ˜o +� +P4/3� +running time. It will be inter- +esting to see whether this running time barrier can be broken, and whether the MTPT problem +can be solved without computing a (max, min)-skewed-convolution. +References +[1] Alfred V. Aho, John E. Hopcroft, and Jeffrey D. Ullman. The Design and Analysis of Computer +Algorithms. Addison-Wesley, 1974. +[2] Karl Bringmann, Nick Fischer, Danny Hermelin, Dvir Shabtay, and Philip Wellnitz. Faster +minimization of tardy processing time on a single machine. arXiv, 2020. +[3] Kim-Manuel Klein, Adam Polak, and Lars Rohwedder. On minimizing tardy processing time, +max-min skewed convolution, and triangular structured ILPs. arXiv, 2022. +[4] Konstantinos Koiliaris and Chao Xu. A faster pseudopolynomial time algorithm for subset +sum. +In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, page +1062–1072, USA, 2017. Society for Industrial and Applied Mathematics. +[5] S. R. Kosaraju. Efficient tree pattern matching. In Proceedings of the 30th Annual Symposium on +Foundations of Computer Science, page 178–183, USA, 1989. IEEE Computer Society. +[6] Eugene L. Lawler and J. M. Moore. A functional equation and its application to resource +allocation and sequencing problems. Management Science, 16:77–84, 1969. +6 + diff --git a/rNE5T4oBgHgl3EQfJw5V/content/tmp_files/load_file.txt b/rNE5T4oBgHgl3EQfJw5V/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1e541a1436c1d77a9c1f71a766068122e185d460 --- /dev/null +++ b/rNE5T4oBgHgl3EQfJw5V/content/tmp_files/load_file.txt @@ -0,0 +1,258 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf,len=257 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content='05460v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content='DS] 13 Jan 2023 Quick Minimization of Tardy Processing Time on a Single Machine Baruch Schieber * Pranav Sitaraman † Abstract We consider the problem of minimizing the total processing time of tardy jobs on a sin- gle machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' This is a classical scheduling problem, first considered by [Lawler and Moore 1969], that also generalizes the Subset Sum problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Recently, it was shown that this prob- lem can be solved efficiently by computing (max, min)-skewed-convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' The running time of the resulting algorithm is equivalent, up to logarithmic factors, to the time it takes to compute a (max, min)-skewed-convolution of two vectors of integers whose sum is O(P), where P is the sum of the jobs’ processing times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' We further improve the running time of the minimum tardy processing time computation by introducing a job “bundling” technique and achieve a ˜O � P2−1/α� running time, where ˜O(Pα) is the running time of a (max, min)-skewed- convolution of vectors of size P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' This results in a ˜O � P7/5� time algorithm for tardy processing time minimization, an improvement over the previously known ˜O � P5/3� time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 1 Introduction The input to the Minimum Tardy Processing Time (MTPT) Problem consists of n jobs each of which is associated with a due date and processing time pi ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Consider a (nonpreemptive) schedule of these jobs on a single machine that can execute only one job at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' A job is tardy if it terminates after its due date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' The MTPT Problem is to find a schedule of the jobs that minimizes the total processing time of the tardy jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' In the standard scheduling notation the MTPT problem is denoted 1|| ∑ pjUj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Consider an instance of MTPT in which all the jobs have the same due date d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Let P = ∑n j=1 pj > d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' The decision whether the total processing time of the tardy jobs is exactly P − d (which is optimal in this case) is equivalent to finding whether there exists a subset of the jobs whose processing time sums to d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' This is equivalent to the Subset Sum problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' It follows that MTPT is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' MTPT is weakly NP-hard and Lawler and Moore [6] gave an O(P · n) time algorithm for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Bringmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' [2] introduced a new convolution variant called a (max, min)-skewed-convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' They gave an algorithm for MTPT that uses (max, min)-skewed-convolutions, and proved that up Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' of Computer Science, New Jersey Institute of Technology, Newark, NJ 07102, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' sbar@njit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content='edu †Edison Academy Magnet School, Edison, NJ 08837, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' sitaraman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content='pranav@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content='com 1 to logarithmic factors, the running time of this algorithm is equivalent to the time it takes to com- pute a (max, min)-skewed-convolution of integers that sum to O(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' They also gave an ˜O � P7/4� time algorithm1 for computing a (max, min)-skewed-convolution of integers that sum to O(P), which results in an ˜O � P7/4� time algorithm for the MTPT problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Klein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' [3] further im- proved the algorithm for computing a (max, min)-skewed-convolution and achieved an ˜O � P5/3� running time, and thus an ˜O � P5/3� time algorithm for the MTPT problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' A natural approach to further improve the MTPT algorithm is by improving the running time of a (max, min)-skewed computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' However, obtaining an ˜o � P3/2� time algorithm for com- puting a (max, min)-skewed-convolution seems difficult as this would imply an improvement to the best known (and decades old) algorithm for computing a (max, min)-convolution [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' We were able to “break” the ˜O � P3/2� barrier by introducing a job “bundling” technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Applying this technique in conjunction with the best known algorithm for computing a (max, min)-skewed- convolution yields an ˜O � P7/5� time algorithm for the MTPT problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' This algorithm outper- forms Lawler and Moore’s algorithm [6] in instances where n = ˜ω � P2/5� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' In general, applying our technique in conjunction with an ˜O(Pα) time algorithm for computing a (max, min)-skewed- convolution yields an ˜O � P2−1/α� time for the MTPT problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' In Section 2 we introduce our notations and describe the prior work that we apply in our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Section 3 specifies our algorithm, and Section 4 has some concluding remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 2 Preliminaries Our notations follow the notations in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' The input to the MTPT problem is a set of n jobs J = {J1, J2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' , Jn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Each job Jj ∈ J has due date ej ∈ N and processing time pj ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Let D# denote the number of distinct due dates, and denote the monotone sequence of distinct due dates by d1 < d2 < d3 < · · · < dD#, with d0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Let Jk ⊆ J be the set of jobs with due date dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Let D = ∑D# i=1 di and P = ∑n i=1 pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' For any I = {i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' , j}, where 1 ≤ i ≤ j ≤ D#, let JI = � i∈I Ji and PI = ∑Ji∈JI pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Recall that the goal is to schedule the jobs in J so that the total processing time of tardy jobs is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Since we only consider non-preemptive schedules, any schedule S corresponds to a permutation σS : {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' , n} → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' , n} of the job indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' The completion time of job Jj ∈ J in schedule S is Cj = ∑σS(i)≤σS(j) pi, and j is tardy in S if Cj > ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Therefore, we can consider that our algorithm seeks to minimize ∑Jj∈J ,Cj>ej pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Next, we recall the definition of convolutions and describe the techniques developed in [2] and used by our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Given two vectors A and B of dimension n + 1 and two binary operations ◦ and •, the (◦, •)-convolution applied on A and B results in a 2n + 1 dimensional vector C, defined as: C[k] = ⃝min{k,n} i=max{0,k−n}A[i] • B[k − i], ∀ k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' , 2n} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' A (max, min)-skewed-convolution applied on A and B results in a 2n + 1 dimensional vector C, defined as: C[k] = min{k,n} max i=max{0,k−n} min {A[i], B[k − i] + k} , ∀ k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' , 2n} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 1The notation ˜O(·) hides all logarithmic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 2 Bringmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' [2] apply an equivalent form of (max, min)-skewed-convolution defined as C[k] = min{k,n} max i=max{0,k−n} min {A[i], B[k − i] − i} , ∀ k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' , 2n} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Below, we use this equivalent form as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Let X and Y be two integral vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Define the sumset X ⊕ Y = {x + y : x ∈ X, y ∈ Y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' It is not difficult to see that the sumset can be inferred from a (+, ·)-convolution of X1 and X2 which can be computed in ˜O(P) time for X, Y ⊆ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' , P} as in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' The set of all subset sums of entries of X, denoted S(X), is defined as S(X) = {∑x∈Z x : Z ⊆ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' These subset sums can be calculated in ˜O(∑x∈X x) time by successive computations of sumsets [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' We note that we always have 0 ∈ S(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Define the t−prefix and t−suffix of S(X) as pref(S, t) = {x ∈ S(X) ∧ x ≤ t} and suff(S, t) = {x ∈ S(X) ∧ x > t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' We say that a subset of jobs J ′ ⊆ J can be scheduled feasibly starting at time t if there exists a schedule of these jobs starting at time t such that all jobs are executed by their due date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Note that it is enough to check whether all jobs in J ′ are executed by their due date in the earliest due date first (EDD) schedule of these jobs starting at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' For a consecutive subset of indices I = {i0, i0 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' , i1}, with 1 ≤ i0 ≤ i1 ≤ D#, define an integral vector M(I) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' The entry M(I)[x] equals −∞ if none of the subsets of jobs in JI with total processing time exactly x can be scheduled feasibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Otherwise, M(I)[x] equals the latest time t starting at which a subset of jobs in JI with total processing time x can be scheduled feasibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Applying the algorithm for (max, min)-skewed-convolutions given in [3], we get an ˜O � PI5/3� time algorithm for computing M(I), where PI = ∑Ji∈JI pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' In addition to the algorithm that uses (max, min)-skewed-convolutions, Bringmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' [2] gave a second algorithm for the MTPT problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' The running time of this algorithm is ˜O(P · D#).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' We use a version of this algorithm in our algorithm and for completeness we describe it in Algo- rithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Algorithm 1 The ˜O(P · D#) time algorithm 1: Let d1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' < dD# denote the different due dates of jobs in J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 2: for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' , D# do 3: Compute Xi = {pj : Jj ∈ Ji} 4: Compute S(Xi) 5: Let S0 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 6: for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' , D# do 7: Compute Si = Si−1 ⊕ S(Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 8: Remove any x ∈ Si with x > di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 9: Return P − x, where x is the maximum value in SD#.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 3 The Algorithm We define job bundles by coloring due dates in red and blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' The blue due dates are the bundled ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 3 Choose some δ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' For each k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' D#, color the due date dk red if ∑Ji∈Jk pi > P1−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' To determine the bundles we repeat the following procedure until all due dates are colored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Let m be the largest index for which due date dm is not yet colored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Find the smallest k < m that satisfies the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Condition 1: None of the due dates dk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' , dm are colored red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Condition 2: ∑m i=k ∑Jj∈Ji pj ≤ P1−δ Color all due dates dk, dk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' , dm blue and “bundle” them into one group, denoted B(k, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' We say that due date dk is the start of the bundle and dm is the end of the bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' The number of red due dates is O � Pδ� and the number of bundles is O � Pδ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Clearly, there can be at most Pδ due dates with ∑Ji∈Jk pi > P1−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Consider a bundle B(k, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Since k < m is the smallest index that satisfies the two conditions above, it is either true that dk−1 is red or ∑m i=k−1 ∑Jj∈Ji pj > P1−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' (i) Since there are at most Pδ red due dates, there can be at most only Pδ bundles B(k, m) for which dk−1 is red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' (ii) Consider the sum ∑m i=k−1 ∑Jj∈Ji pj, for a bundle B(k, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Note that pj of a job Jj ∈ Jk−1 ∪ Jm may appear in at most one more sum that corresponds to a different bundle, while pj of a job Jj ∈ �m−1 i=k Ji cannot appear in any other such sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Thus, the total of all sums cannot exceed 2P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' It follows that there are at most 2Pδ bundles B(k, m) for which ∑m i=k−1 ∑Jj∈Ji pj > P1−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' ■ Algorithm 2 called SOLVE(J ), given below, follows the structure of Algorithm 1 with addi- tional processing of entire bundles that avoids processing each due date in the bundles individ- ually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' We prove later that processing a bundle takes ˜O � P(1−δ)·α + P � time, where ˜O(Pα) is the running time of the algorithm needed for computing a (max, min)-skewed-convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Process- ing each red due date takes ˜O(P) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Substituting δ = 1 − 1 α yields a total running time of ˜O � P · P1−1/α� = ˜O � P2−1/α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Algorithm SOLVE(J ) returns the longest feasible schedule that starts at d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Consider iteration i of Algorithm SOLVE(J ), for an index i such that either di is a red due date or di is the end of some bundle B(k, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' To prove the theorem it suffices to prove that at the end of any such iteration i the set T consists of the processing times of all feasible schedules of jobs in �i j=1 Jj that start at d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' The proof is by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' The basis is trivial since T is initialized to {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Consider such an iteration i and suppose that the claim holds for all iterations i′ < i such that either di′ is a red due date or di′ is the end of some bundle B(k′, i′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' We distinguish two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Case 1: di is a red due date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' In this case it must be that either di−1 is also a red due date or di−1 is the end of some bundle B(k′, i − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Thus, by our induction hypothesis, at the start of iteration i the set T consists of the processing times of all feasible schedules of subsets of jobs in �i−1 j=1 Jj that start at d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Since iteration i sets T = T ⊕ S(Xi) (Line 5), the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 4 Algorithm 2 SOLVE(J ) 1: Let T = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 2: For each red due date di, compute Xi = � pj : ej = di � and S(Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 3: for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' , D# do 4: if di is a red due date then 5: Compute T = T ⊕ S(Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 6: Remove any x ∈ T with x > di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 7: else if di is the end of some bundle B(k, i) then 8: Let I = {k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' , i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 9: Compute the vector M(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 10: Let Si = {x ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' , PI} : M(I)[x] ̸= −∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 11: if dk − PI ≥ 0 then 12: Let T = T ∪ (pref(T, dk − PI) ⊕ Si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 13: Let M′ be an integral vector of dimension dk and initialize M′ = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 14: For each x ∈ suff(T, dk − PI), let M′[x] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 15: for y = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' , dk − 1 + PI do 16: Let C[y] = maxy x=0 min{M′[x], M(I)[y − x] − x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 17: Let Ti = {x ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 2PI − 1} : C[x] = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 18: Let T = T ∪ Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 19: Remove any x ∈ T with x > di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 20: Return P − x, where x is the maximum value in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Case 2: di is the end of some bundle B(k, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' By our induction hypothesis, at the start of iteration i the set T consists of the processing times of all feasible schedules of subsets of jobs in �k−1 j=1 Jj that start at d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Let I = {k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' , i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' The maximum length of any feasible schedule of subsets of jobs in JI is PI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Since the earliest due date of these jobs is dk we are guaranteed that any such feasible schedule can start at any time up to (and including) dk − PI (assuming that dk − PI ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' By the definition of M(I) the set Si = {x ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' , PI} : M(I)[x] ̸= −∞} consists of the processing times of all feasible schedules of subsets of jobs in JI (Line 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' pref(T, dk − PI) consists of the processing times of all feasible schedules of subsets of jobs in �k−1 j=1 Jj that start at d0 and end at any time up to (and including) dk − PI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Since iteration i sets T = T ∪ (pref(T, dk − PI) ⊕ Si) (Line 12), after this line T consists of all the feasible schedules of subsets of jobs in �i j=1 Jj that start at d0 and also satisfy the condition that the sum of the lengths of the jobs in �k−1 j=1 Jj that are scheduled is at most dk − PI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' The set T is still missing the lengths of all the feasible schedules of subsets of jobs in �i j=1 Jj that start at d0 in which the sum of the lengths of the jobs in �k−1 j=1 Jj exceeds dk − PI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' These schedules are added to T in Lines 13–18 of SOLVE(J ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Consider such a feasible schedule of length y in which the length of the jobs in �k−1 j=1 Jj is some x > dk − PI, which implies that M′[x] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' To complement the prefix of this schedule by a feasible schedule of a subset of jobs in JI that starts at x and is of length y − x we must have M(I)[y − x] ≥ x or min{M′[x], M(I)[y − x] − x} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Lines 15–16 of SOLVE(J ) check if such a feasible schedule exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' ■ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' The running time of algorithm SOLVE(J ) is ˜O � P(1−δ)·α + P � Pδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' By Lemma 1 the number of iterations that are not vacuous is Pδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' It is not difficult to see that all operations other than the computation of the vectors M(I) and C take ˜O(P) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' The 5 vector M(I) is computed as in [2] in ˜O(PIα) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' The vector C is also computed via a (max, min)- skewed-convolution and thus its computation time is proportional to the sum of lengths of the vectors M(I) and M′ (up to logarithmic factors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Naively, this sum of lengths is dk + PI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' However, since M′[x] = −∞ for all x ≤ dk − PI, we can ignore these entries and implement the convolution in ˜O(PIα) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Recall that by the definition of bundles PI ≤ P1−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Thus, the lemma is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' ■ 4 Conclusions We have shown a ˜O � P7/5� time algorithm for tardy processing time minimization, an improve- ment over the previously known ˜O � P5/3� time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Improving this bound further is an interesting open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' In general, by applying our job “bundling” technique we can achieve a ˜O � P2−1/α� running time, where ˜O(Pα) is the running time of a (max, min)-skewed-convolution of vectors of size P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Since it is reasonable to assume that computing a (max, min)-skewed-convolution requires ˜Ω � P3/2� time, our technique is unlikely to yield a ˜o � P4/3� running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' It will be inter- esting to see whether this running time barrier can be broken, and whether the MTPT problem can be solved without computing a (max, min)-skewed-convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' References [1] Alfred V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Aho, John E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Hopcroft, and Jeffrey D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Ullman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' The Design and Analysis of Computer Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Addison-Wesley, 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' [2] Karl Bringmann, Nick Fischer, Danny Hermelin, Dvir Shabtay, and Philip Wellnitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Faster minimization of tardy processing time on a single machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' arXiv, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' [3] Kim-Manuel Klein, Adam Polak, and Lars Rohwedder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' On minimizing tardy processing time, max-min skewed convolution, and triangular structured ILPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' arXiv, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' [4] Konstantinos Koiliaris and Chao Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' A faster pseudopolynomial time algorithm for subset sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, page 1062–1072, USA, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Society for Industrial and Applied Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Kosaraju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Efficient tree pattern matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' In Proceedings of the 30th Annual Symposium on Foundations of Computer Science, page 178–183, USA, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' IEEE Computer Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' [6] Eugene L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Lawler and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Moore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' A functional equation and its application to resource allocation and sequencing problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' Management Science, 16:77–84, 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} +page_content=' 6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE5T4oBgHgl3EQfJw5V/content/2301.05460v1.pdf'} diff --git a/rtE3T4oBgHgl3EQf8wvj/content/2301.04811v1.pdf b/rtE3T4oBgHgl3EQf8wvj/content/2301.04811v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6c3132c5eb127e02ee73f11ad55403b66617655d --- /dev/null +++ b/rtE3T4oBgHgl3EQf8wvj/content/2301.04811v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:78ee52d88a6d7df05572098589b72f04a762ba2b5c1e29daf80068b3d92270c3 +size 1211885 diff --git a/rtE3T4oBgHgl3EQf8wvj/vector_store/index.pkl b/rtE3T4oBgHgl3EQf8wvj/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..44580c534a80ddb6c04fde817c3083f39238e112 --- /dev/null +++ b/rtE3T4oBgHgl3EQf8wvj/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d14b123fdfa1eec05d6e7e6dd68b11bbf59a078092e2791467606956623d5842 +size 134617 diff --git a/sNE1T4oBgHgl3EQf3QUB/content/tmp_files/2301.03486v1.pdf.txt b/sNE1T4oBgHgl3EQf3QUB/content/tmp_files/2301.03486v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cdd03163b77b89e1ddcceca9489b436143476aaa --- /dev/null +++ b/sNE1T4oBgHgl3EQf3QUB/content/tmp_files/2301.03486v1.pdf.txt @@ -0,0 +1,472 @@ +arXiv:2301.03486v1 [math.NT] 9 Jan 2023 +A FAMILY OF ELLIPTIC CURVES WITH NON-TRIVIAL +SHAFAREVICH-TATE GROUP +DEBOPAM CHAKRABORTY AND VINODKUMAR GHALE +Abstract +Rank computation of an elliptic curve is one of the most important problems in number +theory due to its importance in various open questions in number theory. A common +way to understand this rank is via understanding the Selmer group and the Shafarevich- +Tate group of an elliptic curve. We construct a family of Heronian elliptic curves of rank +0 and 2-Selmer rank 2. This, in turn, produces a family of elliptic curves such that the +Shafarevich-Tate group is isomorphic to the Klein four-group. +1. Introduction +The rational points E(Q) for an elliptic curve E defined over Q form a finitely generated +abelian group by Mordell-Weil theorem. It is a fundamental question in number theory +to understand the rank r(E/Q) of E(Q), also known as the algebraic or Mordell-Weil +rank. No known algorithm exists to find the rank of an elliptic curve. A full p-descent +method (see [15], Proposition X.4) generates an exact sequence +0 → E(Q)/pE(Q) → Selp(E/Q) → X(E/Q)[p] → 0 +where Selp(E/Q) denotes the p-Selmer group and X(E/Q) denotes the Tate-Shafarevich +group, both described in detail for p = 2 in the next section. The importance of the +Selmer and the Shafarevich-Tate group in the rank computation of an elliptic curve fol- +lows from the above p-descent exact sequence as r(E/Q) = sp(E/Q)−dimFpX(E/Q)[p] +where sp(E/Q) = dimFpSelp(E/Q) − dimFpE(Q)[p] denotes the p-Selmer rank of E. +Due to Cassels-Tate pairing (see [5]), the finiteness of the p-primary part of X(E/Q)[p∞] +would imply that X(E/Q)[p] has even Fp dimension, hence sp(E/Q) and r(E/Q) have +the same parity. The finiteness of X(E/Q)[p∞] implies p-Selmer rank one conjecture +which states that r(E/Q) = 1 whenever sp(E/Q) = 1. This conjecture has been verified +for p ≥ 5 under certain assumptions (see [19], [17], [16], and [3]). +2020 Mathematics Subject Classification. Primary 11G05, 11G07; Secondary 11R29, 51M04. +Key words and phrases. Elliptic curve; Selmer group; Class number; Heron triangle. +1 + +HERON TRIANGLE AND SHAFAREVICH-TATE GROUP +2 +Very little is yet to be known about the p-Selmer rank one conjecture for p = 2 even +though the computation of a full 2-descent is easiest in practice and provides the best +tool as of yet to compute r(E/Q). There has been a growing interest in 2-Selmer group +computation for different families of elliptic curves, as evident through the works of +Klagsbrun-Mazur-Rubin in [10], [11], and the work of Mazur-Rubin in [13]. +D. R. Heath-Brown examined the size of the 2-Selmer group of the congruent number +elliptic curve En : y2 = x3 − n2x in [1] and [2]. As an appendix to [2], Monsky showed +that the 2-Selmer rank s2(En/Q) = 2m − rankF2(M) when m is the number of distinct +odd prime factors of n and M is a 2m×2m matrix known as Monsky matrix. This boils +down to the problem of finding the 2-Selmer group for a congruent elliptic curve into +an intricate application of linear algebra over F2. +An immediate generalization of the congruent number problem is the existence of pos- +itive integers n, which are areas of triangles with rational sides without the constraint +of being a right-angle triangle. Such triangles are called Heron triangles. In [8], Goins +and Maddox showed the existence of an elliptic curve similar to the congruent elliptic +curve that determines whether n is the area of a Heron triangle. This triangle is known +as Heron triangle and is defined later. Even though the Monsly matrix calculation for +2-Selmer rank is unavailable in this setup, recently, some work towards the rank com- +putation of Heronian elliptic curves has been done. In [6], Dujella and Peral showed +elliptic curves of higher ranks associated with Heron triangles. In [9], Halbeisen and +Hungerbühler showed the existence of elliptic curves of rank at least two associated +with Heron triangles. Ghale et al. [7] constructed a family of elliptic curves of rank at +most one from a certain Diophantine equation via Heron triangles. In a recent work +by Chakraborty et al. [4], a family of Heronian elliptic curves with a 2-Selmer rank of +precisely 1 was presented. +In this article, we consider primes p ≡ 1 (mod 8) and q such that 2q = p2 + 1. We +examine the group structure of elliptic curves associated with Heron triangles of area p +having one of the angles as θ such that τ = tan θ +2 = 1 +p. The main result of this work is +the following. +Theorem 1.1. Let p be a prime congruent to 1 modulo 8 and q be a prime such that +2q = p2 +1. Then rank r(Ep) of the elliptic curve Ep : y2 = x(x−1)(x+p2) is precisely +zero, and X(Ep/Q)[2] ∼= Z/2Z × Z/2Z. + +HERON TRIANGLE AND SHAFAREVICH-TATE GROUP +3 +2. Background and notations +We begin this section by recalling the association of a Heron triangle with an elliptic +curve. In [8], Goins and Maddox showed that any triangle ∆ of area n ∈ Z with rational +sides a, b, c and an angle θ is associated with the elliptic curve +E : +y2 = x(x − nτ)(x + nτ −1), where τ = tan θ +2 ∈ Q. +We can identify the elliptic curve Ep in Theorem 1.1 as a Heronian elliptic curve. Let +S be the set consisting of all finite places at which Ep has bad reduction, the infinite +places and the prime 2, i.e., S = {p, q, 2, ∞}. We define +Q(S, 2) = +� +b ∈ Q∗/(Q∗)2 : ordl(b) ≡ 0 (mod 2) for all primes l ̸∈ S +� +(2.1) += {±1, ±2, ±p, ±q, ±2p, ±2q, ±pq, ±2pq} . +By the method of 2-descent (see [15], Proposition X.1.4), there exists an injective ho- +momorphism +φ : Ep(Q)/2Ep(Q) −→ Q(S, 2) × Q(S, 2) +such that A = {(1, 1), (−1, −1), (1, 2q), (−1, −2q)} is the image of Ep(Q)tors under φ. +Moreover, if (b1, b2) ∈ Q(S, 2) × Q(S, 2) is a pair that is not in the image of one of +the three points O, (0, 0), (1, 0), then (b1, b2) is the image of a point P = (x, y) ∈ +Ep(Q)/2Ep(Q) if and only if the equations +b1z2 +1 − b2z2 +2 = 1, +(2.2) +b1z2 +1 − b1b2z2 +3 = −p2, +(2.3) +have a solution (z1, z2, z3) ∈ Q∗ × Q∗ × Q. These smooth curves are called homogeneous +space of Ep defined over Q. Computing Ep(Q)/2Ep(Q) boils down to determining the +existence of Q-rational points in these spaces. Furthermore, the general failure of the +Hasse-Minkowski principle motivates the definition of the Selmer group which studies +the adelic points on the homogeneous spaces (see Remark X.1.2, X.4.9, and Section 3 of +chapter X in [15]). The image of Ep(Q)/2Ep(Q) under the 2-descent map is contained +in a subgroup of Q(S, 2) × Q(S, 2) known as the 2-Selmer group Sel2(Ep/Q), which, as +mentioned earlier, fits into an exact sequence (see Chapter X, [15], Theorem X.4.2) +(2.4) +0 −→ Ep(Q)/2Ep(Q) −→ Sel2(Ep/Q) −→ X(Ep/Q)[2] −→ 0. +The elements in Sel2(Ep/Q) correspond to the pairs (b1, b2) ∈ Q(S, 2) × Q(S, 2) such +that the system of equations (2.2) and (2.3) has non-trivial local solutions in Ql at all + +HERON TRIANGLE AND SHAFAREVICH-TATE GROUP +4 +primes l of Q including infinity. Note that # Ep(Q)/2Ep(Q) = 22+r(Ep). It is customary +to write # Sel2(Ep/Q) = 22+s(Ep), and refer to s(Ep) as the 2-Selmer rank. We have +0 ≤ r(Ep) ≤ s(Ep) always. +A solution for a polynomial f(x) modulo a prime l can be lifted to a solution for f(x) +in Zl through Hensel’s lemma. We state the result below that will be used to compute +Sel2(Ep/Q). +Theorem 2.1 (Hensel’s Lemma (see [18], Theorem A.2)). If f(x) is a polynomial with +coefficients that are l-adic integers and f(x1) ≡ 0 (mod l) for x1 ∈ Z, then if f ′(x1) ̸≡ 0 +(mod l), there exists an l-adic integer x with x ≡ x1 (mod l) such that f(x) = 0 +During the full 2-descent, we use the fact that if K(√α)/K is an unramified quadratic +extension of a number field K, then the class number of K is even. This follows from +the following theorem. +Theorem 2.2 (Hilbert’s Class Field [14]). The Hilbert class field E of a number field +K is the maximal unramified abelian extension of K. Its degree over K equals the class +number of K, and Gal(E/K) is canonically isomorphic to the ideal class group of K. +We conclude this section by noting that +� · +l +� +implies the Legendre symbol for quadratic +residues modulo prime l. +3. Local Solutions for the Homogeneous Spaces +In this section, we examine the properties of the l-adic solutions for (2.2) and (2.3) that +are associated with the 2-Selmer group. We use the well-known fact that any l-adic +number a can be written as a = ln · u where n ∈ Z, u ∈ Z∗ +l . Notice that the l-adic +valuation vl(a) of a is just n. We first prove the following result for all odd primes l. +Lemma 3.1. Suppose (2.2) and (2.3) have a solution (z1, z2, z3) ∈ Ql ×Ql ×Ql for any +prime l. If vl(zi) < 0 for any one i ∈ {1, 2, 3}, then vl(z1) = vl(z2) = vl(z3) = −k < 0 +for some integer k. +Proof. Let zi = lkiui, where ki ∈ Z and ui ∈ Z∗ +l for i = {1, 2, 3}. Then vl(zi) = ki for +all i ∈ {1, 2, 3}. + +HERON TRIANGLE AND SHAFAREVICH-TATE GROUP +5 +Suppose k1 < 0. Then from (2.2) one can get that +b1u2 +1 − b2u2 +2l2(k2−k1) = l−2k1. +If k2 > k1, then l2 must divide b1, a contradiction as b1 is square-free. Hence k2 ≤ k1 < 0. +Now if k2 < k1 < 0 then again from (2.2) we get +b1u2 +1l2(k1−k2) − b2u2 +2 = l−2k2, +which implies l2 must divide b2, a contradiction again. Hence if k1 < 0, then we have +k1 = k2 = −k < 0 for some integer k. For k2 < 0, one similarly gets k1 = k2 = −k < 0. +From (2.3), we have +b1u2 +1 − b1b2u2 +3l2(k3−k1) = −p2 · l−2k1. +If k1 < 0 and k3 > k1, then l2 must divide b1, a contradiction as before. Hence k3 ≤ +k1 < 0 if k1 < 0. For k3 < k1 < 0, we can rewrite the above equation as +(3.1) +b1u2 +1l2(k1−k3) − b1b2u2 +3 = −p2 · l−2k3, +which implies l2 must divide b1b2, i.e., l = 2, p or q. If l = p, then from (3.1) we arrive +at the contradiction that p3 divides b1b2 whereas b1 and b2 are square-free. For l = 2 +and q, one can notice from (2.3) that if k3 ≤ −2, then l3 divides b1b2, a contradiction +again. This in turn implies k3 = −1 and hence k1 ≥ 0 which contradicts the assumption +that k1 < 0. Hence k1 < 0 =⇒ k3 = k1. +Now, suppose k3 < 0. If k1 < 0, then from the previous part we already established +k1 = k2 = k3 = −k for some positive integer k. So without loss of generality, we can +assume k1 ≥ k3. If k3 < k1 and k3 < 0, then as mentioned previously in this proof, one +can get that b1b2 ≡ 0 (mod l2) and l = 2 or q. Now we subtract (2.3) from (2.2) and +observe that +b1b2u2 +3 − b2u2 +2l2(k2−k3) = 2q · l−2k3. +If k2 > k3, we get a contradiction that l3 divides b1b2 for l = 2, q. Therefore, k2 ≤ k3 < 0 +but then by the first part, k1 = k2 ≤ k3, a contradiction to the assumption k1 > k3. +Hence k3 < 0 =⇒ k1 = k3. Together, now we obtain k1 = k2 = k3 = −k < 0 for some +integer k if k1 < 0 or k2 < 0 or k3 < 0. +□ +4. Bounding the Size of the 2-Selmer Group +In this section, we bound the size of the 2-Selmer group of the Heronian elliptic curve Ep. +We limit the size of Sel2(Ep/Q) by ruling out local solutions for homogeneous spaces + +HERON TRIANGLE AND SHAFAREVICH-TATE GROUP +6 +corresponding to certain pairs (b1, b2). Without loss of generality, we can only focus +on the homogeneous spaces corresponding to pairs (b1, b2) such that b1 > 0, b2 > 0 if +b1b2 > 0 which we prove below. This is because every pair (b1, b2) such that b1b2 > +0 will belong to the same coset of (−b1, −b2) in the quotient group Im(φ)/A where +A = {(−1, −1), (1, 2q), (−1, −2q), (1, 1)}. Using the exactly similar argument, without +loss of generality, we can only focus on the local solutions of the homogeneous spaces +corresponding to (b1, b2) such that b2 is odd. +Lemma 4.1. Let (b1, b2) ̸∈ {(1, 1), (1, q), (p, 1), (p, q)}. Then the corresponding homo- +geneous space can not have local solutions for all primes l ≤ ∞. +Proof. Let the homogeneous space corresponding to (b1, b2) ∈ Q(S, 2)×Q(S, 2) have real +solutions. Then b1 > 0 and b2 < 0 implies −p2 > 0 in (2.3), which is absurd. Similarly, +b1 < 0 and b2 > 0 implies 1 < 0 in (2.2), contradiction again. Thus, the homogeneous +space corresponding to (b1, b2) has no l-adic solutions for l = ∞ if b1b2 < 0. +If gcd(b1, b2) ≡ 0 (mod p) and vp(zi) < 0 for any i ∈ {1, 2, 3}, then from Lemma 3.1 +and equation (2.3), one can get p2 divides b1, a contradiction. If gcd(b1, b2) ≡ 0 (mod p) +and vp(zi) ≥ 0 for all i ∈ {1, 2, 3}, then from Lemma 3.1 and equation (2.2), one can +get p divides 1, again a contradiction. Hence gcd(b1, b2) ̸≡ 0 (mod p). Now moreover, +if p divides b2 then vp(zi) ≥ 0 for i ∈ {1, 3} implies p divides b1 or z1 from (2.3), a +contradiction either way. If one of vp(zi) < 0 here, then also from Lemma 3.1 and (2.2), +one gets p divides b1, a contradiction again. +If q divides b1, then from the equation b1b2z2 +3 − b2z2 +2 = 2q, one get that q divides b2 if +vq(z3) ≥ 0 and vq(z2) ≥ 0. This is turn implies q divides 1 from (2.2) and Lemma 3.1, +a contradiction. Otherwise, again from Lemma 3.1 and (2.2), one gets that q divides b2 +and hence from (2.3), b1 ≡ 0 (mod q2), a contradiction. +We now show that for the existence of local solutions everywhere, b1 needs to be odd +always. Otherwise, b1 even and z1, z3 ∈ Z2 implies that p2 ≡ 0 (mod 2), a contradiction. +Else, from Lemma 3.1 and (2.2), one can see that b2 is even, a contradiction from the +assumption made above. +Now we can see for a homogeneous space corresponding to (b1, b2) to have local solution +everywhere b1 ∈ {1, p} and b2 ∈ {1, q}. This concludes the proof. +□ + +HERON TRIANGLE AND SHAFAREVICH-TATE GROUP +7 +5. Mordell-Weil Rank and ideal class group +In this section, we cover the pairs (p, 1) and (1, q). We use the fact that due to Hilbert’s +class field theorem, the existence of an unramified abelian extension of degree n of a +number field K is equivalent to the class number h(K) ≡ 0 (mod n). +Lemma 5.1. The homogeneous spaces corresponding to (p, 1) and (1, q) have no rational +solutions. +Proof. Let zi = +ai +di for i = 1, 2, 3 is a rational solution set for equations (2.2) and +(2.3) where the rational numbers zi are in their lowest form i.e. gcd(ai, di) = 1 for all +i = 1, 2, 3. It can be shown easily that d2 +1 = d2 +2 = d2 +3 = d2 for some integer d. So now we +have the following three equations; +b1a2 +1 − b2a2 +2 = d2, +(5.1) +b1a2 +1 − b1b2a2 +3 = −p2 · d2, +(5.2) +b1b2a2 +3 − b2a2 +2 = 2q · d2, +(5.3) +For the case (p, 1), we first notice that d is odd; hence, ai is odd for each i = 1, 2, 3. +As other wise, from (5.3) and noticing the fact that q ≡ 1 (mod 8), one can observe +a2 +3 − a2 +2 ≡ 2 (mod 8), a contradiction. From (5.2), one can actually see that d2 ≡ +a2 +1 − a2 +3 ≡ 0 (mod 8) =⇒ d ≡ 0 (mod 4). A straightforward calculation shows that +there are no common odd prime factors of a1 + a3 and a1 − a3. Assuming ai ≥ 0 for all +i = 1, 2, 3, equation (5.2) then implies that one of the two possibilities of a1 + a3 and +a1 − a3 is +a1 + a3 = p · 2n1 · m2 +1, a1 − a3 = −2n2 · m2 +2 +where m = m1m2 is odd, n = n1 + n2 ≥ 4 and d2 = 2n · m2. The fact that a3 is odd +and 2a3 = p · 2n1 · m2 +1 + 2n2 · m2 +2 now implies that either a3 = p · 2n−2 · m2 +1 + m2 +2 or +a3 = p · m2 +1 + 2n−2·. In either way, a3 ≡ 1 (mod 4). Same is true for the case when +a1 + a3 = 2n1 · m2 +1, a1 − a3 = −p · 2n2 · m2 +2. Now if one defines α = a3 + d√p ∈ Q(√p), +then from (5.2) we get NK/Q(α) = a2 +1 where K = Q(√p). Because gcd(a1, a3) = 1 can +be proved easily, one can also observe that gcd(α, ¯α) = 1 in OK, the ring of integers +of K, where ¯α = a3 − d√p. This in turn implies that αOK = a2 for some ideal a +which implies no finite primes except possibly primes above 2 ramifies in K(√α)/K. +But α = a3 + d√p ≡ 1 (mod 4) implies 2 also does not ramify in K(√α)/K. It is also +clear that infinite primes also do not ramify in K(√α)/K as K(√α) ⊂ R. Hence from + +HERON TRIANGLE AND SHAFAREVICH-TATE GROUP +8 +Hilbert’s class field theorem, we can conclude K = Q(√p) has an even class number +whenever the homogeneous space corresponding to (p, 1) has a rational solution. But it +is well known that the class number of Q(√p) is always odd. Hence the result follows. +We now look into the case (b1, b2) = (1, q). Similarly to the previous case, we can +show that d ≡ 0 (mod 4) and a3 ≡ 1 (mod 4) again. From (5.3), this implies α = +a3 + d +√ +2 again gives an unramifed abelian quadratic extension K(√α)/K where K = +Q( +√ +2). Hence existence of rational solution to homogeneous space corresponding to +(1, q) implies h(Q( +√ +2)) is even, a contradiction as h(Q( +√ +2)) = 1. +□ +6. Everywhere Local Solution +Now we prove that the homogeneous spaces corresponding to (p, 1) and (1, q) have local +solutions everywhere. +Lemma 6.1. The homogeneous spaces corresponding to (p, 1) and (1, q) have local so- +lutions everywhere for l ≤ ∞. +Proof. For (b1, b2) = (p, 1), first we consider l ≥ 5, l ̸= p. Suppose C is the homogeneous +space given by (2.2) and (2.3) corresponding to the pair (p, 1). Then C is a twist of Ep, +and in particular, it has genus 1. By the Hasse-Weil bound, we have +#C(Fl) ≥ 1 + l − 2 +√ +l ≥ 2 +for l ≥ 5, l ̸= p. +We can choose a solution (z1, z2, z3) ∈ Fl × Fl × Fl such that not all three of them are +zero modulo l. Now z1 ≡ z2 ≡ 0 (mod l) implies l2 divides 1 from (2.2), a contradiction. +Similarly, z1 ≡ z3 ≡ 0 (mod l) implies −p ≡ 0 (mod l2) from (2.3), contradiction again. +One can now suitably choose two of z1, z2 and z3 to convert equations (2.2) and (2.3) into +one single equation of one variable with a simple root over Fl. That common solution +can be lifted to Zl via Hensel’s lemma. +For l = p, we first notice that +� −1 +p +� += 1 as p ≡ 1 (mod 8). Hence there exists a ∈ Z +such that a2 ≡ −1 (mod p). Now fixing z1 = 1 in equation (2.2) and (2.3), we can +immediately see that z2 = a and z3 = 1 are two simple roots of now two single variable +polynomials and hence can be lifted to Zp. +For l = 3 and p ≡ 1 (mod 3), using Lemma 3.1, we look into solutions for the equations +pz2 +1 − z2 +2 = 32k and z2 +1 − z2 +3 = −p · 32k in Z3. Fixing z1 = 1, one can notice that +z2 = z3 = 1 are two simple solutions and hence can be lifted to solution in Z3 for the +equations mentioned above. Diving by 32k, will then gives rise to solutions in Q3 for + +HERON TRIANGLE AND SHAFAREVICH-TATE GROUP +9 +equations (2.2) and (2.3). For p ≡ 2 (mod 3), fixing z2 = 1 and z3 = 0 in (2.2) and +(2.3) respectively will give z1 = 1 as a simple solution modulo 3 and hence can be lifted +to Z3 via Hensel’s lemma. +For the case l = 2, just as in the beginning of the case l = 3, using Lemma 3.1, we +find solutions in Z2 for the equations pz2 +1 − z2 +2 = 22k and z2 +1 − z2 +3 = −p · 22k such that +k ≥ 2. Fixing z2 = z3 = 1 gives rise to z1 ≡ 1 (mod 8) as a solution modulo 8 to both +those equations that can be lifted to a solution in Z2. This ensures a solution for the +homogeneous space corresponding to (p, 1) in Q2 also. +For the case (b1, b2) = (1, q), the proof follows a very similar way as in the case (b1, b2) = +(p, 1). For l ≥ 5, l ̸= p, the homogeneous spaces C corresponding to (1, q) given by +equations (2.2) and (2.3) are of genus 1 and have solutions in Fl by Hasse-Weil bound +and can be lifted to Zl via a similar argument used in the previous case. +For l = p, because 2q ≡ 1 (mod p), one can notice that fixing z1 = 1 and then choosing +z2 = 1 and z3 = a is a solution that can be lifted to Zl via Hensel’s lemma where a2 ≡ 2 +(mod p). +For the case l = 3, fixing z1 = 1 and choosing z2 = z3 = 1 gives rise to solutions for the +equations z2 +1 − qz2 +2 = 32k and z2 +1 − qz2 +2 = −p2 · 32k. This solution can be lifted to Zl and +then give solution for (2.2) and (2.3) in Ql as mentioned in the previous part. +For the case l = 2, noticing the fact p ≡ 1 (mod 8) +=⇒ +q ≡ 1 (mod 8), the proof +follows the same way with the same choice of solutions modulo 8 for the case (b1, b2) = +(p, 1). +□ +7. Mordell-Weil rank and X(Ep/Q)[2] +We now conclude with the proof of Theorem 1.1 using the already established results. +A table is provided below in support of our result. Computations for the table has been +done in Magma [12] software. +Proof of Theorem 1.1. From Lemma 4.1 and Lemma 5.1, it is now obvious that the +rank of Ep : y2 = x(x − 1)(x + p2) is at most one due to the homogeneous space +corresponding to (p, q) which we are yet to rule out. But, Lemma 6.1 guarantees that +both (p, 1), (1, q) ∈ X(Ep/Q)[2]. This, along with the group structure of X(Ep/Q) +implies that (p, q) = (p, 1) × (1, q) ∈ X(Ep/Q)[2] too. Consequently, r(Ep) is exactly +zero. Moreover, due to the Cassels-Tate pairing (see [5]), dimF2(X(Ep/Q))[2] is even +which implies X(Ep/Q)[2] ∼= Z/2Z×Z/2Z. This concludes the proof of the theorem. +□ + +HERON TRIANGLE AND SHAFAREVICH-TATE GROUP +10 +Table 1. Examples +p +Rank of Ep +2- Selmer Rank of Ep +X(Ep/Q)[2] +409 +0 +2 +(Z/2Z)2 +449 +0 +2 +(Z/2Z)2 +521 +0 +2 +(Z/2Z)2 +569 +0 +2 +(Z/2Z)2 +641 +0 +2 +(Z/2Z)2 +Acknowledgement +The first author would like to acknowledge DST-SERB for providing the grant through +the Start-Up Research Grant (SRG/2020/001937) as well as BITS-Pilani, Hyderabad +Campus, for providing amenities. The second author would like to acknowledge the +fellowship (File No:09/1026(0029)/2019-EMR-I) and amenities provided by the Council +of Scientific and Industrial Research, India (CSIR) and BITS-Pilani, Hyderabad. +References +[1] D.R. Heath-Brown, "The size of Selmer groups for the congruent number problem." Inventiones +Mathematicae 111.1 (1993): 171-195. +[2] D.R. Heath-Brown, "The size of Selmer groups for the congruent number problem, II." Inventiones +Mathematicae 118 (1994): 331-370. +[3] C. Francesc, and X. Wan, "Iwasawa Main Conjecture for Heegner Points: Supersingular Case." +arXiv preprint arXiv:1506.02538 (2015). +[4] D. Chakraborty, V. Ghale, and A. Saikia, "Construction of an infinite family of elliptic curves of +2-selmer rank 1 from heron triangles." Research in Number Theory (2022). +[5] L. Chao, "2-Selmer groups, 2-class groups and rational points on elliptic curves." Transactions of +the American Mathematical Society 371, no. 7 (2019): 4631-4653. +[6] A. Dujella, and J.C. Peral, "Elliptic curves coming from Heron triangles." The Rocky Mountain +Journal of Mathematics 44.4 (2014): 1145-1160. +[7] V. Ghale, S. Das, and D. Chakraborty, "A Heron triangle and a Diophantine equation." Periodica +Mathematica Hungarica (2022): 1-8. +[8] E.H. Goins, and D. Maddox, "Heron triangles via elliptic curves." The Rocky Mountain Journal +of Mathematics (2006): 1511-1526. +[9] L. Halbeisen, and H. Norbert, "Heron triangles and their elliptic curves." Journal of Number +Theory 213 (2020): 232-253. +[10] K. Zev, B. Mazur, and K. Rubin, "Disparity in Selmer ranks of quadratic twists of elliptic curves." +Annals of Mathematics (2013): 287-320. +[11] K. Zev, B. Mazur, and K. Rubin. "A Markov model for Selmer ranks in families of twists." +Compositio Mathematica 150, no. 7 (2014): 1077-1106. + +HERON TRIANGLE AND SHAFAREVICH-TATE GROUP +11 +[12] The MAGMA algebra system, available at http://magma.maths.usyd.edu.au/calc/ +[13] B.Mazur, and K. Rubin. "Ranks of twists of elliptic curves and Hilbert’s tenth problem." Inven- +tiones Mathematicae 181, no. 3 (2010): 541-575. +[14] J. Neukirch, “Algebraic number theory". Vol. 322. Springer Science and Business Media, 2013. +[15] J.H. Silverman, "The arithmetic of elliptic curves." Vol. 106. New York: Springer, 2009. +[16] S. Christopher, "A converse to a theorem of Gross, Zagier, and Kolyvagin." Annals of Mathematics +191, no. 2 (2020): 329-354. +[17] W. Xin, "Heegner point Kolyvagin system and Iwasawa main conjecture." Acta Mathematica +Sinica, English Series 37, no. 1 (2021): 104-120. +[18] L.C. Washington, "Elliptic curves: number theory and cryptography." Chapman and Hall/CRC, +2008. +[19] W. Zhang, "Selmer groups and the indivisibility of Heegner points." Cambridge Journal of Math- +ematics 2, no. 2 (2014): 191-253. +Department of Mathematics, BITS-Pilani, Hyderabad campus, Hyderabad, INDIA +Email address: debopam@hyderabad.bits-pilani.ac.in +Department of Mathematics, BITS-Pilani, Hyderabad campus, Hyderabad, INDIA +Email address: p20180465@hyderabad.bits-pilani.ac.in + diff --git a/sNE1T4oBgHgl3EQf3QUB/content/tmp_files/load_file.txt b/sNE1T4oBgHgl3EQf3QUB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..326e41a70b3465cdbfee56d2c95a000a399ecae3 --- /dev/null +++ b/sNE1T4oBgHgl3EQf3QUB/content/tmp_files/load_file.txt @@ -0,0 +1,399 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf,len=398 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='03486v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='NT] 9 Jan 2023 A FAMILY OF ELLIPTIC CURVES WITH NON-TRIVIAL SHAFAREVICH-TATE GROUP DEBOPAM CHAKRABORTY AND VINODKUMAR GHALE Abstract Rank computation of an elliptic curve is one of the most important problems in number theory due to its importance in various open questions in number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' A common way to understand this rank is via understanding the Selmer group and the Shafarevich- Tate group of an elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' We construct a family of Heronian elliptic curves of rank 0 and 2-Selmer rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' This, in turn, produces a family of elliptic curves such that the Shafarevich-Tate group is isomorphic to the Klein four-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Introduction The rational points E(Q) for an elliptic curve E defined over Q form a finitely generated abelian group by Mordell-Weil theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' It is a fundamental question in number theory to understand the rank r(E/Q) of E(Q), also known as the algebraic or Mordell-Weil rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' No known algorithm exists to find the rank of an elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' A full p-descent method (see [15], Proposition X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='4) generates an exact sequence 0 → E(Q)/pE(Q) → Selp(E/Q) → X(E/Q)[p] → 0 where Selp(E/Q) denotes the p-Selmer group and X(E/Q) denotes the Tate-Shafarevich group, both described in detail for p = 2 in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' The importance of the Selmer and the Shafarevich-Tate group in the rank computation of an elliptic curve fol- lows from the above p-descent exact sequence as r(E/Q) = sp(E/Q)−dimFpX(E/Q)[p] where sp(E/Q) = dimFpSelp(E/Q) − dimFpE(Q)[p] denotes the p-Selmer rank of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Due to Cassels-Tate pairing (see [5]), the finiteness of the p-primary part of X(E/Q)[p∞] would imply that X(E/Q)[p] has even Fp dimension, hence sp(E/Q) and r(E/Q) have the same parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' The finiteness of X(E/Q)[p∞] implies p-Selmer rank one conjecture which states that r(E/Q) = 1 whenever sp(E/Q) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' This conjecture has been verified for p ≥ 5 under certain assumptions (see [19], [17], [16], and [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Primary 11G05, 11G07;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Secondary 11R29, 51M04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Elliptic curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Selmer group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Class number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Heron triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' 1 HERON TRIANGLE AND SHAFAREVICH-TATE GROUP 2 Very little is yet to be known about the p-Selmer rank one conjecture for p = 2 even though the computation of a full 2-descent is easiest in practice and provides the best tool as of yet to compute r(E/Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' There has been a growing interest in 2-Selmer group computation for different families of elliptic curves, as evident through the works of Klagsbrun-Mazur-Rubin in [10], [11], and the work of Mazur-Rubin in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Heath-Brown examined the size of the 2-Selmer group of the congruent number elliptic curve En : y2 = x3 − n2x in [1] and [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' As an appendix to [2], Monsky showed that the 2-Selmer rank s2(En/Q) = 2m − rankF2(M) when m is the number of distinct odd prime factors of n and M is a 2m×2m matrix known as Monsky matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' This boils down to the problem of finding the 2-Selmer group for a congruent elliptic curve into an intricate application of linear algebra over F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' An immediate generalization of the congruent number problem is the existence of pos- itive integers n, which are areas of triangles with rational sides without the constraint of being a right-angle triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Such triangles are called Heron triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' In [8], Goins and Maddox showed the existence of an elliptic curve similar to the congruent elliptic curve that determines whether n is the area of a Heron triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' This triangle is known as Heron triangle and is defined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Even though the Monsly matrix calculation for 2-Selmer rank is unavailable in this setup, recently, some work towards the rank com- putation of Heronian elliptic curves has been done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' In [6], Dujella and Peral showed elliptic curves of higher ranks associated with Heron triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' In [9], Halbeisen and Hungerbühler showed the existence of elliptic curves of rank at least two associated with Heron triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Ghale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' [7] constructed a family of elliptic curves of rank at most one from a certain Diophantine equation via Heron triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' In a recent work by Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' [4], a family of Heronian elliptic curves with a 2-Selmer rank of precisely 1 was presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' In this article, we consider primes p ≡ 1 (mod 8) and q such that 2q = p2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' We examine the group structure of elliptic curves associated with Heron triangles of area p having one of the angles as θ such that τ = tan θ 2 = 1 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' The main result of this work is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Let p be a prime congruent to 1 modulo 8 and q be a prime such that 2q = p2 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Then rank r(Ep) of the elliptic curve Ep : y2 = x(x−1)(x+p2) is precisely zero, and X(Ep/Q)[2] ∼= Z/2Z × Z/2Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' HERON TRIANGLE AND SHAFAREVICH-TATE GROUP 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Background and notations We begin this section by recalling the association of a Heron triangle with an elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' In [8], Goins and Maddox showed that any triangle ∆ of area n ∈ Z with rational sides a, b, c and an angle θ is associated with the elliptic curve E : y2 = x(x − nτ)(x + nτ −1), where τ = tan θ 2 ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' We can identify the elliptic curve Ep in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1 as a Heronian elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Let S be the set consisting of all finite places at which Ep has bad reduction, the infinite places and the prime 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=', S = {p, q, 2, ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' We define Q(S, 2) = � b ∈ Q∗/(Q∗)2 : ordl(b) ≡ 0 (mod 2) for all primes l ̸∈ S � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1) = {±1, ±2, ±p, ±q, ±2p, ±2q, ±pq, ±2pq} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' By the method of 2-descent (see [15], Proposition X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='4), there exists an injective ho- momorphism φ : Ep(Q)/2Ep(Q) −→ Q(S, 2) × Q(S, 2) such that A = {(1, 1), (−1, −1), (1, 2q), (−1, −2q)} is the image of Ep(Q)tors under φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Moreover, if (b1, b2) ∈ Q(S, 2) × Q(S, 2) is a pair that is not in the image of one of the three points O, (0, 0), (1, 0), then (b1, b2) is the image of a point P = (x, y) ∈ Ep(Q)/2Ep(Q) if and only if the equations b1z2 1 − b2z2 2 = 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2) b1z2 1 − b1b2z2 3 = −p2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3) have a solution (z1, z2, z3) ∈ Q∗ × Q∗ × Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' These smooth curves are called homogeneous space of Ep defined over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Computing Ep(Q)/2Ep(Q) boils down to determining the existence of Q-rational points in these spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Furthermore, the general failure of the Hasse-Minkowski principle motivates the definition of the Selmer group which studies the adelic points on the homogeneous spaces (see Remark X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='9, and Section 3 of chapter X in [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' The image of Ep(Q)/2Ep(Q) under the 2-descent map is contained in a subgroup of Q(S, 2) × Q(S, 2) known as the 2-Selmer group Sel2(Ep/Q), which, as mentioned earlier, fits into an exact sequence (see Chapter X, [15], Theorem X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='4) 0 −→ Ep(Q)/2Ep(Q) −→ Sel2(Ep/Q) −→ X(Ep/Q)[2] −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' The elements in Sel2(Ep/Q) correspond to the pairs (b1, b2) ∈ Q(S, 2) × Q(S, 2) such that the system of equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3) has non-trivial local solutions in Ql at all HERON TRIANGLE AND SHAFAREVICH-TATE GROUP 4 primes l of Q including infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Note that # Ep(Q)/2Ep(Q) = 22+r(Ep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' It is customary to write # Sel2(Ep/Q) = 22+s(Ep), and refer to s(Ep) as the 2-Selmer rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' We have 0 ≤ r(Ep) ≤ s(Ep) always.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' A solution for a polynomial f(x) modulo a prime l can be lifted to a solution for f(x) in Zl through Hensel’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' We state the result below that will be used to compute Sel2(Ep/Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1 (Hensel’s Lemma (see [18], Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' If f(x) is a polynomial with coefficients that are l-adic integers and f(x1) ≡ 0 (mod l) for x1 ∈ Z, then if f ′(x1) ̸≡ 0 (mod l), there exists an l-adic integer x with x ≡ x1 (mod l) such that f(x) = 0 During the full 2-descent, we use the fact that if K(√α)/K is an unramified quadratic extension of a number field K, then the class number of K is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' This follows from the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2 (Hilbert’s Class Field [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' The Hilbert class field E of a number field K is the maximal unramified abelian extension of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Its degree over K equals the class number of K, and Gal(E/K) is canonically isomorphic to the ideal class group of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' We conclude this section by noting that � · l � implies the Legendre symbol for quadratic residues modulo prime l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Local Solutions for the Homogeneous Spaces In this section, we examine the properties of the l-adic solutions for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3) that are associated with the 2-Selmer group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' We use the well-known fact that any l-adic number a can be written as a = ln · u where n ∈ Z, u ∈ Z∗ l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Notice that the l-adic valuation vl(a) of a is just n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' We first prove the following result for all odd primes l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Suppose (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3) have a solution (z1, z2, z3) ∈ Ql ×Ql ×Ql for any prime l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' If vl(zi) < 0 for any one i ∈ {1, 2, 3}, then vl(z1) = vl(z2) = vl(z3) = −k < 0 for some integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Let zi = lkiui, where ki ∈ Z and ui ∈ Z∗ l for i = {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Then vl(zi) = ki for all i ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' HERON TRIANGLE AND SHAFAREVICH-TATE GROUP 5 Suppose k1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Then from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2) one can get that b1u2 1 − b2u2 2l2(k2−k1) = l−2k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' If k2 > k1, then l2 must divide b1, a contradiction as b1 is square-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Hence k2 ≤ k1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Now if k2 < k1 < 0 then again from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2) we get b1u2 1l2(k1−k2) − b2u2 2 = l−2k2, which implies l2 must divide b2, a contradiction again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Hence if k1 < 0, then we have k1 = k2 = −k < 0 for some integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' For k2 < 0, one similarly gets k1 = k2 = −k < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3), we have b1u2 1 − b1b2u2 3l2(k3−k1) = −p2 · l−2k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' If k1 < 0 and k3 > k1, then l2 must divide b1, a contradiction as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Hence k3 ≤ k1 < 0 if k1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' For k3 < k1 < 0, we can rewrite the above equation as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1) b1u2 1l2(k1−k3) − b1b2u2 3 = −p2 · l−2k3, which implies l2 must divide b1b2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=', l = 2, p or q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' If l = p, then from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1) we arrive at the contradiction that p3 divides b1b2 whereas b1 and b2 are square-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' For l = 2 and q, one can notice from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3) that if k3 ≤ −2, then l3 divides b1b2, a contradiction again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' This in turn implies k3 = −1 and hence k1 ≥ 0 which contradicts the assumption that k1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Hence k1 < 0 =⇒ k3 = k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Now, suppose k3 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' If k1 < 0, then from the previous part we already established k1 = k2 = k3 = −k for some positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' So without loss of generality, we can assume k1 ≥ k3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' If k3 < k1 and k3 < 0, then as mentioned previously in this proof, one can get that b1b2 ≡ 0 (mod l2) and l = 2 or q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Now we subtract (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3) from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2) and observe that b1b2u2 3 − b2u2 2l2(k2−k3) = 2q · l−2k3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' If k2 > k3, we get a contradiction that l3 divides b1b2 for l = 2, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Therefore, k2 ≤ k3 < 0 but then by the first part, k1 = k2 ≤ k3, a contradiction to the assumption k1 > k3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Hence k3 < 0 =⇒ k1 = k3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Together, now we obtain k1 = k2 = k3 = −k < 0 for some integer k if k1 < 0 or k2 < 0 or k3 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Bounding the Size of the 2-Selmer Group In this section, we bound the size of the 2-Selmer group of the Heronian elliptic curve Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' We limit the size of Sel2(Ep/Q) by ruling out local solutions for homogeneous spaces HERON TRIANGLE AND SHAFAREVICH-TATE GROUP 6 corresponding to certain pairs (b1, b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Without loss of generality, we can only focus on the homogeneous spaces corresponding to pairs (b1, b2) such that b1 > 0, b2 > 0 if b1b2 > 0 which we prove below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' This is because every pair (b1, b2) such that b1b2 > 0 will belong to the same coset of (−b1, −b2) in the quotient group Im(φ)/A where A = {(−1, −1), (1, 2q), (−1, −2q), (1, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Using the exactly similar argument, without loss of generality, we can only focus on the local solutions of the homogeneous spaces corresponding to (b1, b2) such that b2 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Let (b1, b2) ̸∈ {(1, 1), (1, q), (p, 1), (p, q)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Then the corresponding homo- geneous space can not have local solutions for all primes l ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Let the homogeneous space corresponding to (b1, b2) ∈ Q(S, 2)×Q(S, 2) have real solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Then b1 > 0 and b2 < 0 implies −p2 > 0 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3), which is absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Similarly, b1 < 0 and b2 > 0 implies 1 < 0 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2), contradiction again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Thus, the homogeneous space corresponding to (b1, b2) has no l-adic solutions for l = ∞ if b1b2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' If gcd(b1, b2) ≡ 0 (mod p) and vp(zi) < 0 for any i ∈ {1, 2, 3}, then from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1 and equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3), one can get p2 divides b1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' If gcd(b1, b2) ≡ 0 (mod p) and vp(zi) ≥ 0 for all i ∈ {1, 2, 3}, then from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1 and equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2), one can get p divides 1, again a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Hence gcd(b1, b2) ̸≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Now moreover, if p divides b2 then vp(zi) ≥ 0 for i ∈ {1, 3} implies p divides b1 or z1 from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3), a contradiction either way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' If one of vp(zi) < 0 here, then also from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2), one gets p divides b1, a contradiction again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' If q divides b1, then from the equation b1b2z2 3 − b2z2 2 = 2q, one get that q divides b2 if vq(z3) ≥ 0 and vq(z2) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' This is turn implies q divides 1 from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Otherwise, again from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2), one gets that q divides b2 and hence from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3), b1 ≡ 0 (mod q2), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' We now show that for the existence of local solutions everywhere, b1 needs to be odd always.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Otherwise, b1 even and z1, z3 ∈ Z2 implies that p2 ≡ 0 (mod 2), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Else, from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2), one can see that b2 is even, a contradiction from the assumption made above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Now we can see for a homogeneous space corresponding to (b1, b2) to have local solution everywhere b1 ∈ {1, p} and b2 ∈ {1, q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' □ HERON TRIANGLE AND SHAFAREVICH-TATE GROUP 7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Mordell-Weil Rank and ideal class group In this section, we cover the pairs (p, 1) and (1, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' We use the fact that due to Hilbert’s class field theorem, the existence of an unramified abelian extension of degree n of a number field K is equivalent to the class number h(K) ≡ 0 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' The homogeneous spaces corresponding to (p, 1) and (1, q) have no rational solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Let zi = ai di for i = 1, 2, 3 is a rational solution set for equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3) where the rational numbers zi are in their lowest form i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' gcd(ai, di) = 1 for all i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' It can be shown easily that d2 1 = d2 2 = d2 3 = d2 for some integer d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' So now we have the following three equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' b1a2 1 − b2a2 2 = d2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1) b1a2 1 − b1b2a2 3 = −p2 · d2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2) b1b2a2 3 − b2a2 2 = 2q · d2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3) For the case (p, 1), we first notice that d is odd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' hence, ai is odd for each i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' As other wise, from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3) and noticing the fact that q ≡ 1 (mod 8), one can observe a2 3 − a2 2 ≡ 2 (mod 8), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2), one can actually see that d2 ≡ a2 1 − a2 3 ≡ 0 (mod 8) =⇒ d ≡ 0 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' A straightforward calculation shows that there are no common odd prime factors of a1 + a3 and a1 − a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Assuming ai ≥ 0 for all i = 1, 2, 3, equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2) then implies that one of the two possibilities of a1 + a3 and a1 − a3 is a1 + a3 = p · 2n1 · m2 1, a1 − a3 = −2n2 · m2 2 where m = m1m2 is odd, n = n1 + n2 ≥ 4 and d2 = 2n · m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' The fact that a3 is odd and 2a3 = p · 2n1 · m2 1 + 2n2 · m2 2 now implies that either a3 = p · 2n−2 · m2 1 + m2 2 or a3 = p · m2 1 + 2n−2·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' In either way, a3 ≡ 1 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Same is true for the case when a1 + a3 = 2n1 · m2 1, a1 − a3 = −p · 2n2 · m2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Now if one defines α = a3 + d√p ∈ Q(√p), then from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2) we get NK/Q(α) = a2 1 where K = Q(√p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Because gcd(a1, a3) = 1 can be proved easily, one can also observe that gcd(α, ¯α) = 1 in OK, the ring of integers of K, where ¯α = a3 − d√p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' This in turn implies that αOK = a2 for some ideal a which implies no finite primes except possibly primes above 2 ramifies in K(√α)/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' But α = a3 + d√p ≡ 1 (mod 4) implies 2 also does not ramify in K(√α)/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' It is also clear that infinite primes also do not ramify in K(√α)/K as K(√α) ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Hence from HERON TRIANGLE AND SHAFAREVICH-TATE GROUP 8 Hilbert’s class field theorem, we can conclude K = Q(√p) has an even class number whenever the homogeneous space corresponding to (p, 1) has a rational solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' But it is well known that the class number of Q(√p) is always odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Hence the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' We now look into the case (b1, b2) = (1, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Similarly to the previous case, we can show that d ≡ 0 (mod 4) and a3 ≡ 1 (mod 4) again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3), this implies α = a3 + d √ 2 again gives an unramifed abelian quadratic extension K(√α)/K where K = Q( √ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Hence existence of rational solution to homogeneous space corresponding to (1, q) implies h(Q( √ 2)) is even, a contradiction as h(Q( √ 2)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Everywhere Local Solution Now we prove that the homogeneous spaces corresponding to (p, 1) and (1, q) have local solutions everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' The homogeneous spaces corresponding to (p, 1) and (1, q) have local so- lutions everywhere for l ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' For (b1, b2) = (p, 1), first we consider l ≥ 5, l ̸= p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Suppose C is the homogeneous space given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3) corresponding to the pair (p, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Then C is a twist of Ep, and in particular, it has genus 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' By the Hasse-Weil bound, we have #C(Fl) ≥ 1 + l − 2 √ l ≥ 2 for l ≥ 5, l ̸= p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' We can choose a solution (z1, z2, z3) ∈ Fl × Fl × Fl such that not all three of them are zero modulo l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Now z1 ≡ z2 ≡ 0 (mod l) implies l2 divides 1 from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Similarly, z1 ≡ z3 ≡ 0 (mod l) implies −p ≡ 0 (mod l2) from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3), contradiction again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' One can now suitably choose two of z1, z2 and z3 to convert equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3) into one single equation of one variable with a simple root over Fl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' That common solution can be lifted to Zl via Hensel’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' For l = p, we first notice that � −1 p � = 1 as p ≡ 1 (mod 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Hence there exists a ∈ Z such that a2 ≡ −1 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Now fixing z1 = 1 in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3), we can immediately see that z2 = a and z3 = 1 are two simple roots of now two single variable polynomials and hence can be lifted to Zp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' For l = 3 and p ≡ 1 (mod 3), using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1, we look into solutions for the equations pz2 1 − z2 2 = 32k and z2 1 − z2 3 = −p · 32k in Z3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Fixing z1 = 1, one can notice that z2 = z3 = 1 are two simple solutions and hence can be lifted to solution in Z3 for the equations mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Diving by 32k, will then gives rise to solutions in Q3 for HERON TRIANGLE AND SHAFAREVICH-TATE GROUP 9 equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' For p ≡ 2 (mod 3), fixing z2 = 1 and z3 = 0 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3) respectively will give z1 = 1 as a simple solution modulo 3 and hence can be lifted to Z3 via Hensel’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' For the case l = 2, just as in the beginning of the case l = 3, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1, we find solutions in Z2 for the equations pz2 1 − z2 2 = 22k and z2 1 − z2 3 = −p · 22k such that k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Fixing z2 = z3 = 1 gives rise to z1 ≡ 1 (mod 8) as a solution modulo 8 to both those equations that can be lifted to a solution in Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' This ensures a solution for the homogeneous space corresponding to (p, 1) in Q2 also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' For the case (b1, b2) = (1, q), the proof follows a very similar way as in the case (b1, b2) = (p, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' For l ≥ 5, l ̸= p, the homogeneous spaces C corresponding to (1, q) given by equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3) are of genus 1 and have solutions in Fl by Hasse-Weil bound and can be lifted to Zl via a similar argument used in the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' For l = p, because 2q ≡ 1 (mod p), one can notice that fixing z1 = 1 and then choosing z2 = 1 and z3 = a is a solution that can be lifted to Zl via Hensel’s lemma where a2 ≡ 2 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' For the case l = 3, fixing z1 = 1 and choosing z2 = z3 = 1 gives rise to solutions for the equations z2 1 − qz2 2 = 32k and z2 1 − qz2 2 = −p2 · 32k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' This solution can be lifted to Zl and then give solution for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='3) in Ql as mentioned in the previous part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' For the case l = 2, noticing the fact p ≡ 1 (mod 8) =⇒ q ≡ 1 (mod 8), the proof follows the same way with the same choice of solutions modulo 8 for the case (b1, b2) = (p, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Mordell-Weil rank and X(Ep/Q)[2] We now conclude with the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1 using the already established results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' A table is provided below in support of our result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Computations for the table has been done in Magma [12] software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' From Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1, it is now obvious that the rank of Ep : y2 = x(x − 1)(x + p2) is at most one due to the homogeneous space corresponding to (p, q) which we are yet to rule out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' But, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1 guarantees that both (p, 1), (1, q) ∈ X(Ep/Q)[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' This, along with the group structure of X(Ep/Q) implies that (p, q) = (p, 1) × (1, q) ∈ X(Ep/Q)[2] too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Consequently, r(Ep) is exactly zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Moreover, due to the Cassels-Tate pairing (see [5]), dimF2(X(Ep/Q))[2] is even which implies X(Ep/Q)[2] ∼= Z/2Z×Z/2Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' This concludes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' □ HERON TRIANGLE AND SHAFAREVICH-TATE GROUP 10 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Examples p Rank of Ep 2- Selmer Rank of Ep X(Ep/Q)[2] 409 0 2 (Z/2Z)2 449 0 2 (Z/2Z)2 521 0 2 (Z/2Z)2 569 0 2 (Z/2Z)2 641 0 2 (Z/2Z)2 Acknowledgement The first author would like to acknowledge DST-SERB for providing the grant through the Start-Up Research Grant (SRG/2020/001937) as well as BITS-Pilani, Hyderabad Campus, for providing amenities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' The second author would like to acknowledge the fellowship (File No:09/1026(0029)/2019-EMR-I) and amenities provided by the Council of Scientific and Industrial Research, India (CSIR) and BITS-Pilani, Hyderabad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' References [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Heath-Brown, "The size of Selmer groups for the congruent number problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='" Inventiones Mathematicae 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='1 (1993): 171-195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Heath-Brown, "The size of Selmer groups for the congruent number problem, II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='" Inventiones Mathematicae 118 (1994): 331-370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Francesc, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Wan, "Iwasawa Main Conjecture for Heegner Points: Supersingular Case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='" arXiv preprint arXiv:1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='02538 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' [4] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Chakraborty, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Ghale, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Saikia, "Construction of an infinite family of elliptic curves of 2-selmer rank 1 from heron triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='" Research in Number Theory (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' [5] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Chao, "2-Selmer groups, 2-class groups and rational points on elliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='" Transactions of the American Mathematical Society 371, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' 7 (2019): 4631-4653.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Dujella, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Peral, "Elliptic curves coming from Heron triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='" The Rocky Mountain Journal of Mathematics 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='4 (2014): 1145-1160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' [7] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Ghale, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Das, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Chakraborty, "A Heron triangle and a Diophantine equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='" Periodica Mathematica Hungarica (2022): 1-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' [8] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Goins, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Maddox, "Heron triangles via elliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='" The Rocky Mountain Journal of Mathematics (2006): 1511-1526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' [9] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Halbeisen, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Norbert, "Heron triangles and their elliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='" Journal of Number Theory 213 (2020): 232-253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' [10] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Zev, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Mazur, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Rubin, "Disparity in Selmer ranks of quadratic twists of elliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='" Annals of Mathematics (2013): 287-320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' [11] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Zev, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Mazur, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Rubin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' "A Markov model for Selmer ranks in families of twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='" Compositio Mathematica 150, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' 7 (2014): 1077-1106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' HERON TRIANGLE AND SHAFAREVICH-TATE GROUP 11 [12] The MAGMA algebra system, available at http://magma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='usyd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='au/calc/ [13] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='Mazur, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Rubin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' "Ranks of twists of elliptic curves and Hilbert’s tenth problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='" Inven- tiones Mathematicae 181, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' 3 (2010): 541-575.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Neukirch, “Algebraic number theory".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' 322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Springer Science and Business Media, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Silverman, "The arithmetic of elliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='" Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' New York: Springer, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Christopher, "A converse to a theorem of Gross, Zagier, and Kolyvagin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='" Annals of Mathematics 191, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' 2 (2020): 329-354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' [17] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Xin, "Heegner point Kolyvagin system and Iwasawa main conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='" Acta Mathematica Sinica, English Series 37, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' 1 (2021): 104-120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' [18] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Washington, "Elliptic curves: number theory and cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='" Chapman and Hall/CRC, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' [19] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Zhang, "Selmer groups and the indivisibility of Heegner points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='" Cambridge Journal of Math- ematics 2, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' 2 (2014): 191-253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content=' Department of Mathematics, BITS-Pilani, Hyderabad campus, Hyderabad, INDIA Email address: debopam@hyderabad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='bits-pilani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='in Department of Mathematics, BITS-Pilani, Hyderabad campus, Hyderabad, INDIA Email address: p20180465@hyderabad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='bits-pilani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} +page_content='in' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQf3QUB/content/2301.03486v1.pdf'} diff --git a/t9E3T4oBgHgl3EQfkAq7/content/tmp_files/2301.04594v1.pdf.txt b/t9E3T4oBgHgl3EQfkAq7/content/tmp_files/2301.04594v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4113cfb12239c5995d5ab9583a7d500c20abe337 --- /dev/null +++ b/t9E3T4oBgHgl3EQfkAq7/content/tmp_files/2301.04594v1.pdf.txt @@ -0,0 +1,1414 @@ +Tunable Three-Dimensional Architecture of Nematic Disclination Lines +Alvin Modin1,∗ Biswarup Ash2,∗ Kelsey Ishimoto1, Robert L. Leheny1, Francesca Serra3, and Hillel Aharoni2 +1Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA +2Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel and +3Department of Physics, Chemistry, and Pharmacy, +University of Southern Denmark, Odense, DK-5230 Denmark +Disclinations lines play a key role in many physical processes, from the fracture of materials to the +formation of the early universe. Achieving versatile control over disclinations is key to developing +novel electro-optical devices, programmable origami, directed colloidal assembly, and controlling +active matter. Here, we introduce a theoretical framework to tailor three-dimensional disclination +architecture in nematic liquid crystals experimentally. +We produce quantitative predictions for +the connectivity and shape of disclination lines found in nematics confined between two thinly +spaced glass substrates with strong planar anchoring. +By drawing an analogy between nematic +liquid crystals and magnetostatics, we find that: i) disclination lines connect defects with the same +topological charge on opposite surfaces, and ii) disclination lines are attracted to regions of the +highest twist. Using polarized light to pattern the in-plane alignment of liquid crystal molecules, +we test these predictions experimentally and identify critical parameters that tune the disclination +lines’ curvature. We verify our predictions with computer simulations and find non-dimensional +parameters enabling us to match experiments and simulations at different length scales. Our work +provides a powerful method to understand and practically control defect lines in nematic liquid +crystals. +Topological singularities link physically distinct phe- +nomena – they mediate phase transitions [1], act as orga- +nizational centers in biological systems [2], and steer the +trajectory of light [3, 4]. Various topological defect con- +figurations are present in nematic liquid crystals (LCs), +fluid-like materials with long-range orientational molecu- +lar order. Disclination lines arise when nematic LCs are +frustrated by incompatible boundary conditions. These +one-dimensional singularities can be facilely formed and +visualized, making nematic LCs an ideal test bed for +studying defect structures and interactions. Manipulat- +ing disclination lines in LCs also has practical applica- +tions in directed self-assembly, [5, 6], tunable photonics +[7], and re-configurable microfluidic devices [8, 9]. +To +effectively utilize the potential of disclinations for these +applications, it is essential to develop a set of fundamen- +tal rules that govern their formation and connectivity. +Recent advances in spatial patterning of liquid crystal +alignment have enabled greater control over the struc- +ture of disclination lines. For example, imprinted nano- +ridges on glass substrates have been used to precisely +shape defect lines, revealing insights into their energy, +structure, and multi-stability [10, 11]. +Using light to +impose LC alignment at photosensitive substrates is an +equally powerful tool. Photo-alignment has enabled the +design of free-standing disclination loops[12–14] and pe- +riodic disclination arrays with different morphologies and +properties [15–20]. +In this work, we introduce a general framework for cre- +ating arbitrarily shaped three-dimensional (3D) disclina- +tion line architecture in nematic liquid crystals. As an +example, we show a structure where the projection of +∗ Authors contributed equally. +disclination lines on a two-dimensional (2D) plane forms +the shape of a heart (Fig. 1). In the experiment, light- +sensitive layers on parallel glass substrates align the ne- +matic at the surfaces in patterns decorated with pairs +of 2D surface defect nucleation sites (Fig.1A). The 2D +defects are characterized by winding numbers - the de- +gree of rotation of the nematic director around the defect +divided by 2π - of +1/2 and −1/2. Aligning opposite- +charged defects on opposing substrates, we observe that +the confined LC forms a pair of disclination lines that pri- +marily run through the mid-plane of the cell to connect +surface defects on the same substrate (Fig. 1B). This +configuration is a stable, equilibrium state, confirmed by +numerical simulations (Fig.1C-D). +To understand the paths that the disclination lines +take, we draw an analogy between the elastic distortion of +a nematic and the magnetostatic field of current-carrying +wires. Using this analogy, we experimentally and numeri- +cally verify two key rules: (i) disclination lines either con- +nect surface defects on opposing substrates with the same +winding number or surface defects on the same substrate +having opposite winding numbers; (ii) the lines’ paths de- +pend on the interplay between forces driving them to re- +gions of maximum twist set by the confining pattern and +the disclination line tension. We utilize these two design +principles to create the heart-shaped disclinations shown +in Fig. 1. Our proposed framework enables the design of +tunable 3D liquid crystal-based disclination networks for +applications in re-configurable optics, photonic devices, +and responsive matter. +The following sections explain the magnetostatic anal- +ogy and its outcomes. Subsequently, we test specific pre- +dictions with experiments and simulations. Finally, we +revisit the structure in Fig. 1 to explain how we designed +the heart-shaped lines and how disclination shapes can +arXiv:2301.04594v1 [cond-mat.soft] 11 Jan 2023 + +2 +FIG. 1. +Designing three-dimensional disclination line +architecture. (A) Schematics of the two-dimensional sur- +face patterns imprinted onto substrates coated with a photo- +sensitive layer (Brilliant Yellow). +Topological defects with +charge +1/2 (red semi-circle) and −1/2 (orange triangle) act +as nucleation sites for disclination lines. The substrates are +aligned so that defects of opposing strength are in registry +with each other. (B) Disclination lines viewed under bright- +field microscopy form a heart-shaped structure (scale bar: +25 µm). This configuration corresponds to a stable equilib- +rium state. Top view (C) and side view (D) of the discli- +nation structure obtained from simulation using the patterns +in (A) as surface boundary conditions. +The observed two- +dimensional projection (black points) is a balance between the +attraction of defect lines to locations where the top and bot- +tom surface patterns are perpendicular (orange lines in (C)) +and their line tension. (D) A side view of the defect configu- +ration obtained in the simulation reveals that the disclination +lines primarily run through the mid-plane of the cell and con- +nect surface defects of opposite charge on the same substrate. +be tailored by varying temperature. +RESULTS AND DISCUSSION +Magnetostatics model +Distortions of the nematic director field are described +by the Frank-Oseen elastic free energy [21], +Fel = +� �K1 +2 (∇ · ˆn)2 + K2 +2 (ˆn · ∇ × ˆn)2 + +K3 +2 (ˆn × ∇ × ˆn)2 +� +dV, +(1) +with ˆn the nematic director and K1,2,3 the splay, twist, +and bend elastic constants, respectively. We consider a +nematic placed between two parallel plates with (suffi- +ciently strong) planar anchoring on them, separated by +a spacing t much smaller than their lateral dimensions. +Under these conditions, we make the following key as- +sumption: in equilibrium, the nematic director is planar +everywhere within the cell, not only at the boundaries. +This is analogous to the Kirchhoff-Love assumptions in +plate elasticity theory [22]. The nematic director field +then takes the form ˆn = (cos θ, sin θ, 0), where θ(x, y, z) +is the director’s azimuthal angle in the xy-plane. +In +addition, we use the two-constant approximation, with +K1 = K3 ≡ K, which is valid near the nematic-isotropic +transition for low molecular weight thermotropic nematic +LCs, particularly 4’-octyl-4-biphenylcarbonitrile (8CB) +[23] used in our experiments. Eq. 1 then assumes the +simple form, +Fel = +� �K +2 +� +(∂xθ)2 + (∂yθ)2� ++ K2 +2 (∂zθ)2 +� +dV. +(2) +Further simplification is obtained by rescaling the z- +axis using ˜z = z +� +K/K2 (defined on a domain of thick- +ness ˜t = t +� +K/K2), and redefining ∇ ≡ (∂x, ∂y, ∂˜z), so +that +Fel = K +2 +� +|∇θ|2 d ˜V . +(3) +The functional in Eq. 3 implies that, in equilibrium, +θ(x, y, ˜z) is a harmonic function. +However, this prop- +erty breaks down along disclination lines; at the defect +core, the nematic order vanishes, and θ is not defined. +Around the defect line, Eq. 3 admits a nontrivial quan- +tized integral, +� +dℓ · ∇θ = 2πq, +q ∈ Z/2. +(4) +Together Eq. 3 and 4 establish an exact mathematical +analogy of the nematic cell to magnetostatics, as previ- +ously identified by de Gennes [21]. In the analogy, the +planar director’s azimuthal angle θ plays the role of a +magnetic scalar potential, whose gradient is the magnetic +field. Disclination lines are current-carrying wires. Their +existence renders θ ambiguous; however, the half-integer +quantization of the current exactly corresponds to the +nematic θ ∼= θ + πZ congruence. +The disclination wires are flexible and stretchy. Each +wire is associated with a line tension γ, the outcome of +melting of the nematic order at the defect core to al- +leviate the diverging elastic energy. +Approximately, γ + +A +B +D3 +is proportional to Kq2; however, there are logarithmic +corrections that depend on the line and cell geometry +[21, 24]. These corrections become significant near the +nematic-isotropic phase transition as the defect core size +diverges. For simplicity, we ignore these corrections and +treat γ as a constant. Similarly to other material param- +eters, namely K, K2, γ may depend on temperature in a +non-trivial way. +Forces on wires +To study the shape of disclination wires, we calculate +the effective forces acting on them (see SI Appendix for +full derivations). There are three forces (per unit length) +acting on the wires: +1. The strong anchoring on the two surfaces acts as +magnetic mirrors. Disclination wires are repelled +by these mirrors (alternatively, by the mirror image +wires) and pushed toward the mid-plane between +the two boundary surfaces by a force +fM = −π2Kq2 +˜t +tan +�π˜z +˜t +� +ˆz. +(5) +2. The anchoring planar angles, θt,b(x, y) on the +top/bottom surfaces, respectively, are analogous to +an external magnetic field that exerts a Lorentz-like +force on disclination wires: +fB = 2πKq +˜t +(θt − θb − qπ) ˆT × ˆz, +(6) +where ˆT is the unit tangent to the defect line. This +force pulls defect lines horizontally towards regions +where the top and bottom are at a ∆θ ≡ θt − θb = +qπ angle difference from each other. +3. The line tension of the wires exerts a force +fγ = γκ ˆN, +(7) +where κ is the curvature of the wire and ˆN its nor- +mal in the Frenet-Serret frame. +The equilibrium shape of a disclination line is obtained +by the balance of fM, fB, and fγ. +This force balance can characterize the geometry of +defect lines connecting to surface defects. A disclination +line emerges perpendicularly from a surface topological +defect due to the magnetic repulsion described by Eq. 5. +For a surface defect |q| > 1 +2 we expect a split into 2|q| +“atomic” lines of magnitude 1 +2 arising from the mutual +repulsion between them (as was observed in [10, 16]). By +the balance of forces, fM and fγ lines emerging from de- +fects turn horizontally into the mid-plane over a typical +length scale ∼ +γ +K ˜t ≡ +γ +√KK2 t. Thus, defect lines whose +lateral span is much larger than this scale traverse within +the mid-plane for the more significant part of their tra- +jectories. +Connectivity of surface defects +We study the topological rules of connecting – with +disclination wires – surface defects patterned on two +confining surfaces. Each surface defect is characterized +by a winding number q, defined by a closed loop Γ +around the defect core as +1 +2π +� +Γ ∇θ · dℓ. +By current +conservation, a disclination line can connect two surface +defects of the same q on opposite surfaces; or two surface +defects of opposite q on the same surface. Alternatively, +disclinations can escape to the sides of the system +or form a closed loop. +Planarity of the director field +forbids connection between a top-surface +1/2 defect +with a bottom-surface −1/2, even though this would be +topologically allowed in a 3D nematic liquid crystal [25]. +Experimental Tests +Test of Design Principles +To verify these connectivity principles experimentally, +we create a LC cell where the bottom and top surfaces +contain a single, isolated q = −1/2 or +1/2 defect, re- +spectively (Fig. 2 A-C), utilizing the custom built photo- +alignment system described in Materials and Methods +and SI Appendix, Fig. +S1 [26–28]. +We shine linearly +polarized light on glass coated with a light-sensitive +alignment layer (Brilliant Yellow). The alignment layer +molecules give planar alignment to the LC, with a di- +rection that is perpendicular to the polarization of the +incident light. By spatially patterning the light polariza- +tion, we imprint half-integer defect nucleation sites onto +confining glass substrates. The defects on each surface +are photo-patterned within a circular patch of diameter +d ≈ 75µm. Under crossed-polarizers, the dichroic prop- +erties of the Brilliant Yellow dye enable us to view the +patterned regions on the confining glass substrates be- +fore filling them with LC. We align the circular patches +on each substrate to overlap, ensuring that defect cores +of opposing topological charges are in registry. Once sub- +strates are secured with epoxy resin, we carefully measure +the cell thickness and inject pre-heated 8CB LC into the +cell, allowing it to slowly cool until it reaches the nematic +phase at ∼ 36◦C. +The resulting defect structure follows the connectivity +rules: rather than a single disclination line connecting +the surface defects as might be expected [25], two discli- +nation lines emerge from the defect cores and escape to +the sides along the mid-plane, as can be seen from a side +view of the numerical simulation in Fig. 2A and from +the top view in experiments (Fig. 2B) and simulations +(Fig. 2C). Indeed, this connectivity rule gives rise to the +two extended lines that make the heart shape in Fig. 1 +rather than two defect lines connecting surface defects di- +rectly facing each other. For verification, we run the same +experiment and simulation with +1/2 surface defects pat- + +4 +FIG. 2. Surface defect connectivity. (A) When two canonical surface defects with ±1/2 charge are imposed as boundary +conditions, the planarity of the director field forbids the connection of the top and bottom surface defects with a single defect +line. The equilibrium state obtained from numerical minimization of Landau de Gennes energy corresponds to two disclination +lines (black points) nucleating at the surface defect cores and extending towards the boundaries.(B) Bright-field microscopy +images where two confining surfaces are photo-aligned with isolated ±1/2 surface defects corresponding to the pattern used +in (A) . Inset: polarized light microscopy image.(C) Two-dimensional projection of the numerical results in (A). Inset: The +expected polarized optical microscopy (POM) texture reconstructed from the director field using Jones calculus. (D) Preserving +the topological charge of the surface defects while altering their geometric structure changes the regions where ∆θ = π/2. The +equilibrium state for these boundary conditions corresponds to two disclination lines now at a relative angle of π/2. +(E) +Bright-field microscopy images where two confining surfaces are photo-aligned with isolated ±1/2 surface defects corresponding +to the pattern used in (D). (F) Two-dimensional projections of the numerical results in (D). Inset: The expected POM texture +reconstructed from the director field using Jones calculus. Scale bars: 25µm. +terned onto confining substrates (see SI Appendix, Fig. +2). A vertical disclination line connects the top and the +bottom surface defects, as is permitted in this case by +current conservation. +The effect of varying the patterned boundary condi- +tions can be seen in Fig. 2(D-F). Here, we preserve the +topological charge of each surface pattern but introduce +a homogeneously aligned region that alters the geometric +structure of the ±1/2 defects. The new surface pattern +modifies the areas where ∆θ (x, y) are orthogonal. The +new regions where fB = 0 lead to a reduced angular sep- +aration of the two defect lines from π (Fig. 2B-C) to π/2 +(Fig. 2E-F). For both designs, the disclination wires do +not bend and κ ≈ 0, implying that fγ has little to no +effect on the positioning of the lines. +Tuning the curvature of disclination architecture +When the imposed surface patterns result in curved +disclination lines, the line tension fγ becomes important. +fγ opposes fB, acting to minimize the wire’s curvature. +The competition between these two forces causes the tra- +jectory of the two disclination lines in Fig. 1D to devi- +ate from regions where ∆θ (x, y) = π/2. The shape of +the disclination lines can then be tuned by changing the +magnitude of fB and fγ, which vary differently with tem- +perature due to the temperature-sensitive behavior of γ +and the elastic constants K, K2. As described below, by +tuning the disclination shape, we measure γ/√KK2 at +various temperatures, enabling us to map our experimen- +tal observations to simulations. +To illustrate the tuning of disclination shape, we con- +struct LC cells whose confining surfaces are each photo- +patterned with a single +1/2 defect. The substrates are +rotated so that the defects are oriented with respect to +one another by an angle θ0 and are translated so that a +horizontal distance ℓ separates the defect cores (Fig. 3A, +B). In this design, the 2D projection of the patterns con- +tains a locus of points where ∆θ = π/2 forms a circular +arc segment with an opening angle 2θ0 connecting the +defects. +Once a cell is filled with 8CB, a disclination +line forms to connect the two defect cores (Fig. 3C).. In +general, the line does not follow the arc with opening + +B +A +C +m =+1/2 +m = -1/2 +D +m = +1/2 +m =-1/25 +FIG. 3. +Measuring the line tension of a disclination. +(A) Diagrams of the geometry used to create arced discli- +nation lines. Initially, +1/2 surface defects are imposed as +anchoring conditions in a nematic cell. +The substrates are +then rotated and displaced with respect to one another. (B)) +When viewed from the top, the defects cores are separated +by a distance ℓ and their lines of symmetry form an angle +θ0. The two-dimensional projection of the resulting disclina- +tion line is a portion of a circular arc with a central angle +α. (C)) Bright-field microscopy overlapped images of discli- +nation lines at two different temperatures. The disclination +line’s curvature depends on the balance of two opposing forces +fγ and fB, whose magnitudes depend on temperature (scale +bar: 10µm, dashed lines are guides for the eye). (D)) Varying +the temperature of the nematic, therefore, results in a change +in the defect line curvature and in the angle α as shown in +the curves obtained for different values of θ0. (E)) When the +measured values of t, ℓ, and θ0 are all accounted for, the im- +plied values of γ/√KK2 all collapse onto the same curve as +a function of temperature. +angle θ0 due to fγ. However, along any circular disclina- +tion arc that passes between the two surface defect cores, +both Eq. 6 and 7 are uniform. Thus, in equilibrium, the +disclination still forms an arc, and finding its curvature +through force balancing is a simple algebraic problem: +0 = (fB + fγ) · ˆN = πK +˜t +�α +2 − θ0 +� ++ γ 2 +ℓ sin α +2 , +(8) +where α is the opening angle of the arc. Rewriting Eq. 8 +in a dimensionless form, we obtain the following tran- +scendental equation: +α +2 + ˜γ sin α +2 = θ0, +(9) +where ˜γ = 2 +π +γ +√KK2 +t +ℓ. As expected, in the limit of van- +ishing line tension, α tends to 2θ0, where fB vanishes. +In the limit of infinite line tension, α tends to zero so +that fγ vanishes. Line tension’s relative importance in +determining the defect line’s contour is described by the +dimensionless parameter ˜γ. +Equation 9 captures the effect of line tension in reduc- +ing the curvature of an arced disclination line. Rearrang- +ing it again, we find that, +γ +√KK2 += +π +2κt +� +θ0 − arcsin +�ℓκ +2 +�� +. +(10) +The +equation +above +links +the +material +parameter +γ/√KK2 to the deviation of the disclination arc’s curva- +ture κ from its zero-line tension limit. Thus, the temper- +ature dependence of γ/√KK2 can be measured directly +in 8CB from the temperature dependence of the line cur- +vature. We track the variation of α = 2 arcsin (ℓκ/2) as +a function of temperature across θ0 ranging from 30◦ to +105◦ (see Materials and Methods for details of the image +analysis). When a disclination line is formed by an ini- +tial θ0 = 105◦, the curvature deep in the nematic phase +(T = 35◦C) is small (Fig. 3C). Increasing the tempera- +ture towards the nematic-isotropic transition, we observe +an increase in κ (and hence α) since fγ decreases more +rapidly than fB on heating (Fig. 3D). Equation 10 is con- +firmed by the collapse in Fig. 3E of measurements held +at different values of t, ℓ and θ0 onto the same curve +that only depends on material properties of the LC. This +affirms the validity of approximating γ with a constant. +Fig. 3E shows the monotonic temperature dependence +of γ/√KK2 in a nematic 8CB, ranging approximately +between 12 and 18. We follow the protocol of Fig. 3 to +also estimate γ/√KK2 in our numerical simulations (see +SI Appendix for details); we analyze arced defect config- +urations for different values of t, ℓ and θ0, and extract α +from which we obtain a mean γ/√KK2 = 3.3±0.1. This +value is not within the experimental range. However, for +every experiment, we can now match a simulation held +at the same value of ˜γ = 2 +π +γ +√KK2 +t +ℓ, by compensating for +the different values of +γ +√KK2 with inversely different val- +ues of the aspect ratio t +ℓ. In simulations, we tweak ˜γ not +with temperature but with aspect ratio. +We now revisit Fig. 1 and the heart-shaped disclina- +tion lines. These are generated using patterns described +in detail in the Materials and Methods. We control the +cusps of the heart by the directions of maximum twist +around the defects as in Fig. 2. When 8CB is cooled by +≈ 6◦C from the nematic-isotropic transition, the increase +in ˜γ constricts the lobes of the heart-shaped disclination +lines (Fig. 4A). We know the value of ˜γ at each tempera- +ture from the thickness of the cell, the lateral separation +between the two surface defects, and Fig. 3E. Simulations +with the same values of ˜γ, obtained by changing the val- +ues of t and ℓ, qualitatively capture a similar change in +the structure of the disclination architecture Fig. 4B. It +is remarkable that despite the experimental uncertainty +and the use of different system sizes in the experiment +and simulation, the resulting defect configurations for the +same values of ˜γ are in good agreement. + +A +D +3T +4 +π-2 +T +4 +E +B +18 +α +K16 +C +T=40°C +14 +29 +60 ° +96° +105 ° +12 +0 +6 +T=35°C +oC6 +FIG. 4. +Tunable disclination line architecture (A) +Bright-field microscopy image of heart-shaped disclination +lines measured for two different temperatures, correspond- +ing to different values of ˜γ (scale bar: 25 µm, dashed lines +are guides for the eye). (B) Defect configurations obtained +in the simulation for two different sets of parameters l, t, cho- +sen such that the values of ˜γ are the same in simulation and +experiment. The change in the structure of the disclination +architecture in both experiment and simulation is captured +by ˜γ. +CONCLUSION +This work introduces a novel framework for creating +and tuning 3D disclination lines in a nematic liquid crys- +tal. +When disclination lines are nucleated by surface +defects, their connectivity and trajectories are analo- +gous to current-carrying wires near a current-free sur- +face. +Whether or not surface defects may connect to +each other can be explained by treating the topologically +charged disclination lines as wires that must conserve +current. +Similarly, substrates imprinted with surface- +anchoring conditions exert a Lorentz-like force on the +wires, pushing them towards regions where the anchoring +conditions on opposing substrates are orthogonal. When +the patterns promote wires to curve, they experience an +additional force from line tension that decreases the cur- +vature. This force can be tuned in both experiments and +simulations by changing a dimensionless parameter, ˜γ. +We verified these connectivity principles through a se- +ries of experiments. +By appropriately designing sur- +face anchoring conditions, we created a three-dimensional +structure whose two-dimensional projection resembles a +heart. We tuned its shape by varying the temperature +and recreated the results using numerical simulations. +Our design principles can be used to interpret similar +results observed in recent experiments with disclination +lines created by patterned surfaces [13–15, 17, 29]. These +principles can further be used to construct more complex +disclination architecture, advancing the design of tunable +3D liquid crystal-based disclination networks for applica- +tions of molecular self-assembly, re-configurable optics, +photonic devices, and responsive matter. Furthermore, +we have shown that the equilibrium shape of disclination +lines depends on temperature and aspect ratio, opening +the door for multi-state systems, switchable by varying +the temperature or thickness of the cell. +MATERIALS AND METHODS +Substrate preparation +Photosensitive material Brilliant Yellow (BY, Sigma- +Aldrich) was mixed with n,n-dimethylformamide (DMF) +solvent at 1 wt.% concentration. Glass substrates (Fisher +Scientific) were washed in an ultrasonic bath with Hell- +manex liquid detergent (Fisherbrand), followed by suc- +cessive washes in acetone, ethanol, and isopropyl alco- +hol, and then dried with N2 gas. The BY-DMF solution +was spin-coated on the substrates at 3000 RPM for 45 +seconds. After spin-coating, the substrates were baked +at 95◦C for 15 minutes. Spin-coating and baking pro- +cesses were conducted at a relative humidity of 35% or +lower[30]. +Patterned Surface Alignment +Surface patterns were created using a custom-built +photo-patterning setup consisting of a polarized LED +source [28] feeding into the side port of a bright-field +inverted microscope body (TI Eclipse TE2000). +Segmented images were generated via a LED-based +projector (Sony MPL-C1A) to a peripheral optical path +(SI Appendix Fig. +S1). +The projector operates using +three time-modulated laser diodes. +To match the ab- +sorption band of the BY-DMF solution, we use the blue +(λ = 445nm) diode. Images generated by the projector +first pass through two aspheric condenser lenses (with fo- +cal lengths f = 32mm, Thorlabs, ACL50832U) before be- +ing expanded with a custom-Keplerian telescope consist- +ing of two convex lenses f = 100mm (Thorlabs, AC508- +100-A-ML) and f = 200mm (Thorlabs, AC508-200-A- +ML), respectively. The expanded image passes through +a linear polarizer before entering the microscope body. +Once inside the body, the image is reflected by a dichroic +mirror, picked up by an infinity-corrected tube lens, and +collected by a microscope objective (20x, Nikon S Plan +Fluor ELWD) that focuses the image onto a BY-coated +substrate. Upon irradiation with linearly polarized light, +the photosensitive azo-dye molecules orient perpendicu- +larly to the plane of polarization, setting the preferred +alignment direction of the nematic director ˆn. +Designed patterns were discretized into pie segments +of fixed polarization with opening angle π/16 and with +the cores of defects located at the center. +Sample preparation +After photo alignment, patterned regions on substrates +were aligned and fixed using epoxy glue (Loctite) to cre- +ate a liquid crystal cell. After cell assembly, we use spec- +troscopic reflectometry to measure the cells’ thickness t, +obtained from the absolute reflectance spectra (Ocean- +view) fit using custom Matlab code. +Cells are subse- + +A +B +Y~0.39 +Y~0.457 +quently filled with 4’-n-octyl-4-cyano-biphenyl (8CB, Ne- +matel GmbH) liquid crystal, pre-heated into the isotropic +phase by capillary flow. After cells are filled, they are +sealed on their ends using UV curable resin (Loon Out- +doors UV Clear Fly Finish). +Polarized optical microscopy +We use a Nikon LV 100N Pol upright microscope to im- +age patterned regions with both 20x and 50x objectives. +Samples are placed on a heating stage (Instec HCS302) +set to 36◦C to keep 8CB in the nematic phase. Optical +microscopy images are captured using a Nikon DS-Ri2 +camera. +Analyzing the curvature of disclination line arcs +Videos +of +disclination +lines +are +captured +using +bright-field microscopy and analyzed using ImageJ, +TrackPy[31], and custom Python code. +The contours +of the disclination lines are detected using a Canny edge +detection algorithm, binarized, and fit to circles using +least squares fitting. For each frame i of the video, the +radius of curvature ri ≡ 1/κ and center of the best-fit +circle +� +xC +i , yC +i +� +are determined. These circles intersect at +the defect cores, corresponding to two unique points. To +find the positions of these points (x, y), we minimize a +cost function Ξ =� +i +� +(x−xc +i )2+(y−yc +i )2) +r2 +i +− 1 +�2 +, where the +sum is over all the frames in the video. The uncertainty +of each defect core’s position is the cost function’s value, +and the distance ℓ between the defect cores is calculated +using the Euclidean distance. +Jones matrix calculations +For qualitative comparison of numerical and experi- +mental director configurations near nucleated disclina- +tion lines, we use Jones calculus to reconstruct the po- +larized optical microscopy (POM) texture of the direc- +tor field obtained from minimization of the Landau de- +Gennes free-energy. +The volume of the numerically- +obtained director field is discretized into volume elements +(voxels) on a 3D grid, with each point at position ρ con- +taining N voxels each of thickness ∆. It is assumed that +variation in n between successive voxels is small com- +pared to the wavelength of incident light λ, so that λ ≪ +1/|∇ˆn|. Each voxel ν is treated as a uniaxial birefrin- +gent optical element, represented by a 2×2 Jones matrix +Mν that depends on both the extraordinary ne and ordi- +nary no indices of refraction of the LC. Light propagating +through a voxel experiences an ne dependant on the polar +angle θν between n and the light’s propagation direction +k0 given by ne(θν) = none/ +� +(no cos θν)2 + (ne sin θν)2. +We choose k0 = ˆz, so that the plane of polarization is the +x−z plane, and write the corresponding Jones matrix as +Mν (ρ) ≡ +� +eine +ν(θν)2π ∆ +λ +0 +0 +eino +ν2π ∆ +λ +� +. +(11) +Using +the +8CB’s +ne +and +no +at +the +exper- +imental +temperature +and +wavelength +λ, +we +compute +Mν (ρ), +constructing +a +single +opera- +tor +γ (ρ) += +�N +ν=1 R (−φν) Mν (ρ) R (φν), +where +R (φν) ≡ +� +cos φν +sin φν +− sin φν cos φν +� +and φν is the azimuthal +component of ˆn in voxel ν. Following [32] and [33], we +construct 2 × 2 Jones matrices for the polarizer P and +analyzer A. Sequential propagation of plane waves E0 +through P,γ (ρ) and A results in a single Jones vector +ET (ρ) = Aν (ρ) PE0. The calculated POM texture is +obtained from the intensity of light transmitted through +each voxel, IT (ρ) = |ET (ρ) |2. +Numerical simulations +The numerical modeling of the nematic liquid crys- +tal is achieved using the lattice-discretized Landau-de +Gennes model implemented in open-Qmin[34]. The con- +figuration of a nematic liquid crystal is represented by +specifying the components of the Q-tensor[34, 35] which +is related to the director ˆn of a uniaxial nematic by +Qij = +3 +2S +� +ninj − 1 +3δij +� +, where i, j ∈ {x, y, z} and S +is the degree of uniaxial nematic order. To simulate a +thin nematic cell, we consider a three-dimensional box +of size L × L × Lz with L ≫ Lz. In the simulation, we +use L = 250 and Lz between 12 and 21, expressed in +units of the number of lattice sites. Note that the thick- +ness t = Lz − 1, since anchoring is imposed on top and +bottom layers. At every lattice point, we start with a +random initial condition for Qij. We impose strong pla- +nar anchoring at the top and bottom surfaces by setting +the anchoring strength W = 50 for the two surfaces. We +use free boundary conditions on the side surfaces of the +simulation box by setting W = 0. We use the Fast In- +ertial Relaxation Engine (FIRE) algorithm within open- +Qmin[34] to minimize the total free energy until the norm +of the residual force vector goes below 10−8 (see SI Ap- +pendix for details). In the energy-minimized configura- +tion, defects are identified locally as lattice sites where +the largest eigenvalue of Q falls below some threshold, +typically 0.95S. +Surface patterns used in experiment and simulation +In the experiment and simulation, we impose a planar +director field, i.e., the nematic director field takes the +form ˆn = (cos θ, sin θ, 0) at the top and bottom surfaces. +In Fig.1 and 4, the surface pattern at the top surface is + +8 +represented by +θt (x, y, z) = 1 +2 +� +tan−1 y − 1 +x +− tan−1 y + 1 +x ++ +x +� +x2 + (y + 1)2 − +x +� +x2 + (y − 1)2 +� +, +(12) +while θb (x, y, z) = −θt (x, y, z). +For the ±1/2 surface defect patterns used in Fig.(2-3) we +have θ (x, y, z) = ± 1 +2 tan−1 � y +x +� +. +Acknowledgements +We greatly acknowledge insights, assistance, and help- +ful discussions with Charles Rosenblatt, David Dolgitzer +and Bastian Pradenas. This research was supported by +a grant from the United States-Israel Binational Science +Foundation (BSF) no. 2018380. RLL acknowledges sup- +port from the NSF (DMR-2104747). +[1] J. M. Kosterlitz and D. J. Thouless, Journal of Physics +C: Solid State Physics 6, 1181 (1973). +[2] Y. +Maroudas-Sacks, +L. +Garion, +L. +Shani-Zerbib, +A. Livshits, E. Braun, and K. Keren, Nature Physics 17, +251 (2021). +[3] E. Brasselet, Physical Review Letters 108, 087801 +(2012). +[4] C. Meng, J.-S. Wu, and I. I. Smalyukh, Nature Materials +22, 64 (2023). +[5] C. Blanc, D. Coursault, and E. Lacaze, Liquid Crystals +Reviews 1, 83 (2013). +[6] Y. Luo, D. A. Beller, G. Boniello, F. Serra, and K. J. +Stebe, Nature Communications 9, 3841 (2018). +[7] H. J. Coles and M. N. Pivnenko, Nature 436, 997 (2005). +[8] A. Sengupta, C. Bahr, and S. Herminghaus, Soft Matter +9, 7251 (2013). +[9] J. J. Sandford O’Neill, P. S. Salter, M. J. Booth, S. J. +Elston, and S. M. Morris, Nature Communications 11, +2203 (2020). +[10] B. S. Murray, R. A. Pelcovits, and C. Rosenblatt, Phys- +ical Review E 90, 052501 (2014). +[11] S. Harkai, B. S. Murray, C. Rosenblatt, and S. Kralj, +Physical Review Research 2, 013176 (2020). +[12] M. Wang, Y. Li, and H. Yokoyama, Nature Communica- +tions 8, 388 (2017). +[13] T. Ouchi, K. Imamura, K. Sunami, H. Yoshida, and +M. Ozaki, Physical Review Letters 123, 097801 (2019). +[14] K. Sunami, K. Imamura, T. Ouchi, H. Yoshida, and +M. Ozaki, Physical Review E 97, 020701 (2018). +[15] Y. Guo, M. Jiang, S. Afghah, C. Peng, R. L. B. Selinger, +O. D. Lavrentovich, and Q. Wei, Advanced Optical Ma- +terials 9, 2170063 (2021). +[16] H. Yoshida, K. Asakura, J. Fukuda, and M. Ozaki, Na- +ture Communications 6, 7180 (2015). +[17] I. Nys, B. Berteloot, J. Beeckman, and K. Neyts, Ad- +vanced Optical Materials 10, 2101626 (2022). +[18] I. Nys, K. Chen, J. Beeckman, and K. Neyts, Advanced +Optical Materials 6, 1701163 (2018). +[19] B. Berteloot, I. Nys, G. Poy, J. Beeckman, and K. Neyts, +Soft Matter 16, 4999 (2020). +[20] J. Jiang, K. Ranabhat, X. Wang, H. Rich, R. Zhang, +and C. Peng, Proceedings of the National Academy of +Sciences 119, e2122226119 (2022). +[21] P.-G. d. Gennes and J. Prost, The Physics of Liquid +Crystals, 2nd ed., International series of monographs on +physics No. 83 (Clarendon Press, Oxford, 2013). +[22] E. Love, Philosophical Transactions of the Royal Society +of London. (A.) 179, 491 (1888). +[23] P. Oswald, J. Baudry, and T. Rondepierre, Phys. Rev. E +70, 041702 (2004). +[24] N. Schopohl and T. J. Sluckin, Phys. Rev. Lett. 59, 2582 +(1987). +[25] N. D. Mermin, Reviews of Modern Physics 51, 591 +(1979). +[26] V. G. Chigrinov, V. M. Kozenkov, and H.-S. Kwok, Pho- +toalignment of liquid crystalline materials: physics and +applications, Wiley SID series in display technology (Wi- +ley, Chichester, England ; Hoboken, NJ, 2008) oCLC: +ocn225429570. +[27] Y. Folwill, Z. Zeitouny, J. Lall, and H. Zappe, Liquid +Crystals 48, 862 (2021). +[28] O. M. Wani, P. Wasylczyk, and A. Priimagi, Adv. Opt. +Mater. 6, 1700949 (2017). +[29] C. Long, M. J. Deutsch, J. Angelo, C. Culbreath, +H. Yokoyama, J. V. Selinger, and R. L. B. Selinger, +Frank-read mechanism in nematic liquid crystals (2022). +[30] J. +Wang, +C. +McGinty, +J. +West, +D. +Bryant, +V. +Finnemeyer, +R. +Reich, +S. +Berry, +H. +Clark, +O. Yaroshchuk, and P. Bos, Liquid Crystals 44, 863 +(2017), https://doi.org/10.1080/02678292.2016.1247479. +[31] D. Allan, T. Caswell, N. Keim, and C. van der Wel, +trackpy: Trackpy v0.3.2 (2016). +[32] E. Hecht, Optics, 5th ed. (Pearson Education, Inc, +Boston, 2017). +[33] P. W. Ellis, E. Pairam, and A. Fern´andez-Nieves, Journal +of Physics D: Applied Physics 52, 213001 (2019). +[34] D. M. Sussman and D. A. Beller, Frontiers in Physics 7, +10.3389/fphy.2019.00204 (2019). +[35] N. J. Mottram and C. J. P. Newton, Introduction to q- +tensor theory (2014). +[36] J. D. Jackson, Classical electrodynamics (Third edition. +New York : Wiley, 1999). +[37] M. Ravnik and S. Zumer, Liquid Crystals 36, 1201 +(2009). + +9 +TUNABLE ARCHITECTURE OF NEMATIC DISCLINATION LINES +SUPPLEMENTARY APPENDIX +OPTICAL SETUP +FIG. S1. Optical setup for photoalignment. A projector with blue LED light generates images focused on substrates at +the sample plane. Images are polarized by a linear polarizer, collimated by aspheric lenses, and then expanded by a relay of +lenses convex lenses. A microscope body contains a dichroic mirror that reflects light toward an infinity-connected tube lens. +Images are picked up by the microscope and focused onto the sample plane. +I. +DISCLINATION LINES CONNECTING SURFACE DEFECTS ON OPPOSING SUBSTRATES +FIG. S2. Defect connectivity with identical boundary conditions (a) A single disclination line runs between surface +defects with identical topological charges. In bright-field microscopy, the defect line appears as a single point at various locations +within the LC cell. Inset: Polarized optical microscopy image. On the side, the schematic of the defect line. The image is +taken in the mid-plane of the cell. Scale bar: 10 µm. (b) Simulation results of a straight disclination line between two surfaces +with +1/2 point defects. + +sample +X-V +stage +obiective +tube lens +convex +aspheric +lenses +lenses +dichroic +mirror +LED +linear +polarizerA +B10 +II. +DERIVATION OF THE FORCES ACTING ON A WIRE ELEMENT +The magnetostatic model emerges from the similarity between equations [3,4] in the main text and their (vacuum) +magnetostatics counterparts [36]: +Fmag = +1 +2µ0 +� +|B|2 d ˜V +(S1) +and +� +dℓ · B = µ0I +(S2) +By comparison, the following pairs are analogous: +Nematic +Magnetic +Field +∇θ +B +Modulus +K +1/µ0 +Current +2πKq +I +Force per unit length on wire in field +2πKq dℓ × ∇θ Idℓ × B +Force per unit length between wires at distance d +2πKq1q2/d +µ0 +2π I1I2/d +The problem (namely, Eq. S2 and the Euler-Lagrange equation associated with Eq. S1) is linear. We may therefore +write the boundary conditions as a sum of contributions, solve them separately, and add up the solutions/forces +exerted in each case. +We start with the method of images. We introduce an infinite ladder of mirror wires outside of our cell, indexed by +m ∈ Z \ {0}. We set the wires parallel to the actual one, located at ˜zm = m˜t + (−1)mδ, all carrying the same current +2πKq. We now sum up the forces per unit length exerted on the wire element by its mirror images: +fM = +∞ +� +m=−∞ +m̸=0 +2πKq2 +δ − ˜zm +ˆz = 2πKq2 +∞ +� +m=1 +m odd +� +ˆz +m˜t + 2δ − +ˆz +m˜t − 2δ +� += −π2Kq2 +˜t +tan +�πδ +˜t +� +ˆz. +(S3) +By construction, the above (real and image) wire setup is symmetric about either of the two boundary plates. +Therefore, ∇θ (induced by this setup alone) is perpendicular to these boundaries, namely the director angles on +both boundaries are constant (henceforth denoted θ0 +t,b). By the integral condition in Eq. S2 and by lateral reflection +symmetry, θ0 +t − θ0 +b = qπ. However, in the experimental/numerical setup discussed in the main text, ∆θ = θt − θb is +an arbitrary function of x, y. +To correct this, the solution to Eqs. S1 and S2 must be the sum of the above solution and a harmonic function +(so that the equations are still satisfied) that makes up for the boundary condition mismatch. The (divergence-free) +gradient of this function can be interpreted as an external magnetic field Bext(x, y, z) that acts regardless of the +exact shape of the wire. It can be written explicitly using Green’s functions. However, we further assume that the +thickness of the cell is much smaller than the lateral gradients of θt,b(x, y), therefore almost everywhere Bext ≈ Bˆz +(this assumption may fail close to surface defects). Being divergence-free, B(x, y, z) must be approximately uniform +in z, and matching the boundary conditions we get B(x, y) = ∆θ−∆θ0 +˜t += 1 +˜t (θt − θb − qπ). Thus, the Lorenz force +exerted on the wire by the external field reads +fB = 2πKq +˜t +(θt − θb − qπ) ˆT × ˆz, +(S4) +where ˆT is the unit tangent to the defect line. +Derivation of the force exerted by line tension is rather straightforward. A line segment of length δ is subject to +tangential forces by its neighboring elements. The force per unit length is therefore, +fγ = lim +δℓ→0 +γ ˆT(ℓ + δ/2) − γ ˆT(ℓ − δ/2) +δ += γ ˙ˆT = γκ ˆN +(S5) +where κ, ˆN are defined with the Frenet-Serret apparatus. + +11 +III. +LANDAU-DE GENNES MODELING OF NEMATIC LIQUID CRYSTALS +In the Landau-de Gennes theory, the phenomenological free energy F of a nematic liquid crystal can be written +as [21, 34] +F = +� +v +(fL + fE) dv + +� +s +fBds, +(S6) +where fL is the Landau free energy density associated with deviation of the nematic order from its equilibrium value +and can be expressed as [21, 34] +fL = a +2tr(Q2) + b +3tr(Q3) + c +4(tr(Q2))2. +(S7) +Here, the phenomenological coefficients a, b, and c are nematic material parameters. To make the free-energy den- +sity dimensionless, all the energy terms are re-scaled by the energy scale |a| in open-Qmin, which implies a non- +dimensionalization of all the elastic constants. In the simulation, we take a = −1.0, b = 12.3, and c = −10.0 [37], +which are commonly used in modeling of 5CB. Note that the values of a, b, and c determine the equilibrium mean-field +value of the nematic order S as [21, 34] +S = −b + +√ +b2 − 24ac +6c +. +Under the two-constant approximation K1 = K3 ≡ K ̸= K2, and assuming strong anchoring at the boundaries (which +render full-derivative terms irrelevant), the elastic free energy density fE is given by [34] +fE = 2K2 +9s2 +∂Qij +xk +∂Qij +xk ++ 4K +9s2 +∂Qij +xj +∂Qik +xk +− 4K2 +9s2 +∂Qik +xj +∂Qij +xk +(S8) +For a broad temperature range within the nematic phase of 5CB, the two-constant approximation is reasonable, and +K2/K ≈ 0.35 [23], which is the value we use in all our simulations. The last term fB in Eq. S6 represents the energy +density associated with the nematic directors at the boundary surface. We set the anchoring strength W ≫ |a| to +achieve strong anchoring. We minimize the total free energy F given in Eq. S6 numerically with the values previously +mentioned for different parameters using the lattice-discretized Landau-de Gennes modeling of nematic liquid crystals +implemented in open-Qmin [34]. +IV. +ESTIMATING +γ +√KK2 FROM THE SIMULATION +While we know t, l, K, and K2 in the simulation, γ is not an input parameter. To estimate γ in the simulation, we +analyze the defect configurations for the boundary condition where two identical + 1 +2 defects are patterned on opposite +surfaces (with a relative rotation of θ0 between the two patterns on the two surfaces) of a thin nematic cell of thickness +t with a separation l between the defect centers (see main text for details). For a given θ0, we start the simulation +with different random initial conditions for different values of l and t keeping K and K2 fixed. For a given l, t, and θ0, +when viewed from the top, the line defect in the energy minimized configuration forms a circular arc having opening +angle α. We determine α by fitting the circular arc with a circle under the constraint that the fitted circle must pass +through the centers of the two defects. Note that sin +� α +2 +� += +l +2R, where R is the radius of curvature of the fitted circle +(see Fig.3B in the main text). Once we know α for a given θ0, we can estimate ˜γ from Eq.(9) mentioned in the main +text. Using the definition of ˜γ = 2 +π +γt +√KK2l in the same equation, we get +γ +√KK2 += π +2 +l +t +θ0 − α +2 +sin +� α +2 +�. +(S9) +Thus, Eq. (S9) allows us to estimate +γ +√KK2 in the simulation for different l, t, and θ0. In the simulation, we consider +a system of size 250 × 250 × Lz with Lz ∈ [15 − 25] and the separation between the defect centers l ∈ [40 − 200]. +Note that thickness t = Lz − 1. We find that +γ +√KK2 varies within a broad range for different l, t, and θ0 as shown in +Fig. S3. +As discussed in the main text, Eq. (S9) is valid when the line defect forms a long horizontal section in the bulk, +i.e., the lateral span (αR) of the line defect in the bulk should be much larger than the typical length scale +γ +√KK2 t + +12 +FIG. S3. Estimation of the parameter +γ +√KK2 in the simulation obtained by analyzing the circular arc-type defect configurations +for a given θ0 and t with different values of l. +The x-axis is in logarithmic scale. +The average value of the parameter +γ +√KK2 = 3.3 ± 0.1 (horizontal dashed line in the plot) is obtained from the mean of all +γ +√KK2 for ˜x = 2 +π +t +l +sin( α +2 ) +α +2 +≤ 0.1. Insets +show the defect configurations from the side view (x-z plane where z-axis is along the thickness of the system) for two sets of +parameters. Inset(I): Formation of a long horizontal line defect around the mid-plane for θ0 = 90◦, l = 145, and t = 14 which +yields α = 2.7, and ˜γ sin +� α +2 +� += 0.2. Such a defect configuration satisfies the condition ˜γ sin +� α +2 +� +≪ +α +2 and thus included in +estimating the simulation parameter +γ +√KK2 . Inset(II): Side view of the defect configuration for θ0 = 60◦, l = 50, and t = 24 +which yields α = 1.2 and ˜γ sin +� α +2 +� += 0.4. Instead of forming a long horizontal section around the mid-plane, the line defect +changes continuously across the thickness of the system. +set by the line tension. Thus, to estimate +γ +√KK2 from Eq. (S9) we need to consider only those defect configurations +for which +γ +√KK2 +t ≪ αR ⇒ 2 +π +t +l +γ +√KK2 +sin +�α +2 +� += ˜γ sin +�α +2 +� +≪ α +2 , +(S10) +where, we have used R = +l +2 sin( α +2 ). Thus, to estimate an optimal value of +γ +√KK2 in the simulation, we need to consider +only those defect configurations for which ˜x = 2 +π +t +l +sin( α +2 ) +α +2 +≪ 1. Fig. S3 shows the dependence of +γ +√KK2 on ˜x for different +t and θ0. To obtain an average value of +γ +√KK2 , we consider the mean of all +γ +√KK2 for ˜x ≤ 0.1 (vertical dashed line in +the plot) which yields +γ +√KK2 = 3.3 ± 0.1 (horizontal dashed line in the plot) for the simulation. As a self-consistency +check, we compute ˜γ sin +� α +2 +� += 2 +π +t +l +γ +√KK2 sin +� α +2 +� +for all the defect configurations considered in Fig. S3 and find that +defect configurations with ˜x ≤ 0.1 satisfy the condition ˜γ sin +� α +2 +� +≪ α +2 (for example see insets of Fig. S3). We find +that a defect configuration with ˜x ≤ 0.1 typically forms a long horizontal section around the mid-plane as shown in +Fig. S3(I) for a particular set of parameters (θ0 = 90◦, l = 145, and t = 14) while a defect configuration which does +not form a horizontal section in the bulk typically yields ˜x > 0.1. This observation allows us to estimate +γ +√KK2 in +the simulation by considering all configurations with ˜x ≤ 0.1. +V. +COMPARISON OF SIMULATION AND EXPERIMENTAL RESULTS +To compare the experimental and simulation results, we use the same value for ˜γ = 2 +π +γt +√KK2l. In the experiment, +t and l are fixed, and we change temperature (T), which changes the value of γ. In the simulation, we don’t have T +directly. Thus, to mimic the role of T (equivalently, γ) in the experiment, we can change t and l in the simulation in +such a way that we have the same value for ˜γ in the experiment and simulation, i.e., we want +2 +π +te +le +� +γ +√KK2 +� +e += 2 +π +ts +ls +� +γ +√KK2 +� +s +, +(S11) + +14 +from simulation +5 +(II) +o = 60°,t = 20 += 120°.t = 1 +4 +0o = 60°,t = 24 + = 120°,t = 20 +9o = 90°,t = 14 +0o = 120°,t = 24 +Qo = 90°,t = 20 +Estimated +(1 ) +9o = 90°,t = 24 +0 +0.05 +0.10 +0.20 +0.50 +Qo = 105°,t = 14 +2t sin α/2 +Qo = 105°,t = 20 +πl(α /2)13 +where the subscript s and e represent the parameters for the simulation and experiment, respectively. Thus, to make +a meaningful comparison, we need to have +�l +t +� +s += +�l +t +� +e +�√KK2 +γ +� +e +� +γ +√KK2 +� +s +(S12) +In the experiment we have le ≈ 61µm ± 0.1µm and te = 2.83µm ± 0.33µm. From the experimental results shown in +Fig. 3(E), we find that for a broad temperature range +� +γ +√KK2 +� +e ∈ [12, 18]. As discussed in the previous section, we have +that +� +γ +√KK2 +� +s = 3.3 ± 0.1 in the simulation. Thus, we find that in the simulation, we need to use +� l +t +� +s ∈ [3.96, 5.94] +which also implies ˜γ ∈ [0.35, 0.53]. +For the heart-shaped pattern, we perform simulation for different l and t with ˜γ ∈ [0.35, 0.53]. In particular, we set +l = 70 lattice units and use t = 13 and 15 which gives ˜γ = 0.39 and 0.45, respectively. To compare the obtained defect +configuration with the experiment, we choose the temperature T (see from Fig.3(E) in the main text) for which we +have the same value of ˜γ. + diff --git a/t9E3T4oBgHgl3EQfkAq7/content/tmp_files/load_file.txt b/t9E3T4oBgHgl3EQfkAq7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..77be675ca41ad098f30d4ce9844519acd9181fdc --- /dev/null +++ b/t9E3T4oBgHgl3EQfkAq7/content/tmp_files/load_file.txt @@ -0,0 +1,674 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf,len=673 +page_content='Tunable Three-Dimensional Architecture of Nematic Disclination Lines Alvin Modin1,∗ Biswarup Ash2,∗ Kelsey Ishimoto1, Robert L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Leheny1, Francesca Serra3, and Hillel Aharoni2 1Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA 2Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel and 3Department of Physics, Chemistry, and Pharmacy, University of Southern Denmark, Odense, DK-5230 Denmark Disclinations lines play a key role in many physical processes, from the fracture of materials to the formation of the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Achieving versatile control over disclinations is key to developing novel electro-optical devices, programmable origami, directed colloidal assembly, and controlling active matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Here, we introduce a theoretical framework to tailor three-dimensional disclination architecture in nematic liquid crystals experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We produce quantitative predictions for the connectivity and shape of disclination lines found in nematics confined between two thinly spaced glass substrates with strong planar anchoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' By drawing an analogy between nematic liquid crystals and magnetostatics, we find that: i) disclination lines connect defects with the same topological charge on opposite surfaces, and ii) disclination lines are attracted to regions of the highest twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Using polarized light to pattern the in-plane alignment of liquid crystal molecules, we test these predictions experimentally and identify critical parameters that tune the disclination lines’ curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We verify our predictions with computer simulations and find non-dimensional parameters enabling us to match experiments and simulations at different length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Our work provides a powerful method to understand and practically control defect lines in nematic liquid crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Topological singularities link physically distinct phe- nomena – they mediate phase transitions [1], act as orga- nizational centers in biological systems [2], and steer the trajectory of light [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Various topological defect con- figurations are present in nematic liquid crystals (LCs), fluid-like materials with long-range orientational molecu- lar order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Disclination lines arise when nematic LCs are frustrated by incompatible boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' These one-dimensional singularities can be facilely formed and visualized, making nematic LCs an ideal test bed for studying defect structures and interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Manipulat- ing disclination lines in LCs also has practical applica- tions in directed self-assembly, [5, 6], tunable photonics [7], and re-configurable microfluidic devices [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' To effectively utilize the potential of disclinations for these applications, it is essential to develop a set of fundamen- tal rules that govern their formation and connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Recent advances in spatial patterning of liquid crystal alignment have enabled greater control over the struc- ture of disclination lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' For example, imprinted nano- ridges on glass substrates have been used to precisely shape defect lines, revealing insights into their energy, structure, and multi-stability [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Using light to impose LC alignment at photosensitive substrates is an equally powerful tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Photo-alignment has enabled the design of free-standing disclination loops[12–14] and pe- riodic disclination arrays with different morphologies and properties [15–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' In this work, we introduce a general framework for cre- ating arbitrarily shaped three-dimensional (3D) disclina- tion line architecture in nematic liquid crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' As an example, we show a structure where the projection of ∗ Authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' disclination lines on a two-dimensional (2D) plane forms the shape of a heart (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' In the experiment, light- sensitive layers on parallel glass substrates align the ne- matic at the surfaces in patterns decorated with pairs of 2D surface defect nucleation sites (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='1A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The 2D defects are characterized by winding numbers - the de- gree of rotation of the nematic director around the defect divided by 2π - of +1/2 and −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Aligning opposite- charged defects on opposing substrates, we observe that the confined LC forms a pair of disclination lines that pri- marily run through the mid-plane of the cell to connect surface defects on the same substrate (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 1B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' This configuration is a stable, equilibrium state, confirmed by numerical simulations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='1C-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' To understand the paths that the disclination lines take, we draw an analogy between the elastic distortion of a nematic and the magnetostatic field of current-carrying wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Using this analogy, we experimentally and numeri- cally verify two key rules: (i) disclination lines either con- nect surface defects on opposing substrates with the same winding number or surface defects on the same substrate having opposite winding numbers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (ii) the lines’ paths de- pend on the interplay between forces driving them to re- gions of maximum twist set by the confining pattern and the disclination line tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We utilize these two design principles to create the heart-shaped disclinations shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Our proposed framework enables the design of tunable 3D liquid crystal-based disclination networks for applications in re-configurable optics, photonic devices, and responsive matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The following sections explain the magnetostatic anal- ogy and its outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Subsequently, we test specific pre- dictions with experiments and simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Finally, we revisit the structure in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 1 to explain how we designed the heart-shaped lines and how disclination shapes can arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='04594v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='soft] 11 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Designing three-dimensional disclination line architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (A) Schematics of the two-dimensional sur- face patterns imprinted onto substrates coated with a photo- sensitive layer (Brilliant Yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Topological defects with charge +1/2 (red semi-circle) and −1/2 (orange triangle) act as nucleation sites for disclination lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The substrates are aligned so that defects of opposing strength are in registry with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (B) Disclination lines viewed under bright- field microscopy form a heart-shaped structure (scale bar: 25 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' This configuration corresponds to a stable equilib- rium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Top view (C) and side view (D) of the discli- nation structure obtained from simulation using the patterns in (A) as surface boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The observed two- dimensional projection (black points) is a balance between the attraction of defect lines to locations where the top and bot- tom surface patterns are perpendicular (orange lines in (C)) and their line tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (D) A side view of the defect configu- ration obtained in the simulation reveals that the disclination lines primarily run through the mid-plane of the cell and con- nect surface defects of opposite charge on the same substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' be tailored by varying temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' RESULTS AND DISCUSSION Magnetostatics model Distortions of the nematic director field are described by the Frank-Oseen elastic free energy [21], Fel = � �K1 2 (∇ · ˆn)2 + K2 2 (ˆn · ∇ × ˆn)2 + K3 2 (ˆn × ∇ × ˆn)2 � dV, (1) with ˆn the nematic director and K1,2,3 the splay, twist, and bend elastic constants, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We consider a nematic placed between two parallel plates with (suffi- ciently strong) planar anchoring on them, separated by a spacing t much smaller than their lateral dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Under these conditions, we make the following key as- sumption: in equilibrium, the nematic director is planar everywhere within the cell, not only at the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' This is analogous to the Kirchhoff-Love assumptions in plate elasticity theory [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The nematic director field then takes the form ˆn = (cos θ, sin θ, 0), where θ(x, y, z) is the director’s azimuthal angle in the xy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' In addition, we use the two-constant approximation, with K1 = K3 ≡ K, which is valid near the nematic-isotropic transition for low molecular weight thermotropic nematic LCs, particularly 4’-octyl-4-biphenylcarbonitrile (8CB) [23] used in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 1 then assumes the simple form, Fel = � �K 2 � (∂xθ)2 + (∂yθ)2� + K2 2 (∂zθ)2 � dV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (2) Further simplification is obtained by rescaling the z- axis using ˜z = z � K/K2 (defined on a domain of thick- ness ˜t = t � K/K2), and redefining ∇ ≡ (∂x, ∂y, ∂˜z), so that Fel = K 2 � |∇θ|2 d ˜V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (3) The functional in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 3 implies that, in equilibrium, θ(x, y, ˜z) is a harmonic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' However, this prop- erty breaks down along disclination lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' at the defect core, the nematic order vanishes, and θ is not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Around the defect line, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 3 admits a nontrivial quan- tized integral, � dℓ · ∇θ = 2πq, q ∈ Z/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (4) Together Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 3 and 4 establish an exact mathematical analogy of the nematic cell to magnetostatics, as previ- ously identified by de Gennes [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' In the analogy, the planar director’s azimuthal angle θ plays the role of a magnetic scalar potential, whose gradient is the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Disclination lines are current-carrying wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Their existence renders θ ambiguous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' however, the half-integer quantization of the current exactly corresponds to the nematic θ ∼= θ + πZ congruence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The disclination wires are flexible and stretchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Each wire is associated with a line tension γ, the outcome of melting of the nematic order at the defect core to al- leviate the diverging elastic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Approximately, γ A B D3 is proportional to Kq2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' however, there are logarithmic corrections that depend on the line and cell geometry [21, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' These corrections become significant near the nematic-isotropic phase transition as the defect core size diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' For simplicity, we ignore these corrections and treat γ as a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Similarly to other material param- eters, namely K, K2, γ may depend on temperature in a non-trivial way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Forces on wires To study the shape of disclination wires, we calculate the effective forces acting on them (see SI Appendix for full derivations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' There are three forces (per unit length) acting on the wires: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The strong anchoring on the two surfaces acts as magnetic mirrors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Disclination wires are repelled by these mirrors (alternatively, by the mirror image wires) and pushed toward the mid-plane between the two boundary surfaces by a force fM = −π2Kq2 ˜t tan �π˜z ˜t � ˆz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (5) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The anchoring planar angles, θt,b(x, y) on the top/bottom surfaces, respectively, are analogous to an external magnetic field that exerts a Lorentz-like force on disclination wires: fB = 2πKq ˜t (θt − θb − qπ) ˆT × ˆz, (6) where ˆT is the unit tangent to the defect line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' This force pulls defect lines horizontally towards regions where the top and bottom are at a ∆θ ≡ θt − θb = qπ angle difference from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The line tension of the wires exerts a force fγ = γκ ˆN, (7) where κ is the curvature of the wire and ˆN its nor- mal in the Frenet-Serret frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The equilibrium shape of a disclination line is obtained by the balance of fM, fB, and fγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' This force balance can characterize the geometry of defect lines connecting to surface defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' A disclination line emerges perpendicularly from a surface topological defect due to the magnetic repulsion described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' For a surface defect |q| > 1 2 we expect a split into 2|q| “atomic” lines of magnitude 1 2 arising from the mutual repulsion between them (as was observed in [10, 16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' By the balance of forces, fM and fγ lines emerging from de- fects turn horizontally into the mid-plane over a typical length scale ∼ γ K ˜t ≡ γ √KK2 t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Thus, defect lines whose lateral span is much larger than this scale traverse within the mid-plane for the more significant part of their tra- jectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Connectivity of surface defects We study the topological rules of connecting – with disclination wires – surface defects patterned on two confining surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Each surface defect is characterized by a winding number q, defined by a closed loop Γ around the defect core as 1 2π � Γ ∇θ · dℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' By current conservation, a disclination line can connect two surface defects of the same q on opposite surfaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' or two surface defects of opposite q on the same surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Alternatively, disclinations can escape to the sides of the system or form a closed loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Planarity of the director field forbids connection between a top-surface +1/2 defect with a bottom-surface −1/2, even though this would be topologically allowed in a 3D nematic liquid crystal [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Experimental Tests Test of Design Principles To verify these connectivity principles experimentally, we create a LC cell where the bottom and top surfaces contain a single, isolated q = −1/2 or +1/2 defect, re- spectively (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 2 A-C), utilizing the custom built photo- alignment system described in Materials and Methods and SI Appendix, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' S1 [26–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We shine linearly polarized light on glass coated with a light-sensitive alignment layer (Brilliant Yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The alignment layer molecules give planar alignment to the LC, with a di- rection that is perpendicular to the polarization of the incident light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' By spatially patterning the light polariza- tion, we imprint half-integer defect nucleation sites onto confining glass substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The defects on each surface are photo-patterned within a circular patch of diameter d ≈ 75µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Under crossed-polarizers, the dichroic prop- erties of the Brilliant Yellow dye enable us to view the patterned regions on the confining glass substrates be- fore filling them with LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We align the circular patches on each substrate to overlap, ensuring that defect cores of opposing topological charges are in registry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Once sub- strates are secured with epoxy resin, we carefully measure the cell thickness and inject pre-heated 8CB LC into the cell, allowing it to slowly cool until it reaches the nematic phase at ∼ 36◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The resulting defect structure follows the connectivity rules: rather than a single disclination line connecting the surface defects as might be expected [25], two discli- nation lines emerge from the defect cores and escape to the sides along the mid-plane, as can be seen from a side view of the numerical simulation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 2A and from the top view in experiments (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 2B) and simulations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 2C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Indeed, this connectivity rule gives rise to the two extended lines that make the heart shape in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 1 rather than two defect lines connecting surface defects di- rectly facing each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' For verification, we run the same experiment and simulation with +1/2 surface defects pat- 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Surface defect connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (A) When two canonical surface defects with ±1/2 charge are imposed as boundary conditions, the planarity of the director field forbids the connection of the top and bottom surface defects with a single defect line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The equilibrium state obtained from numerical minimization of Landau de Gennes energy corresponds to two disclination lines (black points) nucleating at the surface defect cores and extending towards the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (B) Bright-field microscopy images where two confining surfaces are photo-aligned with isolated ±1/2 surface defects corresponding to the pattern used in (A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Inset: polarized light microscopy image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (C) Two-dimensional projection of the numerical results in (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Inset: The expected polarized optical microscopy (POM) texture reconstructed from the director field using Jones calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (D) Preserving the topological charge of the surface defects while altering their geometric structure changes the regions where ∆θ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The equilibrium state for these boundary conditions corresponds to two disclination lines now at a relative angle of π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (E) Bright-field microscopy images where two confining surfaces are photo-aligned with isolated ±1/2 surface defects corresponding to the pattern used in (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (F) Two-dimensional projections of the numerical results in (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Inset: The expected POM texture reconstructed from the director field using Jones calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Scale bars: 25µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' terned onto confining substrates (see SI Appendix, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' A vertical disclination line connects the top and the bottom surface defects, as is permitted in this case by current conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The effect of varying the patterned boundary condi- tions can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 2(D-F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Here, we preserve the topological charge of each surface pattern but introduce a homogeneously aligned region that alters the geometric structure of the ±1/2 defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The new surface pattern modifies the areas where ∆θ (x, y) are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The new regions where fB = 0 lead to a reduced angular sep- aration of the two defect lines from π (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 2B-C) to π/2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 2E-F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' For both designs, the disclination wires do not bend and κ ≈ 0, implying that fγ has little to no effect on the positioning of the lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Tuning the curvature of disclination architecture When the imposed surface patterns result in curved disclination lines, the line tension fγ becomes important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' fγ opposes fB, acting to minimize the wire’s curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The competition between these two forces causes the tra- jectory of the two disclination lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 1D to devi- ate from regions where ∆θ (x, y) = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The shape of the disclination lines can then be tuned by changing the magnitude of fB and fγ, which vary differently with tem- perature due to the temperature-sensitive behavior of γ and the elastic constants K, K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' As described below, by tuning the disclination shape, we measure γ/√KK2 at various temperatures, enabling us to map our experimen- tal observations to simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' To illustrate the tuning of disclination shape, we con- struct LC cells whose confining surfaces are each photo- patterned with a single +1/2 defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The substrates are rotated so that the defects are oriented with respect to one another by an angle θ0 and are translated so that a horizontal distance ℓ separates the defect cores (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 3A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' In this design, the 2D projection of the patterns con- tains a locus of points where ∆θ = π/2 forms a circular arc segment with an opening angle 2θ0 connecting the defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Once a cell is filled with 8CB, a disclination line forms to connect the two defect cores (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 3C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='. In general, the line does not follow the arc with opening B A C m =+1/2 m = -1/2 D m = +1/2 m =-1/25 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Measuring the line tension of a disclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (A) Diagrams of the geometry used to create arced discli- nation lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Initially, +1/2 surface defects are imposed as anchoring conditions in a nematic cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The substrates are then rotated and displaced with respect to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (B)) When viewed from the top, the defects cores are separated by a distance ℓ and their lines of symmetry form an angle θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The two-dimensional projection of the resulting disclina- tion line is a portion of a circular arc with a central angle α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (C)) Bright-field microscopy overlapped images of discli- nation lines at two different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The disclination line’s curvature depends on the balance of two opposing forces fγ and fB, whose magnitudes depend on temperature (scale bar: 10µm, dashed lines are guides for the eye).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (D)) Varying the temperature of the nematic, therefore, results in a change in the defect line curvature and in the angle α as shown in the curves obtained for different values of θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (E)) When the measured values of t, ℓ, and θ0 are all accounted for, the im- plied values of γ/√KK2 all collapse onto the same curve as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' angle θ0 due to fγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' However, along any circular disclina- tion arc that passes between the two surface defect cores, both Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 6 and 7 are uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Thus, in equilibrium, the disclination still forms an arc, and finding its curvature through force balancing is a simple algebraic problem: 0 = (fB + fγ) · ˆN = πK ˜t �α 2 − θ0 � + γ 2 ℓ sin α 2 , (8) where α is the opening angle of the arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Rewriting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 8 in a dimensionless form, we obtain the following tran- scendental equation: α 2 + ˜γ sin α 2 = θ0, (9) where ˜γ = 2 π γ √KK2 t ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' As expected, in the limit of van- ishing line tension, α tends to 2θ0, where fB vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' In the limit of infinite line tension, α tends to zero so that fγ vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Line tension’s relative importance in determining the defect line’s contour is described by the dimensionless parameter ˜γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Equation 9 captures the effect of line tension in reduc- ing the curvature of an arced disclination line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Rearrang- ing it again, we find that, γ √KK2 = π 2κt � θ0 − arcsin �ℓκ 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (10) The equation above links the material parameter γ/√KK2 to the deviation of the disclination arc’s curva- ture κ from its zero-line tension limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Thus, the temper- ature dependence of γ/√KK2 can be measured directly in 8CB from the temperature dependence of the line cur- vature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We track the variation of α = 2 arcsin (ℓκ/2) as a function of temperature across θ0 ranging from 30◦ to 105◦ (see Materials and Methods for details of the image analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' When a disclination line is formed by an ini- tial θ0 = 105◦, the curvature deep in the nematic phase (T = 35◦C) is small (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 3C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Increasing the tempera- ture towards the nematic-isotropic transition, we observe an increase in κ (and hence α) since fγ decreases more rapidly than fB on heating (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 3D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Equation 10 is con- firmed by the collapse in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 3E of measurements held at different values of t, ℓ and θ0 onto the same curve that only depends on material properties of the LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' This affirms the validity of approximating γ with a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 3E shows the monotonic temperature dependence of γ/√KK2 in a nematic 8CB, ranging approximately between 12 and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We follow the protocol of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 3 to also estimate γ/√KK2 in our numerical simulations (see SI Appendix for details);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' we analyze arced defect config- urations for different values of t, ℓ and θ0, and extract α from which we obtain a mean γ/√KK2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' This value is not within the experimental range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' However, for every experiment, we can now match a simulation held at the same value of ˜γ = 2 π γ √KK2 t ℓ, by compensating for the different values of γ √KK2 with inversely different val- ues of the aspect ratio t ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' In simulations, we tweak ˜γ not with temperature but with aspect ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We now revisit Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 1 and the heart-shaped disclina- tion lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' These are generated using patterns described in detail in the Materials and Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We control the cusps of the heart by the directions of maximum twist around the defects as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' When 8CB is cooled by ≈ 6◦C from the nematic-isotropic transition, the increase in ˜γ constricts the lobes of the heart-shaped disclination lines (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 4A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We know the value of ˜γ at each tempera- ture from the thickness of the cell, the lateral separation between the two surface defects, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 3E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Simulations with the same values of ˜γ, obtained by changing the val- ues of t and ℓ, qualitatively capture a similar change in the structure of the disclination architecture Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' It is remarkable that despite the experimental uncertainty and the use of different system sizes in the experiment and simulation, the resulting defect configurations for the same values of ˜γ are in good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' A D 3T 4 π-2 T 4 E B 18 α K16 C T=40°C 14 29 60 ° 96° 105 ° 12 0 6 T=35°C oC6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Tunable disclination line architecture (A) Bright-field microscopy image of heart-shaped disclination lines measured for two different temperatures, correspond- ing to different values of ˜γ (scale bar: 25 µm, dashed lines are guides for the eye).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (B) Defect configurations obtained in the simulation for two different sets of parameters l, t, cho- sen such that the values of ˜γ are the same in simulation and experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The change in the structure of the disclination architecture in both experiment and simulation is captured by ˜γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' CONCLUSION This work introduces a novel framework for creating and tuning 3D disclination lines in a nematic liquid crys- tal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' When disclination lines are nucleated by surface defects, their connectivity and trajectories are analo- gous to current-carrying wires near a current-free sur- face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Whether or not surface defects may connect to each other can be explained by treating the topologically charged disclination lines as wires that must conserve current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Similarly, substrates imprinted with surface- anchoring conditions exert a Lorentz-like force on the wires, pushing them towards regions where the anchoring conditions on opposing substrates are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' When the patterns promote wires to curve, they experience an additional force from line tension that decreases the cur- vature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' This force can be tuned in both experiments and simulations by changing a dimensionless parameter, ˜γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We verified these connectivity principles through a se- ries of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' By appropriately designing sur- face anchoring conditions, we created a three-dimensional structure whose two-dimensional projection resembles a heart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We tuned its shape by varying the temperature and recreated the results using numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Our design principles can be used to interpret similar results observed in recent experiments with disclination lines created by patterned surfaces [13–15, 17, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' These principles can further be used to construct more complex disclination architecture, advancing the design of tunable 3D liquid crystal-based disclination networks for applica- tions of molecular self-assembly, re-configurable optics, photonic devices, and responsive matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Furthermore, we have shown that the equilibrium shape of disclination lines depends on temperature and aspect ratio, opening the door for multi-state systems, switchable by varying the temperature or thickness of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' MATERIALS AND METHODS Substrate preparation Photosensitive material Brilliant Yellow (BY, Sigma- Aldrich) was mixed with n,n-dimethylformamide (DMF) solvent at 1 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='% concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Glass substrates (Fisher Scientific) were washed in an ultrasonic bath with Hell- manex liquid detergent (Fisherbrand), followed by suc- cessive washes in acetone, ethanol, and isopropyl alco- hol, and then dried with N2 gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The BY-DMF solution was spin-coated on the substrates at 3000 RPM for 45 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' After spin-coating, the substrates were baked at 95◦C for 15 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Spin-coating and baking pro- cesses were conducted at a relative humidity of 35% or lower[30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Patterned Surface Alignment Surface patterns were created using a custom-built photo-patterning setup consisting of a polarized LED source [28] feeding into the side port of a bright-field inverted microscope body (TI Eclipse TE2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Segmented images were generated via a LED-based projector (Sony MPL-C1A) to a peripheral optical path (SI Appendix Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The projector operates using three time-modulated laser diodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' To match the ab- sorption band of the BY-DMF solution, we use the blue (λ = 445nm) diode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Images generated by the projector first pass through two aspheric condenser lenses (with fo- cal lengths f = 32mm, Thorlabs, ACL50832U) before be- ing expanded with a custom-Keplerian telescope consist- ing of two convex lenses f = 100mm (Thorlabs, AC508- 100-A-ML) and f = 200mm (Thorlabs, AC508-200-A- ML), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The expanded image passes through a linear polarizer before entering the microscope body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Once inside the body, the image is reflected by a dichroic mirror, picked up by an infinity-corrected tube lens, and collected by a microscope objective (20x, Nikon S Plan Fluor ELWD) that focuses the image onto a BY-coated substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Upon irradiation with linearly polarized light, the photosensitive azo-dye molecules orient perpendicu- larly to the plane of polarization, setting the preferred alignment direction of the nematic director ˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Designed patterns were discretized into pie segments of fixed polarization with opening angle π/16 and with the cores of defects located at the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Sample preparation After photo alignment, patterned regions on substrates were aligned and fixed using epoxy glue (Loctite) to cre- ate a liquid crystal cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' After cell assembly, we use spec- troscopic reflectometry to measure the cells’ thickness t, obtained from the absolute reflectance spectra (Ocean- view) fit using custom Matlab code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Cells are subse- A B Y~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='39 Y~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='457 quently filled with 4’-n-octyl-4-cyano-biphenyl (8CB, Ne- matel GmbH) liquid crystal, pre-heated into the isotropic phase by capillary flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' After cells are filled, they are sealed on their ends using UV curable resin (Loon Out- doors UV Clear Fly Finish).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Polarized optical microscopy We use a Nikon LV 100N Pol upright microscope to im- age patterned regions with both 20x and 50x objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Samples are placed on a heating stage (Instec HCS302) set to 36◦C to keep 8CB in the nematic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Optical microscopy images are captured using a Nikon DS-Ri2 camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Analyzing the curvature of disclination line arcs Videos of disclination lines are captured using bright-field microscopy and analyzed using ImageJ, TrackPy[31], and custom Python code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The contours of the disclination lines are detected using a Canny edge detection algorithm, binarized, and fit to circles using least squares fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' For each frame i of the video, the radius of curvature ri ≡ 1/κ and center of the best-fit circle � xC i , yC i � are determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' These circles intersect at the defect cores, corresponding to two unique points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' To find the positions of these points (x, y), we minimize a cost function Ξ =� i � (x−xc i )2+(y−yc i )2) r2 i − 1 �2 , where the sum is over all the frames in the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The uncertainty of each defect core’s position is the cost function’s value, and the distance ℓ between the defect cores is calculated using the Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Jones matrix calculations For qualitative comparison of numerical and experi- mental director configurations near nucleated disclina- tion lines, we use Jones calculus to reconstruct the po- larized optical microscopy (POM) texture of the direc- tor field obtained from minimization of the Landau de- Gennes free-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The volume of the numerically- obtained director field is discretized into volume elements (voxels) on a 3D grid, with each point at position ρ con- taining N voxels each of thickness ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' It is assumed that variation in n between successive voxels is small com- pared to the wavelength of incident light λ, so that λ ≪ 1/|∇ˆn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Each voxel ν is treated as a uniaxial birefrin- gent optical element, represented by a 2×2 Jones matrix Mν that depends on both the extraordinary ne and ordi- nary no indices of refraction of the LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Light propagating through a voxel experiences an ne dependant on the polar angle θν between n and the light’s propagation direction k0 given by ne(θν) = none/ � (no cos θν)2 + (ne sin θν)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We choose k0 = ˆz, so that the plane of polarization is the x−z plane, and write the corresponding Jones matrix as Mν (ρ) ≡ � eine ν(θν)2π ∆ λ 0 0 eino ν2π ∆ λ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (11) Using the 8CB’s ne and no at the exper- imental temperature and wavelength λ, we compute Mν (ρ), constructing a single opera- tor γ (ρ) = �N ν=1 R (−φν) Mν (ρ) R (φν), where R (φν) ≡ � cos φν sin φν − sin φν cos φν � and φν is the azimuthal component of ˆn in voxel ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Following [32] and [33], we construct 2 × 2 Jones matrices for the polarizer P and analyzer A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Sequential propagation of plane waves E0 through P,γ (ρ) and A results in a single Jones vector ET (ρ) = Aν (ρ) PE0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The calculated POM texture is obtained from the intensity of light transmitted through each voxel, IT (ρ) = |ET (ρ) |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Numerical simulations The numerical modeling of the nematic liquid crys- tal is achieved using the lattice-discretized Landau-de Gennes model implemented in open-Qmin[34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The con- figuration of a nematic liquid crystal is represented by specifying the components of the Q-tensor[34, 35] which is related to the director ˆn of a uniaxial nematic by Qij = 3 2S � ninj − 1 3δij � , where i, j ∈ {x, y, z} and S is the degree of uniaxial nematic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' To simulate a thin nematic cell, we consider a three-dimensional box of size L × L × Lz with L ≫ Lz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' In the simulation, we use L = 250 and Lz between 12 and 21, expressed in units of the number of lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Note that the thick- ness t = Lz − 1, since anchoring is imposed on top and bottom layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' At every lattice point, we start with a random initial condition for Qij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We impose strong pla- nar anchoring at the top and bottom surfaces by setting the anchoring strength W = 50 for the two surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We use free boundary conditions on the side surfaces of the simulation box by setting W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We use the Fast In- ertial Relaxation Engine (FIRE) algorithm within open- Qmin[34] to minimize the total free energy until the norm of the residual force vector goes below 10−8 (see SI Ap- pendix for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' In the energy-minimized configura- tion, defects are identified locally as lattice sites where the largest eigenvalue of Q falls below some threshold, typically 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='95S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Surface patterns used in experiment and simulation In the experiment and simulation, we impose a planar director field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=', the nematic director field takes the form ˆn = (cos θ, sin θ, 0) at the top and bottom surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='1 and 4, the surface pattern at the top surface is 8 represented by θt (x, y, z) = 1 2 � tan−1 y − 1 x − tan−1 y + 1 x + x � x2 + (y + 1)2 − x � x2 + (y − 1)2 � , (12) while θb (x, y, z) = −θt (x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' For the ±1/2 surface defect patterns used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (2-3) we have θ (x, y, z) = ± 1 2 tan−1 � y x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Acknowledgements We greatly acknowledge insights, assistance, and help- ful discussions with Charles Rosenblatt, David Dolgitzer and Bastian Pradenas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' This research was supported by a grant from the United States-Israel Binational Science Foundation (BSF) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 2018380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' RLL acknowledges sup- port from the NSF (DMR-2104747).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Kosterlitz and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Thouless, Journal of Physics C: Solid State Physics 6, 1181 (1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [2] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Maroudas-Sacks, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Garion, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Shani-Zerbib, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Livshits, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Braun, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Keren, Nature Physics 17, 251 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [3] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Brasselet, Physical Review Letters 108, 087801 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Meng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Wu, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Smalyukh, Nature Materials 22, 64 (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Blanc, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Coursault, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Lacaze, Liquid Crystals Reviews 1, 83 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [6] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Luo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Beller, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Boniello, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Serra, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Stebe, Nature Communications 9, 3841 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [7] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Coles and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Pivnenko, Nature 436, 997 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Sengupta, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Bahr, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Herminghaus, Soft Matter 9, 7251 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Sandford O’Neill, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Salter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Booth, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Elston, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Morris, Nature Communications 11, 2203 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [10] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Murray, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Pelcovits, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Rosenblatt, Phys- ical Review E 90, 052501 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Harkai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Murray, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Rosenblatt, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Kralj, Physical Review Research 2, 013176 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Li, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Yokoyama, Nature Communica- tions 8, 388 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [13] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Ouchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Imamura, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Sunami, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Yoshida, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Ozaki, Physical Review Letters 123, 097801 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [14] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Sunami, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Imamura, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Ouchi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Yoshida, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Ozaki, Physical Review E 97, 020701 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [15] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Guo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Jiang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Afghah, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Peng, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Selinger, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Lavrentovich, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Wei, Advanced Optical Ma- terials 9, 2170063 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [16] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Yoshida, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Asakura, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Fukuda, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Ozaki, Na- ture Communications 6, 7180 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [17] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Nys, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Berteloot, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Beeckman, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Neyts, Ad- vanced Optical Materials 10, 2101626 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [18] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Nys, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Beeckman, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Neyts, Advanced Optical Materials 6, 1701163 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [19] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Berteloot, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Nys, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Poy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Beeckman, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Neyts, Soft Matter 16, 4999 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Jiang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Ranabhat, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Rich, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Zhang, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Peng, Proceedings of the National Academy of Sciences 119, e2122226119 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [21] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Gennes and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Prost, The Physics of Liquid Crystals, 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=', International series of monographs on physics No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 83 (Clarendon Press, Oxford, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [22] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Love, Philosophical Transactions of the Royal Society of London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=') 179, 491 (1888).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [23] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Oswald, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Baudry, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Rondepierre, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' E 70, 041702 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [24] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Schopohl and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Sluckin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 59, 2582 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [25] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Mermin, Reviews of Modern Physics 51, 591 (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [26] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Chigrinov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Kozenkov, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Kwok, Pho- toalignment of liquid crystalline materials: physics and applications, Wiley SID series in display technology (Wi- ley, Chichester, England ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Hoboken, NJ, 2008) oCLC: ocn225429570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [27] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Folwill, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Zeitouny, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Lall, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Zappe, Liquid Crystals 48, 862 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [28] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Wani, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Wasylczyk, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Priimagi, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 6, 1700949 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [29] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Long, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Deutsch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Angelo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Culbreath, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Yokoyama, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Selinger, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Selinger, Frank-read mechanism in nematic liquid crystals (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [30] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' McGinty, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' West, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Bryant, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Finnemeyer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Reich, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Berry, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Clark, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Yaroshchuk, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Bos, Liquid Crystals 44, 863 (2017), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='1080/02678292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='1247479.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [31] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Allan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Caswell, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Keim, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' van der Wel, trackpy: Trackpy v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='2 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [32] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Hecht, Optics, 5th ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (Pearson Education, Inc, Boston, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [33] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Ellis, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Pairam, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Fern´andez-Nieves, Journal of Physics D: Applied Physics 52, 213001 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [34] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Sussman and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Beller, Frontiers in Physics 7, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='3389/fphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='00204 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [35] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Mottram and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Newton, Introduction to q- tensor theory (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [36] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Jackson, Classical electrodynamics (Third edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' New York : Wiley, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' [37] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Ravnik and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Zumer, Liquid Crystals 36, 1201 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 9 TUNABLE ARCHITECTURE OF NEMATIC DISCLINATION LINES SUPPLEMENTARY APPENDIX OPTICAL SETUP FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Optical setup for photoalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' A projector with blue LED light generates images focused on substrates at the sample plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Images are polarized by a linear polarizer, collimated by aspheric lenses, and then expanded by a relay of lenses convex lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' A microscope body contains a dichroic mirror that reflects light toward an infinity-connected tube lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Images are picked up by the microscope and focused onto the sample plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' DISCLINATION LINES CONNECTING SURFACE DEFECTS ON OPPOSING SUBSTRATES FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Defect connectivity with identical boundary conditions (a) A single disclination line runs between surface defects with identical topological charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' In bright-field microscopy, the defect line appears as a single point at various locations within the LC cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Inset: Polarized optical microscopy image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' On the side, the schematic of the defect line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The image is taken in the mid-plane of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Scale bar: 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (b) Simulation results of a straight disclination line between two surfaces with +1/2 point defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' sample X-V stage obiective tube lens convex aspheric lenses lenses dichroic mirror LED linear polarizerA B10 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' DERIVATION OF THE FORCES ACTING ON A WIRE ELEMENT The magnetostatic model emerges from the similarity between equations [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='4] in the main text and their (vacuum) magnetostatics counterparts [36]: Fmag = 1 2µ0 � |B|2 d ˜V (S1) and � dℓ · B = µ0I (S2) By comparison,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' the following pairs are analogous: Nematic Magnetic Field ∇θ B Modulus K 1/µ0 Current 2πKq I Force per unit length on wire in field 2πKq dℓ × ∇θ Idℓ × B Force per unit length between wires at distance d 2πKq1q2/d µ0 2π I1I2/d The problem (namely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' S2 and the Euler-Lagrange equation associated with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' S1) is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We may therefore write the boundary conditions as a sum of contributions, solve them separately, and add up the solutions/forces exerted in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We start with the method of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We introduce an infinite ladder of mirror wires outside of our cell, indexed by m ∈ Z \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We set the wires parallel to the actual one, located at ˜zm = m˜t + (−1)mδ, all carrying the same current 2πKq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We now sum up the forces per unit length exerted on the wire element by its mirror images: fM = ∞ � m=−∞ m̸=0 2πKq2 δ − ˜zm ˆz = 2πKq2 ∞ � m=1 m odd � ˆz m˜t + 2δ − ˆz m˜t − 2δ � = −π2Kq2 ˜t tan �πδ ˜t � ˆz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (S3) By construction, the above (real and image) wire setup is symmetric about either of the two boundary plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Therefore, ∇θ (induced by this setup alone) is perpendicular to these boundaries, namely the director angles on both boundaries are constant (henceforth denoted θ0 t,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' By the integral condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' S2 and by lateral reflection symmetry, θ0 t − θ0 b = qπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' However, in the experimental/numerical setup discussed in the main text, ∆θ = θt − θb is an arbitrary function of x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' To correct this, the solution to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' S1 and S2 must be the sum of the above solution and a harmonic function (so that the equations are still satisfied) that makes up for the boundary condition mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The (divergence-free) gradient of this function can be interpreted as an external magnetic field Bext(x, y, z) that acts regardless of the exact shape of the wire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' It can be written explicitly using Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' However, we further assume that the thickness of the cell is much smaller than the lateral gradients of θt,b(x, y), therefore almost everywhere Bext ≈ Bˆz (this assumption may fail close to surface defects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Being divergence-free, B(x, y, z) must be approximately uniform in z, and matching the boundary conditions we get B(x, y) = ∆θ−∆θ0 ˜t = 1 ˜t (θt − θb − qπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Thus, the Lorenz force exerted on the wire by the external field reads fB = 2πKq ˜t (θt − θb − qπ) ˆT × ˆz, (S4) where ˆT is the unit tangent to the defect line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Derivation of the force exerted by line tension is rather straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' A line segment of length δ is subject to tangential forces by its neighboring elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The force per unit length is therefore, fγ = lim δℓ→0 γ ˆT(ℓ + δ/2) − γ ˆT(ℓ − δ/2) δ = γ ˙ˆT = γκ ˆN (S5) where κ, ˆN are defined with the Frenet-Serret apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 11 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' LANDAU-DE GENNES MODELING OF NEMATIC LIQUID CRYSTALS In the Landau-de Gennes theory, the phenomenological free energy F of a nematic liquid crystal can be written as [21, 34] F = � v (fL + fE) dv + � s fBds, (S6) where fL is the Landau free energy density associated with deviation of the nematic order from its equilibrium value and can be expressed as [21, 34] fL = a 2tr(Q2) + b 3tr(Q3) + c 4(tr(Q2))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (S7) Here, the phenomenological coefficients a, b, and c are nematic material parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' To make the free-energy den- sity dimensionless, all the energy terms are re-scaled by the energy scale |a| in open-Qmin, which implies a non- dimensionalization of all the elastic constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' In the simulation, we take a = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='0, b = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='3, and c = −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='0 [37], which are commonly used in modeling of 5CB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Note that the values of a, b, and c determine the equilibrium mean-field value of the nematic order S as [21, 34] S = −b + √ b2 − 24ac 6c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Under the two-constant approximation K1 = K3 ≡ K ̸= K2, and assuming strong anchoring at the boundaries (which render full-derivative terms irrelevant), the elastic free energy density fE is given by [34] fE = 2K2 9s2 ∂Qij xk ∂Qij xk + 4K 9s2 ∂Qij xj ∂Qik xk − 4K2 9s2 ∂Qik xj ∂Qij xk (S8) For a broad temperature range within the nematic phase of 5CB, the two-constant approximation is reasonable, and K2/K ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='35 [23], which is the value we use in all our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The last term fB in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' S6 represents the energy density associated with the nematic directors at the boundary surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We set the anchoring strength W ≫ |a| to achieve strong anchoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We minimize the total free energy F given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' S6 numerically with the values previously mentioned for different parameters using the lattice-discretized Landau-de Gennes modeling of nematic liquid crystals implemented in open-Qmin [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' ESTIMATING γ √KK2 FROM THE SIMULATION While we know t, l, K, and K2 in the simulation, γ is not an input parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' To estimate γ in the simulation, we analyze the defect configurations for the boundary condition where two identical + 1 2 defects are patterned on opposite surfaces (with a relative rotation of θ0 between the two patterns on the two surfaces) of a thin nematic cell of thickness t with a separation l between the defect centers (see main text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' For a given θ0, we start the simulation with different random initial conditions for different values of l and t keeping K and K2 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' For a given l, t, and θ0, when viewed from the top, the line defect in the energy minimized configuration forms a circular arc having opening angle α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We determine α by fitting the circular arc with a circle under the constraint that the fitted circle must pass through the centers of the two defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Note that sin � α 2 � = l 2R, where R is the radius of curvature of the fitted circle (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='3B in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Once we know α for a given θ0, we can estimate ˜γ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (9) mentioned in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Using the definition of ˜γ = 2 π γt √KK2l in the same equation, we get γ √KK2 = π 2 l t θ0 − α 2 sin � α 2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (S9) Thus, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (S9) allows us to estimate γ √KK2 in the simulation for different l, t, and θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' In the simulation, we consider a system of size 250 × 250 × Lz with Lz ∈ [15 − 25] and the separation between the defect centers l ∈ [40 − 200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Note that thickness t = Lz − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We find that γ √KK2 varies within a broad range for different l, t, and θ0 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' As discussed in the main text, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (S9) is valid when the line defect forms a long horizontal section in the bulk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=', the lateral span (αR) of the line defect in the bulk should be much larger than the typical length scale γ √KK2 t 12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Estimation of the parameter γ √KK2 in the simulation obtained by analyzing the circular arc-type defect configurations for a given θ0 and t with different values of l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The x-axis is in logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' The average value of the parameter γ √KK2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='1 (horizontal dashed line in the plot) is obtained from the mean of all γ √KK2 for ˜x = 2 π t l sin( α 2 ) α 2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Insets show the defect configurations from the side view (x-z plane where z-axis is along the thickness of the system) for two sets of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Inset(I): Formation of a long horizontal line defect around the mid-plane for θ0 = 90◦, l = 145, and t = 14 which yields α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='7, and ˜γ sin � α 2 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Such a defect configuration satisfies the condition ˜γ sin � α 2 � ≪ α 2 and thus included in estimating the simulation parameter γ √KK2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Inset(II): Side view of the defect configuration for θ0 = 60◦, l = 50, and t = 24 which yields α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='2 and ˜γ sin � α 2 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Instead of forming a long horizontal section around the mid-plane, the line defect changes continuously across the thickness of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' set by the line tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Thus, to estimate γ √KK2 from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' (S9) we need to consider only those defect configurations for which γ √KK2 t ≪ αR ⇒ 2 π t l γ √KK2 sin �α 2 � = ˜γ sin �α 2 � ≪ α 2 , (S10) where, we have used R = l 2 sin( α 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Thus, to estimate an optimal value of γ √KK2 in the simulation, we need to consider only those defect configurations for which ˜x = 2 π t l sin( α 2 ) α 2 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' S3 shows the dependence of γ √KK2 on ˜x for different t and θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' To obtain an average value of γ √KK2 , we consider the mean of all γ √KK2 for ˜x ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='1 (vertical dashed line in the plot) which yields γ √KK2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='1 (horizontal dashed line in the plot) for the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' As a self-consistency check, we compute ˜γ sin � α 2 � = 2 π t l γ √KK2 sin � α 2 � for all the defect configurations considered in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' S3 and find that defect configurations with ˜x ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='1 satisfy the condition ˜γ sin � α 2 � ≪ α 2 (for example see insets of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' We find that a defect configuration with ˜x ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='1 typically forms a long horizontal section around the mid-plane as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' S3(I) for a particular set of parameters (θ0 = 90◦, l = 145, and t = 14) while a defect configuration which does not form a horizontal section in the bulk typically yields ˜x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' This observation allows us to estimate γ √KK2 in the simulation by considering all configurations with ˜x ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' COMPARISON OF SIMULATION AND EXPERIMENTAL RESULTS To compare the experimental and simulation results, we use the same value for ˜γ = 2 π γt √KK2l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' In the experiment, t and l are fixed, and we change temperature (T), which changes the value of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' In the simulation, we don’t have T directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Thus, to mimic the role of T (equivalently, γ) in the experiment, we can change t and l in the simulation in such a way that we have the same value for ˜γ in the experiment and simulation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=', we want 2 π te le � γ √KK2 � e = 2 π ts ls � γ √KK2 � s , (S11) 14 from simulation 5 (II) o = 60°,t = 20 = 120°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='t = 1 4 0o = 60°,t = 24 = 120°,t = 20 9o = 90°,t = 14 0o = 120°,t = 24 Qo = 90°,t = 20 Estimated (1 ) 9o = 90°,t = 24 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='50 Qo = 105°,t = 14 2t sin α/2 Qo = 105°,t = 20 πl(α /2)13 where the subscript s and e represent the parameters for the simulation and experiment, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Thus, to make a meaningful comparison, we need to have �l t � s = �l t � e �√KK2 γ � e � γ √KK2 � s (S12) In the experiment we have le ≈ 61µm ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='1µm and te = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='83µm ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='33µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' From the experimental results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' 3(E), we find that for a broad temperature range � γ √KK2 � e ∈ [12, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' As discussed in the previous section, we have that � γ √KK2 � s = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='1 in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' Thus, we find that in the simulation, we need to use � l t � s ∈ [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='96, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='94] which also implies ˜γ ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='35, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' For the heart-shaped pattern, we perform simulation for different l and t with ˜γ ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='35, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' In particular, we set l = 70 lattice units and use t = 13 and 15 which gives ˜γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='39 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='45, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content=' To compare the obtained defect configuration with the experiment, we choose the temperature T (see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} +page_content='3(E) in the main text) for which we have the same value of ˜γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E3T4oBgHgl3EQfkAq7/content/2301.04594v1.pdf'} diff --git a/ttE3T4oBgHgl3EQfkApM/content/tmp_files/2301.04593v1.pdf.txt b/ttE3T4oBgHgl3EQfkApM/content/tmp_files/2301.04593v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1b9ae80172b909629d2b8515f56c30a8d5d4573c --- /dev/null +++ b/ttE3T4oBgHgl3EQfkApM/content/tmp_files/2301.04593v1.pdf.txt @@ -0,0 +1,2341 @@ +Draft version January 12, 2023 +Typeset using LATEX twocolumn style in AASTeX63 +Inhomogeneous enrichment of radioactive nuclei in the Galaxy: Deposition of live 53Mn, 60Fe, 182Hf, +and 244Pu into deep-sea archives. Surfing the wave? +Benjamin Wehmeyer,1, 2 Andr´es Yag¨ue L´opez +,1, 3 Benoit Cˆot´e +,4, 1 Maria K. Pet˝o,1 Chiaki Kobayashi +,2 and +Maria Lugaro +1, 5, 6 +1Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, E¨otv¨os Lor´and Research Network (ELKH), +Excellence Centre of the Hungarian Academy of Sciences, +Konkoly-Thege Mikl´os ´ut 15-17, H-1121 Budapest, Hungary +2Centre for Astrophysics Research, University of Hertfordshire, +College Lane, Hatfield AL10 9AB, UK +3Computer, Computational and Statistical Sciences (CCS) Division, Center for Theoretical Astrophysics, Los Alamos National +Laboratory, Los Alamos, NM 87545, USA +4Department of Physics and Astronomy, University of Victoria, BC, V8W 2Y2, Canada +5ELTE E¨otv¨os Lor´and University, Institute of Physics, Budapest 1117, Hungary +6School of Physics and Astronomy, Monash University, VIC 3800, Australia +(Received 2022 April 28; Revised 2022 December 31; Accepted 2023 January 02) +Submitted to ApJ +ABSTRACT +While modelling the galactic chemical evolution (GCE) of stable elements provides insights to the +formation history of the Galaxy and the relative contributions of nucleosynthesis sites, modelling the +evolution of short-lived radioisotopes (SLRs) can provide supplementary timing information on recent +nucleosynthesis. To study the evolution of SLRs, we need to understand their spatial distribution. +Using a 3-dimensional GCE model, we investigated the evolution of four SLRs: +53Mn, 60Fe, 182Hf, +and 244Pu with the aim of explaining detections of recent (within the last ≈ 1 − 20 Myr) deposition +of live 53Mn, 60Fe, and 244Pu of extrasolar origin into deep-sea reservoirs. We find that core-collapse +supernovae (CCSNe) are the dominant propagation mechanism of SLRs in the Galaxy. This results +in the simultaneously arrival of these four SLRs on Earth, although they could have been produced in +different astrophysical sites, which can explain why live extrasolar 53Mn, 60Fe, and 244Pu are found +within the same, or similar, layers of deep-sea sediments. We predict that 182Hf should also be found +in such sediments at similar depths. +Keywords: astrochemistry — Galaxy: abundances — ISM: abundances — local interstellar matter — +nuclear reactions, nucleosynthesis, abundances +1. INTRODUCTION +Studying the galactic chemical evolution (GCE) of +stable elements provides insights on the formation of +the Galaxy and the nucleosynthesis processes that pro- +duced the chemical elements (e.g., Audouze & Tinsley +1976; Matteucci & Greggio 1986; Gibson et al. 2003; +Nomoto et al. 2013; Kobayashi et al. 2020a; Prantzos +et al. 2020). The abundance of short-lived radioactive +isotopes (SLRs), with half-lives of less than a few 100 +Corresponding author: Benjamin Wehmeyer +benjamin.wehmeyer@csfk.org +Myrs, instead follows the trend of star formation rate, +and is determined by the balance between their pro- +duction and decay (the steady-state equilibrium, e.g., +Clayton 1984). SLRs can be observed live in the inter- +stellar medium (ISM, e.g., Diehl et al. 2010), in Earth +deep-sea sediments accumulating in the last 10 Myr with +a constant rate (Wallner et al. 2015, 2016; Korschinek +et al. 2020; Wallner et al. 2021) and extinct in me- +teorites (Dauphas & Chaussidon 2011; Lugaro et al. +2018), and offer additional information on the timing +of processes and our Solar System. Comparing the SLR +abundances in meteorites to the abundances predicted +by GCE models can constrain the last stellar events con- +tributing SLRs to the molecular cloud in which the Solar +arXiv:2301.04593v1 [astro-ph.GA] 11 Jan 2023 + +ID2 +Wehmeyer et al. +System formed, and determine the time the Solar Sys- +tem matter remained isolated from the Galactic medium +into such molecular cloud (e.g., Clayton 1984; Meyer & +Clayton 2000; Huss et al. 2009; Lugaro et al. 2014, 2018; +Cˆot´e et al. 2019a, 2021; Trueman et al. 2022). Further- +more, comparing SLR abundances in the ISM at any +two points in time (e.g., at the time of Solar System for- +mation from meteorites and today from deep-sea sedi- +ments) to GCE models can allow us to constrain the ori- +gin of a given SLR based on differences in the event rates +of stellar processes producing such a given SLR (e.g., Ho- +tokezaka et al. 2015). In addition, the SLR abundances +are reported in deep-sea sediments with high resolution +temporal profiles (with 200 kyr sampling), which showed +live 60Fe (half-life t1/2 = 2.62 Myr) originating from +the ISM together with resolvable 244Pu (t1/2 = 80.0 +Myr) and 53Mn (t1/2 = 3.74 Myr) anomalies (e.g., Wall- +ner et al. 2015, 2016; Korschinek et al. 2020; Wallner +et al. 2021). These profiles identified at least 2 distinct +signals of SLR deposition to Earth, 2.5 ± 0.5 and at +5.4 ± 0.7 Myr ago. The temporal overlap of the 53Mn +and 60Fe signals supports the involvement of a core- +collapse supernova (CCSN) origin of these signals, while +the temporal overlap with 244Pu signals may indicate +a rare supernova event or a more complicated trans- +port history of material in the ISM (e.g., Hotokezaka +et al. 2015). +To best interpret all these observations, +we need to better understand the dependence of SLR +abundances on GCE model parameters, the temporal +and spatial stochasticity of enrichment events from the +different stellar sources, and the transport mechanism +of material in the ISM. To accommodate SLRs in more +recent GCE models, Cˆot´e et al. (2019a) derived the be- +haviour of the ratio of SLRs to their stable reference +isotopes over the lifetime of the Galaxy using the two- +zone GCE model Omega+ (Cˆot´e et al. 2018). +This +study explained and quantified the effects of galactic +inflows and outflows, delay-time distributions of enrich- +ment events from different stellar sources, the Galactic +star formation history, and the gas-to-star mass ratio, +assuming a continuous production rate of SLRs and a +homogeneous ISM. This quantification of uncertainties +on the average composition of GCE models was recently +extended to consider inhomogeneities in the ISM due to +the temporal stochasticity of stellar enrichment events. +Cˆot´e et al. (2019b) developed a Monte Carlo simula- +tion for the temporal evolution of SLRs in a local region +of the interstellar gas. The study developed a general +statistical framework to quantify the uncertainty (prob- +ability distributions) of SLR abundances based on the +stochastic delay time between star formation and en- +richment, and explained the dependence of SLR abun- +dances on the average time between enriching events, +the delay time distribution, and the mean life of the +SLRs. +Cˆot´e et al. (2019b) also quantified the proba- +bility of whether an SLR could sample only one stellar +event or whether a particular SLR abundance represents +a steady-state in the ISM, where frequent production of +nuclides is balanced out by their decay. Yag¨ue L´opez +et al. (2021) studied the abundance ratios of two SLRs +using the same statistical framework, which also depend +on the relative mean life of the SLRs. The ratios of SLRs +with similar, and short enough half-lives are largely in- +dependent of GCE model uncertainties, and therefore +are unique tools to study the nucleosynthesis at their +formation sites, provided that their production occurs +synchronously. The theoretical considerations in Yag¨ue +L´opez et al. (2021) have been successfully applied to +the 129I/247Cm ratio to understand the physical condi- +tion of the last rapid neutron capture process event that +contributed to the Solar System inventory prior to its +formation (Cˆot´e et al. 2021). The focus of the present +study is to use a full inhomogeneous GCE code to con- +sider not only temporal discretization (as done in the +papers mentioned above) but also spatial discretization +of SLRs and the role of inhomogeneities in the ISM to +develop a better understanding of transport of matter +in our Galaxy. +In particular, we aim to explain the +possibly synchronous delivery of live 53Mn, 60Fe, and +244Pu onto the ocean floor as recorded in the deep-sea +sediments deposited in the last 10 Myr. So far, it has +been difficult to draw conclusions about the impact of +these detections using GCE models that assume homo- +geneous mixing, since spatial discretization effects have +a significant impact on the abundances of the detected +SLRs. Since these problems are difficult or impossible +to study with 1-dimensional models, here we use the 3- +dimensional GCE model from Wehmeyer et al. (2019). +We simulate the spatial and temporal evolution of four +SLR abundances over the lifetime of the Galaxy: 53Mn, +60Fe, 182Hf (t1/2 = 8.90 Myr), and 244Pu to draw con- +clusions about their most recent and ongoing (within the +last Myrs) deposition to Earth. The comparison of our +model results to SLR abundances in the early Solar Sys- +tem derived from meteorites and its implications will be +published in a separate study. This paper is organized +as follows. In Section 2, we summarize the main model- +ing parameters. In Section 3.1 we discuss the abundance +evolution of the selected SLRs over the entire lifetime of +the Galaxy. In Section 3.2, we zoom in on the more re- +cent time period closer to present day, and compare our +simulation results with deep-sea sediment detections. In +Sections 4.1 and 4.2, we compare the propagation of the +different SLRs in the simulation volume and derive the + +SLRs: Surfing the wave? +3 +dominant transport mechanism for SLRs. We present a +schematic interpretation of our findings in Section 4.3, +discuss the impact of yields in Section 4.4 and provide +our conclusions in Section 5. +2. THE MODEL +For this study, we use the 3-dimensional GCE model +described in Wehmeyer et al. (2015, 2019). Below, we +recall the most important modelling assumptions, and +highlight relevant updates to the model. +2.1. General setup +A simulation cube of (2kpc)3 is divided into 403 sub- +cubes (or cells) with an edge length of 50 pc each. Dur- +ing each calculated time step of 1 Myr, the following +operations are performed. +1. Gas with primordial composition falls into the sim- +ulation volume according to the prescription of +Wehmeyer et al. (2015), which permits for a lin- +ear rise of infalling material until 2 Gyr, and then +an exponential decrease of the infall rate. When +falling into the simulation volume, the gas is ho- +mogeneously distributed among all sub-cubes. +2. The total gas mass of the simulation volume is +used to determine the number of stars to be born +based on a Schmidt law with power α = 1.5 +(Schmidt 1959; Kennicutt 1998; Larson 1991). +The mass of the new born stars is sampled from +a Salpeter initial mass function (IMF) (Salpeter +1955) with a slope of −2.35, and mass limits of +0.1 M⊙ ≤ M < 1 M⊙1 for low-mass stars (LMS), +1 M⊙ ≤ M < 10 M⊙ for intermediate-mass stars +(IMS), and 10 M⊙ ≤ M ≤ 50M⊙ for high-mass +stars (HMS). The newly born stars inherit the +chemical composition of the gas out of which they +were formed. +3. Once the number and masses of the new born stars +are known, their birth location is chosen randomly. +Due to supernova explosions (Section 2.2.3), the +gas density distribution becomes more and more +inhomogeneous with time (i.e., the density distri- +bution function steepens), and preference of star +formation is given to cells with higher gas den- +sities due to the Schmidt law. +To prevent mis- +sampling of the IMF, only cells containing at least +50 M⊙ of gas are permitted to form stars. This +constraint is also a limiting factor to increase the +1 Concerning star masses, we refer to the zero age main sequence +(ZAMS) mass of the star throughout this manuscript. +resolution of the sub-cells: every time step, a suffi- +ciently large number of sub-cells has to be available +for star formation, i.e., has to fulfil this minimum +mass requirement. +If the resolution of the sub- +cells was increased, not enough sub-cells available +for star formation would be found during a time +step. This would alter the star formation rate and +lead to other problems (e.g., with the applicability +of the model on the GCE of α-elements). Decreas- +ing the spatial resolution (i.e., using larger sub-cell +size) is instead feasible and we tested the depen- +dence of our model on this in Appendix A: We +find that decreasing the sub-cell resolution results +in the abundance spectrum getting smaller (i.e., +converging towards a line), which resembles rather +a one-zone model behavior. +4. The lifetime for every newly born star is calcu- +lated using the formula by the Geneva group (see +Schaller et al. 1992; Schaerer et al. 1993a,b; Char- +bonnel et al. 1993): +log(t) = (3.79 + 0.24Z) − (3.10 + 0.35Z)log(M) ++ (0.74 + 0.11Z)log2(M), +(1) +where t is the expected lifetime of a star in Myr, +Z is the metallicity with respect to solar, and M +the stellar mass in solar masses. +5. If a time step contains stars that have reached the +end of their lifetime, stellar death is simulated fol- +lowing the description detailed in the next section. +2.2. Stellar deaths +2.2.1. LMS and IMS +During their lifetime, LMS & IMS add significant +amounts of C and N to the galactic inventory (e.g., +Kobayashi et al. 2011). +They do not reach burning +stages more advanced than He burning, and thus do not +contribute to iron-group elements significantly, however +they produce slow neutron capture (s-process) elements +such as Sr, Y, Zr, Ba, La, Ce, and Pb, as well as Hf of +interest here during their thermally-pulsing asymptotic +giant branch (AGB) phase (e.g., K¨appeler et al. 2011; +Bisterzo et al. 2014; Kobayashi et al. 2020a). +When +LMS & IMS die, they eject some of their initial abun- +dances, plus their nucleosynthesis products via stellar +winds, except for the fraction that remains locked in the +degenerate core. Their death may result in a planetary +nebula, and then a white dwarf remnant, and it is far less +violent/energetic compared to the death of a HMS be- +cause it is wind-driven instead of explosion-driven. The + +4 +Wehmeyer et al. +ejecta of IMS typically pollute only volumes with di- +ameters of the order of light years (e.g., the Cat’s eye +nebula NGC 6543 has a radius of 0.1 light year, Reed +et al. 1999, and the Helix nebula NGC 7293 a radius +of 1.43 light years, O’Dell et al. 2004). +LMS & IMS +have two main functions in our simulation: to lock up +gas during their lifetime, and to produce heavy isotopes +via the s-process, including 182Hf. These isotopes are +injected at the location of the source, since the energy +injection by the site is negligible. +2.2.2. Thermonuclear supernovae +Since many stars are born in binary systems (e.g., +Duchˆene & Kraus 2013), a significant fraction of IMS +interact with a companion and undergo a supernova of +type Ia (SNIa), which are the dominant source of Fe +in the galactic disk (e.g., Matteucci & Greggio 1986). +To include SNeIa, we use the analytical prescription of +Greggio (2005), which reduces all stellar and binary evo- +lution parameters to the factor PSNIa = 6 · 10−3, repre- +senting the probability of an IMS to be born in a system +that fulfills all necessary prerequisites to later end up in +a SNIa. When the system has reached the end of its +lifetime, we eject stable isotopes in the amounts calcu- +lated by Iwamoto et al. (1999, model CDD2), together +with 10−4 M⊙ of 53Mn, at the same location (which is in +agreement with, e.g., Seitenzahl et al. 2013; Kobayashi +et al. 2020b). +When a supernova explosion occurs, a +shock wave pushes the ejecta into the ISM; we model +this by moving the gas mass of the inner cells into a +shell, with an enclosed (pre-explosion) mass of 5 · 104 +M⊙. This mass corresponds to an explosion energy of +1 Bethe, according to Sedov-Taylor blast wave theory +(Ryan et al. 1996; Shigeyama & Tsujimoto 1998), which +implies that the radius of a remnant depends strongly on +the ISM density surrounding the explosion. The pushed +out gas is distributed in a chemically homogeneous shell +around the remnant, leaving behind a “bubble” in the +ISM. We follow the approach of Wehmeyer et al. (2015, +2019) and eject a constant yield of elements per SNIa, +independent of metallicity. While this approximation is +somewhat inaccurate (e.g., Timmes et al. 2003; Thiele- +mann et al. 2004; Travaglio et al. 2005; Bravo et al. +2010; Seitenzahl et al. 2013; Kobayashi et al. 2020b), it +does not strongly affect the outcomes of our simulations, +which are focused on SLRs that are mostly influenced +by solar metallicity yields. +2.2.3. HMS +HMSs experience every stellar burning stage and pro- +duce significant amounts of α- as well as iron-group el- +ements (e.g., Woosley & Weaver 1995; Kobayashi et al. +2006; Limongi & Chieffi 2018; Ritter et al. 2018). When +a HMS has reached the end of its life time, we let it ex- +plode as a CCSN, analogous to the explosion of a SNIa: +We eject stable elements (yields from Thielemann et al. +1996; Nomoto et al. 1997) and SLRs (according to Sec- +tion 2.2.5), and move the surrounding gas into a shell +around the explosion, depending on the injected kinetic +energy. +As described in Section 2.2.6, we consider a +range of different explosion energies and remnant ge- +ometries to also account for the potential effect of hy- +pernovae (e.g., Nomoto et al. 2004, 2013) and altered +CCSN remnant geometries (e.g., Fry et al. 2020). For +our purposes, HMS and their CCSNe are the exclusive +contributors of the SLR 60Fe. +2.2.4. Neutron Star Mergers +Most HMS are born in binary systems (e.g., Sana et al. +2012; Duchˆene & Kraus 2013), where both stars eventu- +ally undergo CCSN explosions, leaving behind two neu- +tron stars (NSs). There is a possibility that these two +NSs are still gravitationally bound after the two CC- +SNe (e.g., Tauris et al. 2017). +If in a suitable orbit, +these two objects reduce their separation distance via +the emission of gravitational waves, until they coalesce. +Such a merger event provides conditions to synthesize r- +process elements (e.g., Freiburghaus et al. 1999; Panov +et al. 2008; Korobkin et al. 2012; Bauswein et al. 2013; +Rosswog 2013; Rosswog et al. 2013, 2014; Wanajo et al. +2014; Eichler et al. 2015; Just et al. 2015; Vassh et al. +2019). +For the purpose of our simulation, we reduce +all the mentioned probabilities to a factor PNSM, which +represents the fraction of HMSs that fulfil all the needed +prerequisites to later undergo a merger event. This ap- +proach simplifies the detailed physics of population syn- +thesis, the explosion dynamics of CCSNe, and binary +survival probabilities, and allows us to reduce all these +details to one free parameter. We choose PNSM = 0.04 +as in Wehmeyer et al. (2019). Using a Salpeter initial +mass function with an integrated slope of −1.35, and +a standard cosmic star formation history with constant +NSM delay times (see Cˆot´e et al. 2017 for details), this +probability translates to 1.03·10−4 NSM events per unit +solar mass of stars formed, which would produce a the- +oretical gravitational wave event rate of ∼ 1800 Gpc−3 +yr−1. Although this is ∼ 2 times higher than the latest +upper limit of 810 Gpc−3 yr−1 derived by LIGO/Virgo +(Abbott et al. 2021), our main conclusions are not af- +fected by the exact choice of NSM rate (see discussion +in Section 4.2). For our purposes, NSMs are the only +source of 244Pu in the Galaxy. Analogous to the explo- +sion of a SNIa or a CCSN, when an NSM occurs in our +model, 244Pu is ejected to a spherical shell around the +source, assuming an explosion energy of 1 Bethe. + +SLRs: Surfing the wave? +5 +CCSNe mostly occur asymmetrically, resulting in a +natal kick for the newly born NS, which could lead to +offsets in the NSM locations from the original CCSNe +that formed the two NSs (e.g., van de Voort et al. 2022). +However, various studies (e.g., Beniamini & Piran 2016; +Tauris et al. 2017) have shown that the second born NS +in the majority of NS binaries in our Galaxy were formed +by much weaker explosions (possibly ultra-stripped SNe, +see Tauris et al. 2015; Mor et al. 2023) that resulted in +very weak kicks. +Indeed, Perets & Beniamini (2021) +showed that the offset locations of short GRBs — when +divided according to galaxy type — support the idea +that kicks play a subdominant role in setting binary +neutron star merger offsets. In addition, if binary NS +formation is often preceded by such weak explosions, +this could lead to the amount of swept up mass by such +explosions to be significantly smaller than obtained for +standard CCSNe. +Further, the coalescence time of two NSs can be approx- +imated with a t−x distribution (e.g., Belczynski et al. +2016; Cˆot´e et al. 2017), which could also potentially +lead to a larger spectrum in abundances (see Cˆot´e et al. +2019b). Here, we use instead a constant coalescence time +of 108 years for newly born NSs to merge. In this re- +spect, our work will mostly highlight the “surfing” effect +on SLRs. +2.2.5. SLR sources and yields +In this work, we focus on four SLRs: +53Mn, 60Fe, +182Hf, and 244Pu. For a quick overview over these iso- +topes’ origins in our model, and their half-lives, see Ta- +ble 1. The deep-sea detections of three of these isotopes +(53Mn, 60Fe, and 244Pu) can be translated into their +corresponding ISM density at time of deposition into +the deep-sea archive (see Wallner et al. 2021, Supple- +mental Material). In our simulation, the four isotopes +are produced in the four separate individual sites de- +scribed above exclusively: SNIa for 53Mn, CCSNe for +60Fe, IMSs for 182Hf, and NSMs for 244Pu. These asso- +ciations are reasonable in first order approximation (see +details below), furthermore, the advantage of assigning +each of these isotopes to a different site is that we are +able to use them as unique tracers of each site. Hence we +are able to study the production parameters of each site +(e.g., occurrence frequency, yields) almost completely +independent of the production parameters of the other +sites. We adopted the following yields. +• For 53Mn, 10−4 M⊙ is ejected per SNIa. The cal- +culations of Seitenzahl et al. (2013) resulted in a +range of 3.06 · 10−5 M⊙ up to 3.95 · 10−4 M⊙ of +53Mn produced by a SNIa, so our chosen value +is well within these limits. +It is important to +note that the Seitenzahl et al. (2013) results were +obtained using Chandra-mass models, while our +double-degenerate models would probably not pro- +duce as much 53Mn, which would lead to a possible +over-estimation of 53Mn in our model. Although +53Mn can in principle also be produced by CC- +SNe (Lugaro et al. 2016), the expected yields are +rather low and the production is not necessarily +efficient around solar metallicities in the galaxy +(e.g., Kobayashi et al. 2015, 2020a). Hence, we as- +sume that no 53Mn is ejected from CCSNe in our +model, and test a simplified approach where 53Mn +is ejected by SNIa exclusively. +• For 60Fe from CCSNe, we used progenitor mass- +dependent yields from the solar metallicity mod- +els by Limongi & Chieffi (2006). +60Fe might also +be produced in electron-capture supernovae (e.g., +Wanajo 2013); at solar metallicity, however, the +predicted rate for this site is expected to be low +(∼ 1 − 5%) in comparison to CCSNe (e.g., Poe- +larends et al. 2008; Doherty et al. 2015; Kobayashi +et al. 2020a), and the ejecta mass is ∼ 100 times +lower than that of CCSNe. Therefore, we do not +consider it as a source of 60Fe in our model. +• For 182Hf from IMSs, we used progenitor mass- +dependent yields from the solar metallicity mod- +els by Lugaro et al. (2014). +182Hf is also pro- +duced in NSMs, but since their occurrence fre- +quency is much lower than IMSs, they are far less +relevant for the overall production of 182Hf, and we +thus simplify the model to produce 182Hf only in +IMSs. Further, this simplification allows us to test +whether 182Hf would still arrive in deep-sea sedi- +ments conjointly with the CCSN produced 60Fe +and the NSM produced 244Pu, even though it is +not produced by either of them in our model. +• The yields for 244Pu from NSMs are highly un- +certain. If we assume the order of 10−2 M⊙ to- +tal mass ejection by a single NSM (e.g., Korobkin +et al. 2012), and a mass fraction of 10−6 for 244Pu +(which is within the range predicted by Eichler +et al. 2015 using different fission fragment and nu- +clear mass models), we obtain 10−8 M⊙ +244Pu +ejected by a single NSM, the value used for our +simulations. +Black hole – neutron star mergers +might also produce 244Pu if the mass of the black +hole is low enough, or its spin is sufficiently strong, +however, this site is probably more relevant at +early galactic stages. At later galactic stages, the +occurrence rate of NSMs likely exceeds the oc- +currence rate of black hole – neutron star merg- + +6 +Wehmeyer et al. +Isotope +Half-life t1/2 +Source +53Mn +3.74 Myr +SNIa +60Fe +2.62 Myr +CCSN +182Hf +8.90 Myr +IMS +244Pu +80.0 Myr +NSM +Table 1. Overview of the isotopes, their half-lives, and their +sources in our model. See Section 2.2.5 for details. +ers (e.g., Wehmeyer et al. 2019). This is consis- +tent with population synthesis models which typ- +ically predict that NSMs occur more frequently +than black hole – neutron star mergers at high +metallicities (e.g., Dominik et al. 2012; Chruslin- +ska et al. 2019). +Hence, in our simulations, we +assume that 244Pu is dominated by NSMs. In this +work, we omit also other possible r-process sites +such as magnetorotationally driven CCSNe (e.g., +Winteler et al. 2012; Nishimura et al. 2017; M¨osta +et al. 2018; Reichert et al. 2021) and rare, peculiar +CCSNe, e.g., quark-hadron phase transition CC- +SNe (Fischer et al. 2020), as these sites still lack +observational confirmation. +We comment on the dependence of the results on CCSN, +IMS, and NSM yields in Section 4.4. +2.2.6. Supernova ejecta dynamics +To estimate the influence of hypernovae, featuring +substantially higher explosion energies than regular CC- +SNe (e.g., Nomoto et al. 2004, 2013), and the effect +of a varied CCSN bubble remnant geometry and SLR +distribution within the explosion shell due to magneto- +hydrodynamical effects (e.g., Fry et al. 2020), we set +up four different scenarios (Table 2) to study the im- +plications of different assumptions for the interaction of +CCSN ejecta with the ISM. +1. Standard case (as described in Section 2.2.3). All +CCSNe explode with a kinetic energy of 1 Bethe, +and therefore pollute 5·104 M⊙ of ISM. We assume +that all SLRs are deposited on a chemically well +mixed shell located on the edge of the blast wave. +2. Increased explosion energy case (hypernova model, +HN). All CCSNe have an increased explosion en- +ergy and pollute 2 · 105 M⊙ of ISM, this estimates +the effect on the SLR abundance evolution if all +CCSNe exploded as hypernovae. The remnant ge- +ometry is the same as in the standard case. +3. Modified geometry case (PINBALL). To estimate +the impact of a potential “pinball model”-style +remnant geometry (Fry et al. 2020), where SLRs +are reflected backwards towards the center of the +explosion after the remnant bubble has halted, all +CCSNe explode with a kinetic energy of 1 Bethe, +but we assume that 1% of the swept-up ISM (and +therefore SLRs) contained therein will remain ho- +mogeneously distributed inside the explosion bub- +ble. +4. Combination of increased explosion energy and +modified geometry (HN PINBALL). As a combi- +nation of models HN and PINBALL, all CCSNe +pollute 2 · 105 M⊙ of ISM and 1% of the swept- +up ISM are distributed homogeneously inside the +explosion bubble. +Once these values/scenarios are chosen at the beginning +of a simulation, they remain constant throughout the en- +tire run. This is a simplification because supernova (SN) +explosion radii are sound speed dependent, and thus +also dependent on the local density (e.g., Chamandy +& Shukurov 2020), which in a galaxy strongly fluctu- +ates, especially in its earlier evolution stages. +How- +ever, for our modelling, we use a Sedov-Taylor approach, +which simplifies these assumptions to an almost con- +stant swept-up mass mostly determined by the CCSN +explosion energy (e.g., Shigeyama & Tsujimoto 1998, +who employed a constant sound speed of 10km/s), this +swept-up mass parameter remains constant during a run +once it is chosen in the beginning. This approach might +underestimate the spectrum of the SLR densities, espe- +cially at earlier galactic stages, but we chose to keep this +swept-up mass parameter constant to be able to focus +on identifying the dependence of SLR density on mixing +caused by explosions. Further, our approach limits the +time resolution of the model. We deliberately chose a +time step size of 1 Myr because this allows us to sim- +plify all thermo- and hydrodynamic processes into the +value of the swept-up mass. One Myr after explosion, +the shock front has stopped, and the ejected and swept- +up material has cooled down sufficiently to be available +for star formation in the following time step. With bet- +ter time resolution (with shorter time steps), the ex- +plosion ejecta and the swept-up material would have +to be closely followed, which would require a detailed +thermo- and hydro-dynamic modeling, not available in +our model. We tested, instead, the dependence of our +model on a decreased rather than increased time resolu- +tion (i.e., longer time steps) in Appendix A. +3. RESULTS +In this section, we report on the evolution of SLRs over +the lifetime of the Galaxy (Section 3.1), and compare our +results to deep-sea sediment abundances (Section 3.2). + +SLRs: Surfing the wave? +7 +Model name +ISM polluted +Remnant geometry +Standard +5 · 104 M⊙ +Standard +HN +2 · 105 M⊙ +Standard +PINBALL +5 · 104 M⊙ +PINBALL model +HN PINBALL +2 · 105 M⊙ +PINBALL model +Table 2. Overview of the different models. The left column +states the model name, the middle column how much ISM +is polluted by a single CCSN, and the third column how +the swept-up material is distributed after the event. +See +Section 2.2.6 for details. +Model name +t +53Mn +1/100 +t +60Fe +1/100 +t +182Hf +1/100 +t +244Pu +1/100 +Standard +1790 +254 +585 +432 +HN +138 +192 +480 +370 +PINBALL +192 +228 +580 +426 +HN PINBALL +152 +194 +483 +360 +Table 3. Delay time (tSLR +1/100, in Myr) for each SLR between +the onset of the simulation and the time when the median of +the abundance of a given SLR reaches 1% of its maximum +within the first 3 Gyr. +3.1. Evolution of SLRs +At every time step, we store and analyse the gas con- +tents and the SLR abundances in every cell. Figure 1 +shows the statistical distribution of the abundances of +the four isotopes over the lifetime of the Galaxy for the +four cases described in Section 2.2.6. Overall, the ISM +densities of the four SLRs follow the galactic star forma- +tion rate with an early rise and slow decrease, although +each of them shows a different shift to later times relative +to the peak of the star formation rate. The magnitude of +these shifts (where we define tSLR +1/100 as the delay between +the onset of the simulation and the time when the me- +dian of the abundance of a given SLR reaches 1% of its +maximum within the first 3 Gyr) is linked to the delay +time between the formation of the progenitor stars and +the enrichment events. We have summarized the values +of tSLR +1/100 for each model in Table 3. +In our simulations, CCSNe (ejecting 60Fe) have the +shortest delay times, because they are produced from +HMSs with the shortest life times. +This leads to the +60Fe curve having the smallest shift towards later times +(t +60Fe +1/100 = 254 Myr). +244Pu has the second lowest de- +lay time(t +244Pu +1/100 = 432 Myr), since this SLR is ejected +as soon as two short-living HMSs have died and the +two resulting NSs have spiraled inwards towards their +common center of mass for the coalescence time. 182Hf +has a longer delay time than 244Pu (t +182Hf +1/100 = 585 Myr), +because IMSs stars live longer than HMSs plus our as- +sumed coalescence time. Since we require two IMSs in a +binary system to have reached the end of their lives for a +SNIa to occur, the time scale to produce the SNIa ejecta +will always be longer than the lifetime of the secondary +star in the binary system. Thus, 53Mn has the largest +shift of all SLRs (t +53Mn +1/100 = 1.79 Gy). +Because of the large variations of gas contents and SLR +abundances among the cells, we present the SLR densi- +ties in the cells for every time step as statistical distri- +butions, where the shaded areas represent 100% (light +grey), 95% (dark grey) and 68% (black) of the SLR den- +sities. For 60Fe, the 68% cell-to-cell fluctuation span two +orders of magnitude at almost all times. The spectrum +of possible 60Fe densities is the most narrow right after +the time of highest star formation (at the point of high- +est number of HMS deaths) at t≈ 3.5 Gy, and broadens +with later times. The reason why many stellar deaths +lead to a narrower spectrum in abundances stems from +the fact that the SLR has less time to decay between +enrichment events, therefore the minimum abundance +value immediately before a subsequent nucleosynthesis +event is closer to the maximum value of that abundance +just after the nucleosynthesis event, as compared to a +case when stellar deaths are more apart in time. In this +latter case, the SLR has much more time to decay before +the subsequent injection of that SLR into the ISM, and +hence decays to lower values than in the former case. A +secondary effect is that if more stars die in a given time +span, there is also more production of 60Fe, which leads +to a higher overall abundance of 60Fe. A corresponding +behavior can also be observed for the other SLRs, at +the respective points in time when the highest number +of stars contribute to each SLR. +We show the implications of the four shock remnant +models introduced in Section 2.2.6 in Figure 1: Stronger +explosions (HN model) lead to a larger spectrum of SLR +density. This effect can be well observed when compar- +ing the difference in e.g., the 68% band of 60Fe between +the standard and the HN model. In the latter, more +cells are affected by a single SN explosion. +Because +of the larger radius of the remnant in the HN model, +there are more cells inside the remnant, and these are +cleared of their SLR content, which leads to a larger +number of lower SLR density cells in every time step. +For 244Pu, this also leads to a significantly lower density +median, since NSMs are very rare, the additional num- +ber of cells with low 244Pu content strongly affects the +median. In the HN model, instead, more cells are pol- +luted per time step and thus the mixing is more efficient. +Therefore, also the delay for the SLRs to approach their +steady-state value is lower than in the Standard model +(t +53Mn +1/100 = 138 Myr, t +60Fe +1/100 = 192 Myr, t +182Hf +1/100 = 480 Myr, + +8 +Wehmeyer et al. +10 +11 +10 +7 +Mn-53 +10 +8 +10 +6 +Fe-60 +10 +11 +10 +8 +Hf-182 +0 +2 +4 +6 +8 +10 +12 +14 +Time in Gyr +10 +14 +10 +12 +Pu-244 +(a) Standard model +10 +11 +10 +7 +Mn-53 +10 +8 +10 +6 +Fe-60 +10 +11 +10 +8 +Hf-182 +0 +2 +4 +6 +8 +10 +12 +14 +Time in Gyr +10 +14 +10 +12 +Pu-244 +(b) HN model +10 +11 +10 +7 +Mn-53 +10 +8 +10 +6 +Fe-60 +10 +11 +10 +8 +Hf-182 +0 +2 +4 +6 +8 +10 +12 +14 +Time in Gyr +10 +14 +10 +12 +Pu-244 +(c) PINBALL model +10 +11 +10 +7 +Mn-53 +10 +8 +10 +6 +Fe-60 +10 +11 +10 +8 +Hf-182 +0 +2 +4 +6 +8 +10 +12 +14 +Time in Gyr +10 +14 +10 +12 +Pu-244 +(d) HN PINBALL model +Figure 1. SLR densities in g/cm3 in the simulation volume, where the distribution at each time step represents the spatial +distribution of SLR densities in the (40)3 sub-cells, for the four cases of Table 1, as indicated below each panel. The median is +shown as a solid yellow line. The black (dark grey, light grey) shaded areas represent the 68% (95%, 100%) distribution. + +SLRs: Surfing the wave? +9 +t +244Pu +1/100 = 370 Myr). +The PINBALL model case leads to ejecta distributed +more homogeneously inside the supernova bubble, how- +ever, the effect appears to influence the SLR density +statistics only marginally. The delays of the SLRs to +approach their steady-state value (t +53Mn +1/100 = 192 Myr, +t +60Fe +1/100 = 228 Myr, t +182Hf +1/100 = 580 Myr, t +244Pu +1/100 = 426 Myr) +are all longer than in the HN model case, but show only a +slight correction when compared to the Standard model. +Even when the fraction of reflected material is increased +to a much higher value (50 %), it only affects SLR densi- +ties locally in sub-cells, but not their overall abundance +statistics, as described in Appendix B. +In the combined case with both PINBALL geometry and +high explosion energy (HN PINBALL model), instead +the flattening effect of the more homogeneous distribu- +tion of SLRs (due to the PINBALL treatment) combined +with the larger explosion bubbles (due to the HN treat- +ment) becomes more prominent. In this model, ejecta +from every nucleosynthesis site are distributed through- +out the entire, larger explosion bubble, which results in +more cells being polluted by an SLR per nucleosynthe- +sis event than in any other model. For the more rare +sites (SNeIa and NSMs producing 53Mn and 244Pu, re- +spectively), this also leads to larger fluctuations in the +median between time steps (compared to the standard +model), since the SLRs produced in these rare events are +distributed throughout the volume much quicker. This +behavior can also be observed in the delay times for the +SLRs to approach the steady-state value (t +53Mn +1/100 = 152 +Myr, t +60Fe +1/100 = 194 Myr, t +182Hf +1/100 = 483 Myr, t +244Pu +1/100 = 360 +Myr). +This means that the ISM homogenizes faster, +eliminating quickly the spikes in SLR densities gener- +ated by a single rare nucleosynthesis event. +In other +words, since this last model homogenizes the ISM so +quickly, the median behaves more like a one-zone-model +instead of a single cell in the simulation volume, the +latter will be discussed in the following section. +3.2. Recent evolution +Figure 2 shows a zoom-in on the evolution of the SLRs, +closer to the current day. To the figure, we added the +ISM densities inferred from deep-sea sediment detec- +tions from Wallner et al. (2016), Korschinek et al. (2020) +and Wallner et al. (2021), as well as the SLR density +evolution in one of the sub-cells of the simulation which +best fits the ISM densities derived from the deep-sea de- +tections. We introduced a factor λ for a vertical shift for +all derived ISM densities of every isotope of the deep-sea +detections in every model for fitting, since we were more +interested in fitting the shape of the detection curves, +rather than the actual values. Further, we introduced a +Model name +λ53Mn +λ60Fe +λ244Pu +∆t (Myr) +Standard +4.07 +0.211 +16.5 +174.63 +HN +4.15 +0.125 +1.01 +57.63 +PINBALL +4.24 +0.02 +0.0439 +431.63 +HN PINBALL +4.23 +0.476 +0.669 +240.63 +Table 4. Vertical (λ) and horizontal (time) shift (∆t) factors +for the vertical and the horizontal shift of the ISM densities +of the four isotopes as inferred by their deep-sea detections +used for our fitting. +factor ∆t to account for a horizontal (time) shift in the +ISM densities. The values for the two factors for each +model can be found in Table 4. We test the possibility +for a sub-cell to fit the detection data in Appendix C. +First, we consider the top left panel in Figure 2 (the +standard case). If we follow the density evolution of the +best-fitting cell, the effects of radioactive decay of the +SLRs in that cell are visible (e.g., around 13330 Myr +in the 60Fe evolution). +However, another effect often +dominates: since we assume that SN ejecta behave in +a Sedov-Taylor-like expansion pattern, any stellar ex- +plosion clears its neighboring cells completely of their +contents as the blast wave travels through those cells. +Together with the gas contents, also the SLRs are car- +ried away from those cells to pile up in the SN remnant +shell. This results in a very low or even zero gas and +isotope content of these cells, which is the main cause of +the major discontinuities seen in the green lines in the +figure. +Further, the sudden increases in the green lines can also +be explained by blast waves. Again, all cells within the +radius of a shock wave are emptied (or almost emptied in +the PINBALL models), and their gas and isotope con- +tent from before the explosion is deposited on a shell +around the explosion. If an observed cell is located on +the shell around an explosion, its gas and isotope con- +tent thus strongly increases, which leads to an upward +jump in isotope abundances in the cell. +This effect of blast waves is observed in all SLRs in our +models and might explain the data of Wallner et al. +(2021), who found that 60Fe and 244Pu arrive on Earth +synchronously. If in our model 60Fe suddenly arrives in +an ISM cell due to a nearby stellar explosion, it is very +likely that also some 244Pu arrives synchronously in that +cell. +In the top right panel of Figure 2 (increased explosion +energy), the sweep-up effect discussed above is more fre- +quent as demonstrated by the fact that the lower 68% +confidence band has a lower boundary), which happens +because more space (i.e., more simulation cells) is af- +fected by a each CCSN, therefore, the clearing out ef- + +10 +Wehmeyer et al. +fect occurs more frequently than in the standard case. +For the PINBALL model case (bottom left panel of Fig- +ure 2), we observe the same as in the previous Section, +that this model does not have a strong effect on the +statistics of the SLR densities. A difference to the stan- +dard model, however, can be observed here when con- +sidering the evolution in a single cell (green line), which +is more variable than in the standard model case (when +the radioactive decay effect is subtracted from the evolu- +tion of the line). This is because more cells are affected +per CCSN explosion. This leads to all cells being af- +fected more often in a given time interval, which leads +to the density in each cell varying at a higher frequency +as compared to the standard case. +In the lower right panel of Figure 2 (HN PINBALL +model), the evolution of the single cell (green line) oscil- +lates even stronger in comparison to all the other mod- +els, because the most cells are affected by each single +CCSN. This results in any given cell being affected by +SLR density changes much more often in a given time +span, which results in the density curve oscillating at the +highest frequency. In this model, the SLR density in a +given cell is completely determined by external events, +rather than by radioactive decay, whereas decay is the +dominant effect in the standard model. +In all four models, it is possible to find cells that repro- +duce the shape of the measurement data reasonably well. +The underlying reason for the observation that all four +isotopes show a synchronous increase in a given cell is +that CCSNe are the dominant propagation mechanism +for all SLRs. Even if e.g., an NSM (ejecting 244Pu) hap- +pened far away from a given location in the ISM, CCSNe +will be responsible for the ejecta propagation, as we will +see in the following Sections. +4. DISCUSSION +In this section, we analyze the pollution intervals to +a given parcel of the ISM expected for different SLRs +(Section 4.1), and the propagation of SLRs in the ISM +(Section 4.2). +We suggest a schematic interpretation +of the findings of Sections 4.1 and 4.2 in Section 4.3. +Further, we test the impact of varying yields and event +occurrence frequencies on our results in Section 4.4. +4.1. Pollution intervals +We have noted in the previous section that the in- +creases in different SLR densities often coincide (green +lines in Figure 2), although they are produced by differ- +ent nucleosynthesis sites with very distinct occurrence +frequencies. To investigate the cause of this behavior, +we examine how often a given gas cell is polluted with +a given SLR (or equivalently, how much time elapses +between two consecutive pollution events in that cell), +and compare this to the time that elapses between two +consecutive positive changes in the gas mass of the cell2. +To exclude infalling gas (see Section 2.1) from triggering +a positive gas mass change event in a cell, we set a fidu- +cial threshold for such a positive change in gas to 100 +solar masses of gas or more (which is higher than the +highest amount of infalling gas at all times). We mon- +itored the time between two consecutive SLR and gas +mass enrichment events, δSLR (i.e., δ53Mn, δ60Fe, δ182Hf, +or δ244Pu), and δmass, respectively, in every cell. Then, +we calculate the average of each of these pollution times +( < δSLR > for every SLR) in each cell over the entire +lifetime of the Galaxy, and generate a histogram over +the averages of all cells, which is shown in Figure 3. All +the histogram curves show a Gaussian behavior (due to +the central limit theorem) and overlap with each other +at large portions. This means that all the SLRs arrive +at a given location at a very similar frequency, and in +conjunction with every arrival of gas. This is further +evidence that the probability of all SLRs to arrive syn- +chronously at a given location is very high. +4.2. The dominant propagation mechanism +To further examine the cause of the synchronous ar- +rival of SLRs, we explore the contributions of the dif- +ferent nucleosynthesis sites towards the propagation of +matter throughout our simulation volume and the age of +the Galaxy. For every nucleosynthesis event, we monitor +the coordinates of all the cells that are affected. From +this, we calculate the time that elapses between two con- +secutive events that affect each given cell for each type +of explosion, δsite (i.e., δCCSN, δNSM, and δSNIa). +In +other words, we examine the time that elapses between +the cell being affected by the shock fronts of two dif- +ferent explosions of the same type of event. We then +calculate the mean of these elapsed times for each ex- +ploding nucleosynthesis site, < δCCSN >, < δNSM >, +and < δSNIa >, and plot these medians in Figure 4, +together with < δmass > from Figure 3. None of the +< δsite > distributions is congruent with < δmass >, but +< δCCSN > is the closest to < δmass >, which means +that CCSNe affect the most cells, relative to the other +types of explosions. +We therefore conclude that CC- +SNe are the dominant propagation mechanism of mass. +When considering that < δmass >≈< δSLR > from Sec- +tion 4.1, we further conclude that CCSNe are the dom- +inant propagation mechanism of all SLRs in our model. +This means that the ejecta from all the nucleosynthesis +2 We examine this for the standard model only, although the con- +clusions drawn could be extended to the other models as well. + +SLRs: Surfing the wave? +11 +(a) Standard model +(b) HN model +(c) PINBALL model +(d) HN PINBALL model +Figure 2. Same as Figure 1, but zoomed-in to ±10 Myr around the respective ∆t. In addition to what is shown in Figure 1, +ISM densities for 53Mn, 60Fe, and 244Pu, inferred from deep-sea sediment detections are shown as red symbols with error bars +(shifted by factors λ and ∆t), and the abundance evolution of the one simulation cell in each model that best fits these deep-sea +detection ISM abundances is shown in green. +sites included in our model travel as dynamically as the +ejecta of CCSNe, even if their occurrence rate is much +lower than that of CCSNe. +4.3. Schematic interpretation +As seen in the previous sections, although each of the +SLRs is produced in a different nucleosynthesis site, the +density of all SLRs most likely increase synchronously +in each given cell. This is consistent with the observa- +tion of the three SLRs detected in deep-sea sediments, +53Mn (Korschinek et al. 2020), 60Fe (Wallner et al. 2016) +and 244Pu (Wallner et al. 2021). Especially for the latter +two, it has been shown in Wallner et al. (2021) that they +are both deposited in deep-sea sediments of the same +depth, which leads to the conclusion that they arrived +synchronously, even if they could have been produced in +separate, independent nucleosynthesis sites. This could +be interpreted in the following way, for the example of +60Fe from HMS and 244Pu from NSMs (Figure 5). An +NSM explodes in an area of the Galaxy that is relatively +close to the Solar System, however, it is separated from +it by a region of high density, which could have been cre- +ated by, e.g., a previous CCSN or a superbubble. Once +the NSM explodes, the shock wave of the explosion, and +hence the ejecta, will stop at the high density gas region, +so none of the ejecta of this explosion event reaches the +Solar System. If there is a HMS inside the region be- +hind the shock wave (i.e., within the explosion bubble) +of that NSM, that star will end its life in a CCSN. Once +it explodes, it will create a second shock wave, carrying + +10-6 +3 +5 +10-7 +G-u +10-8 +10-9 +> +10-10 +10-11 +10-6 +510-7 +L +10-9 +10-9 +182 +Hf-1 +10-11 +Pu-244 +10-12 +10-13 +10-14 +13.305 +13.310 +13.315 +13.320 +13.325 +13.330 +time [Gyr]10-6 +10-7 +5 +10-8 +-u +10-9 +> +10-10 +10-11 +10-6 +0 +10-7 +e +10-8 +L +10-9 +2 +10-9 +8 +-H +10-11 +Pu-244 +10-12 +10-13 +10-14 +13.425 +13.430 +13.435 +13.440 +13.445 +13.450 +time [Gyr]10-7 +3 +5 +10-9 +Mn-5 +10-11 +10-13 +10-7 +e 10-8 +10-9 +2 +Hf-182 +10-10 +10-12 +10-11 +Pu-244 +10-13 +10-15 +13.050 +13.055 +13.060 +13.065 +13.070 +13.075 +time [Gyr]10-6 +3 +10-7 +5 +10-8 +10-9 +10-10 +> +10-11 +10-6 +10-7 +10-8 +e +F 10-9 +2 +8 +10-10 +10-12 +Pu-244 +10-12 +10-13 +10-14 +13.215 +13.220 +13.225 +13.230 +13.235 +13.240 +time [Gyr]12 +Wehmeyer et al. +9.5 +10.0 +10.5 +11.0 +11.5 +12.0 +Myr +100 +101 +102 +103 +104 +N +Figure 3. Distribution of the means of the elapsed time of +arrival of the different focus SLRs in all cells: < δ53Mn > +(red), < δ60Fe > (yellow), < δ182Hf > (green), < δ244Pu > +(magenta), and < δmass > (blue). +Figure 4. Cell distribution of the means of the elapsed time +between two consecutive events of the same kind: < δCCSN > +(blue), < δNSM > (yellow), < δSNIa > (green), and < δmass > +(magenta, same as blue color curve in Figure 3). +with it its CCSN ejecta. This second shock wave might +just have enough energy to push the high density region +(where the NSM ejecta were stopped earlier) towards the +Solar System. Since the NSM ejecta are still conserved +in this stopped high density area which is now pushed +further by the subsequent CCSN, both ejecta from NSM +and CCSN are conjointly pushed towards the Solar Sys- +tem. The result is a synchronous arrival of CCSN and +NSM ejecta on Earth, hence 60Fe and 244Pu in Earth’s +inventory increase simultaneously, as observed. In other +words, NSM and other ejecta could “surf the wave” of +the CCSN explosion shock fronts. Since we have seen +in Sections 4.1 and 4.2 that all SLRs show a very sim- +ilar < δSLR >, this propagation argument can also be +made for the other rare nucleosynthesis site ejecta, 53Mn +from SNIa, and also for the more locally deposited 182Hf +from IMSs. In order to determine how often this “surf- +ing” effect occurs, we calculate the fraction of CCSNe +that sweep up significant amounts of SLRs in their blast +wave: 97.35% / 99.94% / 99.89% / 99.81% of all CCSNe +carry 53Mn, 60Fe, 182Hf, and 244Pu, respectively, which +means that almost all CCSNe contribute significantly +to the propagation of all SLRs. To examine how this +number is correlated with each nucleosynthesis site, we +also consider a model where we lowered the frequency +of NSMs by a factor of 100 (with PNSM = 4 · 10−4, +which would correspond to a theoretical gravitational +wave rate of ∼ 18 Gpc−3 yr−1, roughly four times be- +low the current uncertainties of the LIGO/Virgo de- +tections). +In this model, the fraction of CCSNe that +sweep up significant amounts of the NSM ejecta 244Pu +decreases to 88.15%, while the other values remain al- +most unchanged. +4.4. Effects of varying yields and frequency of the +events +To test the impact of different yields and event fre- +quencies on the results, first, we tested a case where the +mass dependency of the yields was kept, but all their +values were uniformly decreased by a factor of 10. The +result confirms that a linear change of a factor of ≈ 10 is +reflected in the whole distribution. Second, we quanti- +fied the effect of the progenitor mass-dependency of the +60Fe and 182Hf yields on the ISM evolution of the SLRs. +To this aim we changed the 60Fe and 182Hf yields from +our fiducial yields from the literature (as described in +Section 2.2), to a case where we took a constant yield +for each progenitor equal to the IMF-weighted average +over the entire ZAMS mass range. +The results are shown in Figure 6. At the end of the +simulation at time 13.475 Gyr, stellar births and deaths +are almost in equilibrium, and a comparison made at +this point allows us to minimize the impact of stochas- +tic star formation or deaths. Using the constant yields, +the 60Fe and 182Hf median increase by factor of ≈ 1.2 +and 3, respectively, at this late time of the evolution. +The overall trend however differs: in the case of 60Fe +the median calculated with the constant yields is always +above the fiducial case, while for 182Hf the two lines +cross at a time of roughly 4.5 Gyr. The reason is that +for the 182Hf ejected by IMSs, in the fiducial case the +more massive IMSs eject more of this SLR, they also die +earlier and therefore more 182Hf per IMS is ejected at +earlier galactic stages, as compared to the model where +all IMSs eject the averaged yield. At later stages, in- + +104 +mass> +< CCSN> +103 +< NSM> +< SNla> + 102 +101 +100 +101 +102 +MyrSLRs: Surfing the wave? +13 +Solar System +Bubble of higher + gas density +NSM progenitor +Solar System +NSM remnant +bubble +CCSN +progenitor +Solar System +CCSN remnant +bubble +Figure 5. Schematic interpretation of the the result that all +SLRs arrive at the Solar System conjointly. Top: The Solar +System and NSM progenitor separated from each other by +an area of high density ISM, e.g., the edges of a previous +SN bubble or a superbubble, which is created when multiple +CCSNe share the same shock front. Middle: The NSM pro- +genitor explodes and pollutes the higher density area with its +ejecta. Bottom: A CCSN explodes inside the bubble, push- +ing the NSM ejecta into the Solar System. CCSN and NSM +ejecta arrive simultaneously at the Solar System location. +stead, the lower mass IMSs become more predominant, +and in the case of constant yields, they eject more 182Hf +than in the fiducial case resulting in a higher final 182Hf +median. +The upper limit of the full abundance statistics also in- +creases for 182Hf, however, the increase is lower, ≈ 1.2, +than that of the median value. This is due to a clump- +ing effect more significantly present in the case of the +fiducial yield. The more massive IMSs that contribute +relatively more 182Hf are actually fewer than their lower- +mass counterparts, therefore their ejections are less ho- +mogeneously distributed than in the case where all IMSs +all eject the same amount of 182Hf. +For 60Fe from HMSs, the effect is weaker because the +life time of the most massive HMSs compared to the +least massive HMSs is not so different than the differ- +ence in life times of the most and least massive IMSs. +In the fiducial yields case, the most massive HMSs eject +far more 60Fe than the lower mass ones, which leads +to a larger fluctuation of abundances (e.g., larger upper +limit of the full abundance distribution) in the former +case. However, the median of abundances is lower in +the former case, because the majority of HMSs are (due +to the IMF) at the lower mass end of the HMS mass +spectrum. This leads to a lower abundance median in +this case compared to the case where all HMSs eject the +same, IMF-averaged yield of 60Fe. +As discussed in Cˆot´e et al. (2019b) and Yag¨ue L´opez +et al. (2021), the occurrence frequency of stellar events +influences how much time an SLR has to decay before a +subsequent nucleosynthesis event. Since this frequency +is governed by the rate of an event, this implies that also +the event rates play a major role for the evolution of the +SLR abundances. Although the rates of HMSs, IMSs, +and SNIa have been determined by observations, the ob- +servational error bar on the occurrence rate of NSMs as +suggested by gravitational wave detections is uncertain +within a factor of ≈10. We illustrate the effect of chang- +ing the NSM rate on 244Pu. +In Figure 7, we present the evolution of 244Pu in our fidu- +cial model versus a model with NSM occurrence rates +reduced by a factor of 10. The median of 244Pu ISM +densities is reduced by a factor of ≈ 20, i.e., it is two +times lower than the case when the yields where reduced +by a factor of 10. The reason is that a decrease in the +rate results in less cells polluted by NSM ejecta per time +step. +A decreased yield, instead, results in the same +amount of cells polluted per time step, but with a lower +244Pu density. Since the median is more sensitive to the +amount of cells being polluted than the amount of 244Pu +in the cells, the median value of the abundance is then +lower in the former case. + +14 +Wehmeyer et al. +0 +2 +4 +6 +8 +10 +12 +14 +Time in Gyr +10 +8 +10 +7 +10 +6 +Fe-60 +Figure 6. Effect of varied 182Hf (left) and 60Fe (right) yields. The solid lines represent the abundance medians and the shaded +areas the full abundance distribution. The fiducial case is in blue and the case with the constant IMF-averaged yields is in +magenta. +Figure 7. Same as Figure 6, but showing the effect of de- +creasing the occurrence frequency of NSMs by a factor of 10 +(magenta), relative to the fiducial case in blue. +5. CONCLUSIONS +We simulated the evolution of four SLRs from four +separate nucleosynthesis sites: +53Mn from SNIa, 60Fe +from CCSNe, 182Hf from IMSs, and 244Pu from NSMs, +over the lifetime of our Galaxy using a 3-dimensional +GCE model. We conclude that: +1. The integrated (over the entire simulation volume) +median ISM density of the four SLRs follows the +Galactic star formation rate over time (with an +individual delay for each SLR), but their density +at a given location fluctuates strongly. +2. 60Fe has the lowest delay since HMSs have the +shortest lifetimes. Second is 244Pu since the life- +times of two HMSs plus the coalescence time scale +is still lower than the lifetime of IMSs (ejecting +182Hf, second longest delay time). +53Mn has the +longest delay time since it always requires the +longer-lived IMS in a IMS binary to die to pro- +duce that SLR. +3. Even though the SLRs are produced at different +sites, their arrival at a given location in the ISM +often coincides, because their propagation is dom- +inantly governed by the frequency of CCSNe. +4. This explains why three different SLRs, +53Mn, +60Fe, and 244Pu, produced in three different nucle- +osynthesis sites (SNIa, HMSs, and NSMs) could be +detected in layers of similar depth in deep-sea sed- +iments, which suggest that they arrived conjointly +on Earth. Our model can robustly reproduce these +deep-sea detections. +5. Neither assuming a PINBALL model, nor that all +CCSNe explode as HNe strongly influences the +overall evolution of SLRs. +6. Reducing the occurrence frequency of NSMs re- +duces the median abundances of 244Pu. Reduced +yields reduce the median abundances proportion- +ally. Decreasing CCSN and IMS yields decreases +the median abundances of 60Fe and 182Hf. +Effects that still need to be addressed in future work are: +Those deriving from a different IMF, which would alter +the lifetimes of stars and thus change the distribution of +the pollution intervals, and from the existence of islands +of explodability of massive stars (e.g., Sukhbold et al. + +10-! +10-10 +10-12 +10-13 +0 +2 +4 +6 +8 +10 +12 +14 +Time in Gyr10-9 +10-10 +244 +Pu- +10-11 +10-12 +10-13 +0 +2 +4 +6 +8 +10 +12 +14 +Time in GyrSLRs: Surfing the wave? +15 +2016; Ertl et al. 2016; Ebinger et al. 2019; Curtis et al. +2019). The presence of failed CCSNe may reduce their +dominance in being the major propagation mechanism +for SLRs. Further, additional sources for the four SLRs +should be addressed, e.g., rare sub-types of CCSNe as a +source of 244Pu (e.g., Fischer et al. 2020). Also, further +propagation mechanisms not currently included in our +model could have an effect on the propagation of SLRs: +• Diffusion/turbulence: +Hotokezaka et al. (2015) +found that the discrepancies between early Solar +System abundance and more recent deep-sea de- +tection of 244Pu can be explained by a model that +assumes a purely diffusion/turbulence-dominated +propagation. This model was later expanded by +Beniamini & Hotokezaka (2020), who concluded +that this approach could also applied to r-process +elements. We did not include diffusion/turbulence +in our model. +If included, diffusion/turbulence +might reduce (increase) the SLR densities in those +cells with the highest (lowest) SLR abundances, +which would lead to a reduced spectrum of SLR +densities. However, these considerations would go +beyond the scope of this work, where we wanted to +highlight the “surfing” effect of SLRs. We plan to +work on a comparison between CCSN-dominated, +and diffusion-dominated propagation for SLRs in +the future. +• Superbubbles: A large bubble in the ISM can be +created by multiple CCSN explosions (e.g., Krause +et al. 2018). +The effect of superbubbles is very +difficult to estimate in our GCE model with lim- +ited spatial and time resolution. The implemen- +tation of superbubbles requires the implementa- +tion of thermodynamics/fluid dynamics (as done +in e.g., Vasileiadis et al. 2013; Fujimoto et al. +2018), but this would go beyond the scope of +this work. Nonetheless, more detailed effects that +could be expected in hydrodynamical simulations +can be estimated with our set of models with vary- +ing parameters, and all of our models show the surf +effect. +• Galactic outflows: +Galactic outflows could po- +tentially built up a reservoir of gas outside the +Galaxy that may be unaffected by GCE for a pe- +riod of time (e.g., as done with Omega+ in Cˆot´e +et al. 2018), or may be enriched differently in hy- +drodynamical simulations (Vincenzo & Kobayashi +2020). +The implications of this effect on SLR +abundances are difficult to estimate in our model. +Although SLR abundances (due to the absence +of nucleosynthesis sites increasing the SLR abun- +dances in the extragalactic reservoir) would poten- +tially decrease exponentially in that extragalactic +reservoir due to radioactive decay, gas being incor- +porated back from the reservoir into the Galaxy +might slightly enhance the abundance of SLRs in +the simulation volume. However, since our main +aim for this work was to showcase the “surfing” +effect for SLRs, we decided to use primordial in- +flow into the simulation volume as described in +Section 2.1. +Future further detection of live radioisotopes in the +deep-sea floor might provide further constraints on the +propagation mechanism of SLRs (e.g., Wang et al. +2021a,b). +This will also be of further interest for +the GCE of r-process elements, since it is yet unclear +whether the behaviour of different classes of elements +(e.g., iron group and r-process elements) is coupled or +not (Beniamini & Hotokezaka 2020). A direct compar- +ison between different or combined propagation mecha- +nisms would therefore help to further confirm or rule out +whether the propagation of SLRs and r-process elements +is CCSN- or diffusion-dominated. Our study will also be +extended towards a full cosmological zoom-in simulation +of a galaxy (Kobayashi & Nakasato 2011; Vincenzo & +Kobayashi 2020), which will allow for additional sub- +galactic-scale mixing effects and mechanisms to be ad- +dressed (Wehmeyer et al., in prep.). Additionally, our +models can be applied to study the abundances of SLRs +at the time of the formation of the Solar System. +ACKNOWLEDGEMENTS +The authors thank Anton Wallner for providing the +deep-sea measurement data relevant for this work. We +further thank Adrienne Ertel and Jesse Miller for con- +structive discussion with regard to the PINBALL model +at the 2019 JINA-CEE Frontiers meeting. This work +is supported by the ERC Consolidator Grant (Hun- +gary) funding scheme (Project RADIOSTAR, G.A. no. +724560). We also thank the COST actions “ChETEC” +(G. A. no. 16117) and “ChETEC-INFRA” (G. A. no. +101008324). +MKP received funding from the Euro- +pean Union’s Horizon 2020 research and innovation pro- +gramme under the Marie Sklodowska-Curie grant agree- +ment No 753276. +BC and BW acknowledge support +from the National Science Foundation (NSF, USA) un- +der Grant no. PHY-1430152 (JINA Center for the Evo- +lution of the Elements) and Grant no. OISE-1927130 +(IReNA). CK acknowledges funding from the UK Sci- +ence and Technology Facility Council (STFC) through +grant ST/R000905/1, & ST/V000632/1. +The work +of AYL was supported by the US Department of En- + +16 +Wehmeyer et al. +ergy through the Los Alamos National Laboratory. Los +Alamos National Laboratory is operated by Triad Na- +tional Security, LLC, for the National Nuclear Security +Administration of U.S. Department of Energy (Contract +No. 89233218CNA000001). +We thank an anonymous +referee for valuable comments that helped improving +this paper. +REFERENCES +Abbott, R., Abbott, T. D., Abraham, S., et al. 2021, ApJL, +913, L7, doi: 10.3847/2041-8213/abe949 +Audouze, J., & Tinsley, B. M. 1976, ARA&A, 14, 43, +doi: 10.1146/annurev.aa.14.090176.000355 +Bauswein, A., Goriely, S., & Janka, H. T. 2013, ApJ, 773, +78, doi: 10.1088/0004-637X/773/1/78 +Belczynski, K., Repetto, S., Holz, D. E., et al. 2016, ApJ, +819, 108, doi: 10.3847/0004-637X/819/2/108 +Beniamini, P., & Hotokezaka, K. 2020, MNRAS, 496, 1891, +doi: 10.1093/mnras/staa1690 +Beniamini, P., & Piran, T. 2016, MNRAS, 456, 4089, +doi: 10.1093/mnras/stv2903 +Bisterzo, S., Travaglio, C., Gallino, R., Wiescher, M., & +K¨appeler, F. 2014, ApJ, 787, 10, +doi: 10.1088/0004-637X/787/1/10 +Bravo, E., Dom´ınguez, I., Badenes, C., Piersanti, L., & +Straniero, O. 2010, ApJL, 711, L66, +doi: 10.1088/2041-8205/711/2/L66 +Chamandy, L., & Shukurov, A. 2020, Galaxies, 8, 56, +doi: 10.3390/galaxies8030056 +Charbonnel, C., Meynet, G., Maeder, A., Schaller, G., & +Schaerer, D. 1993, A&AS, 101, 415 +Chruslinska, M., Nelemans, G., & Belczynski, K. 2019, +MNRAS, 482, 5012, doi: 10.1093/mnras/sty3087 +Clayton, D. D. 1984, ApJ, 285, 411, doi: 10.1086/162518 +Cˆot´e, B., Belczynski, K., Fryer, C. L., et al. 2017, ApJ, 836, +230, doi: 10.3847/1538-4357/aa5c8d +Cˆot´e, B., Lugaro, M., Reifarth, R., et al. 2019a, ApJ, 878, +156, doi: 10.3847/1538-4357/ab21d1 +Cˆot´e, B., Silvia, D. W., O’Shea, B. W., Smith, B., & Wise, +J. H. 2018, ApJ, 859, 67, doi: 10.3847/1538-4357/aabe8f +Cˆot´e, B., Yag¨ue, A., Vil´agos, B., & Lugaro, M. 2019b, ApJ, +887, 213, doi: 10.3847/1538-4357/ab5a88 +Cˆot´e, B., Eichler, M., Yag¨ue L´opez, A., et al. 2021, Science, +371, 945, doi: 10.1126/science.aba1111 +Curtis, S., Ebinger, K., Fr¨ohlich, C., et al. 2019, ApJ, 870, +2, doi: 10.3847/1538-4357/aae7d2 +Dauphas, N., & Chaussidon, M. 2011, Annual Review of +Earth and Planetary Sciences, 39, 351, +doi: 10.1146/annurev-earth-040610-133428 +Diehl, R., Lang, M. G., Martin, P., et al. 2010, A&A, 522, +A51, doi: 10.1051/0004-6361/201014302 +Doherty, C. L., Gil-Pons, P., Siess, L., Lattanzio, J. C., & +Lau, H. H. B. 2015, MNRAS, 446, 2599, +doi: 10.1093/mnras/stu2180 +Dominik, M., Belczynski, K., Fryer, C., et al. 2012, ApJ, +759, 52, doi: 10.1088/0004-637X/759/1/52 +Duchˆene, G., & Kraus, A. 2013, ARA&A, 51, 269, +doi: 10.1146/annurev-astro-081710-102602 +Ebinger, K., Curtis, S., Fr¨ohlich, C., et al. 2019, ApJ, 870, +1, doi: 10.3847/1538-4357/aae7c9 +Eichler, M., Arcones, A., Kelic, A., et al. 2015, ApJ, 808, +30, doi: 10.1088/0004-637X/808/1/30 +Ertl, T., Janka, H. T., Woosley, S. E., Sukhbold, T., & +Ugliano, M. 2016, ApJ, 818, 124, +doi: 10.3847/0004-637X/818/2/124 +Fischer, T., Wu, M.-R., Wehmeyer, B., et al. 2020, ApJ, +894, 9, doi: 10.3847/1538-4357/ab86b0 +Freiburghaus, C., Rosswog, S., & Thielemann, F. K. 1999, +ApJL, 525, L121, doi: 10.1086/312343 +Fry, B. J., Fields, B. D., & Ellis, J. R. 2020, ApJ, 894, 109, +doi: 10.3847/1538-4357/ab86bf +Fujimoto, Y., Krumholz, M. R., & Tachibana, S. 2018, +MNRAS, 480, 4025, doi: 10.1093/mnras/sty2132 +Gibson, B. K., Fenner, Y., Renda, A., Kawata, D., & Lee, +H.-c. 2003, PASA, 20, 401, doi: 10.1071/AS03052 +Greggio, L. 2005, A&A, 441, 1055, +doi: 10.1051/0004-6361:20052926 +Hotokezaka, K., Piran, T., & Paul, M. 2015, Nature +Physics, 11, 1042, doi: 10.1038/nphys3574 +Huss, G. R., Meyer, B. S., Srinivasan, G., Goswami, J. N., +& Sahijpal, S. 2009, GeoCoA, 73, 4922, +doi: 10.1016/j.gca.2009.01.039 +Iwamoto, K., Brachwitz, F., Nomoto, K., et al. 1999, ApJS, +125, 439, doi: 10.1086/313278 +Just, O., Bauswein, A., Ardevol Pulpillo, R., Goriely, S., & +Janka, H. T. 2015, MNRAS, 448, 541, +doi: 10.1093/mnras/stv009 +K¨appeler, F., Gallino, R., Bisterzo, S., & Aoki, W. 2011, +Reviews of Modern Physics, 83, 157, +doi: 10.1103/RevModPhys.83.157 +Kennicutt, Robert C., J. 1998, ApJ, 498, 541, +doi: 10.1086/305588 +Kobayashi, C., Karakas, A. I., & Lugaro, M. 2020a, ApJ, +900, 179, doi: 10.3847/1538-4357/abae65 + +SLRs: Surfing the wave? +17 +Kobayashi, C., Karakas, A. I., & Umeda, H. 2011, MNRAS, +414, 3231, doi: 10.1111/j.1365-2966.2011.18621.x +Kobayashi, C., Leung, S.-C., & Nomoto, K. 2020b, ApJ, +895, 138, doi: 10.3847/1538-4357/ab8e44 +Kobayashi, C., & Nakasato, N. 2011, ApJ, 729, 16, +doi: 10.1088/0004-637X/729/1/16 +Kobayashi, C., Nomoto, K., & Hachisu, I. 2015, ApJL, 804, +L24, doi: 10.1088/2041-8205/804/1/L24 +Kobayashi, C., Umeda, H., Nomoto, K., Tominaga, N., & +Ohkubo, T. 2006, ApJ, 653, 1145, doi: 10.1086/508914 +Korobkin, O., Rosswog, S., Arcones, A., & Winteler, C. +2012, MNRAS, 426, 1940, +doi: 10.1111/j.1365-2966.2012.21859.x +Korschinek, G., Faestermann, T., Poutivtsev, M., et al. +2020, PhRvL, 125, 031101, +doi: 10.1103/PhysRevLett.125.031101 +Krause, M. G. H., Burkert, A., Diehl, R., et al. 2018, A&A, +619, A120, doi: 10.1051/0004-6361/201732416 +Larson, R. B. 1991, in Astronomical Society of the Pacific +Conference Series, Vol. 20, Frontiers of Stellar Evolution, +ed. D. L. Lambert, 571 +Limongi, M., & Chieffi, A. 2006, ApJ, 647, 483, +doi: 10.1086/505164 +—. 2018, ApJS, 237, 13, doi: 10.3847/1538-4365/aacb24 +Lugaro, M., Ott, U., & Kereszturi, ´A. 2018, Progress in +Particle and Nuclear Physics, 102, 1, +doi: 10.1016/j.ppnp.2018.05.002 +Lugaro, M., Pignatari, M., Ott, U., et al. 2016, Proceedings +of the National Academy of Science, 113, 907, +doi: 10.1073/pnas.1519344113 +Lugaro, M., Heger, A., Osrin, D., et al. 2014, Science, 345, +650, doi: 10.1126/science.1253338 +Matteucci, F., & Greggio, L. 1986, A&A, 154, 279 +Meyer, B. S., & Clayton, D. D. 2000, SSRv, 92, 133, +doi: 10.1023/A:1005282825778 +Mor, R., Livne, E., & Piran, T. 2023, MNRAS, 518, 623, +doi: 10.1093/mnras/stac2775 +M¨osta, P., Roberts, L. F., Halevi, G., et al. 2018, ApJ, 864, +171, doi: 10.3847/1538-4357/aad6ec +Nishimura, N., Sawai, H., Takiwaki, T., Yamada, S., & +Thielemann, F. K. 2017, ApJL, 836, L21, +doi: 10.3847/2041-8213/aa5dee +Nomoto, K., Hashimoto, M., Tsujimoto, T., et al. 1997, +NuPhA, 616, 79, doi: 10.1016/S0375-9474(97)00076-6 +Nomoto, K., Kobayashi, C., & Tominaga, N. 2013, +ARA&A, 51, 457, +doi: 10.1146/annurev-astro-082812-140956 +Nomoto, K., Maeda, K., Mazzali, P. A., et al. 2004, in +Astrophysics and Space Science Library, Vol. 302, +Astrophysics and Space Science Library, ed. C. L. Fryer, +277–325, doi: 10.1007/978-0-306-48599-2 10 +O’Dell, C. R., McCullough, P. R., & Meixner, M. 2004, AJ, +128, 2339, doi: 10.1086/424621 +Panov, I. V., Korneev, I. Y., & Thielemann, F. K. 2008, +Astronomy Letters, 34, 189, +doi: 10.1007/s11443-008-3006-1 +Perets, H. B., & Beniamini, P. 2021, MNRAS, 503, 5997, +doi: 10.1093/mnras/stab794 +Poelarends, A. J. T., Herwig, F., Langer, N., & Heger, A. +2008, ApJ, 675, 614, doi: 10.1086/520872 +Prantzos, N., Abia, C., Cristallo, S., Limongi, M., & Chieffi, +A. 2020, MNRAS, 491, 1832, doi: 10.1093/mnras/stz3154 +Reed, D. S., Balick, B., Hajian, A. R., et al. 1999, AJ, 118, +2430, doi: 10.1086/301091 +Reichert, M., Obergaulinger, M., Eichler, M., Aloy, M. ´A., +& Arcones, A. 2021, MNRAS, 501, 5733, +doi: 10.1093/mnras/stab029 +Ritter, C., Herwig, F., Jones, S., et al. 2018, MNRAS, 480, +538, doi: 10.1093/mnras/sty1729 +Rosswog, S. 2013, Nature, 500, 535, doi: 10.1038/500535a +Rosswog, S., Korobkin, O., Arcones, A., Thielemann, F. K., +& Piran, T. 2014, MNRAS, 439, 744, +doi: 10.1093/mnras/stt2502 +Rosswog, S., Piran, T., & Nakar, E. 2013, MNRAS, 430, +2585, doi: 10.1093/mnras/sts708 +Ryan, S. G., Norris, J. E., & Beers, T. C. 1996, ApJ, 471, +254, doi: 10.1086/177967 +Salpeter, E. E. 1955, ApJ, 121, 161, doi: 10.1086/145971 +Sana, H., de Mink, S. E., de Koter, A., et al. 2012, Science, +337, 444, doi: 10.1126/science.1223344 +Schaerer, D., Charbonnel, C., Meynet, G., Maeder, A., & +Schaller, G. 1993a, A&AS, 102, 339 +Schaerer, D., Meynet, G., Maeder, A., & Schaller, G. +1993b, A&AS, 98, 523 +Schaller, G., Schaerer, D., Meynet, G., & Maeder, A. 1992, +A&AS, 96, 269 +Schmidt, M. 1959, ApJ, 129, 243, doi: 10.1086/146614 +Seitenzahl, I. R., Ciaraldi-Schoolmann, F., R¨opke, F. K., +et al. 2013, MNRAS, 429, 1156, +doi: 10.1093/mnras/sts402 +Shigeyama, T., & Tsujimoto, T. 1998, ApJL, 507, L135, +doi: 10.1086/311699 +Sukhbold, T., Ertl, T., Woosley, S. E., Brown, J. M., & +Janka, H. T. 2016, ApJ, 821, 38, +doi: 10.3847/0004-637X/821/1/38 +Tauris, T. M., Langer, N., & Podsiadlowski, P. 2015, +MNRAS, 451, 2123, doi: 10.1093/mnras/stv990 + +18 +Wehmeyer et al. +Tauris, T. M., Kramer, M., Freire, P. C. C., et al. 2017, +ApJ, 846, 170, doi: 10.3847/1538-4357/aa7e89 +Thielemann, F. K., Brachwitz, F., H¨oflich, P., +Martinez-Pinedo, G., & Nomoto, K. 2004, NewAR, 48, +605, doi: 10.1016/j.newar.2003.12.038 +Thielemann, F.-K., Nomoto, K., & Hashimoto, M.-A. 1996, +ApJ, 460, 408, doi: 10.1086/176980 +Timmes, F. X., Brown, E. F., & Truran, J. W. 2003, ApJL, +590, L83, doi: 10.1086/376721 +Travaglio, C., Hillebrandt, W., & Reinecke, M. 2005, A&A, +443, 1007, doi: 10.1051/0004-6361:20052883 +Trueman, T. C. L., Cˆot´e, B., Yag¨ue L´opez, A., et al. 2022, +ApJ, 924, 10, doi: 10.3847/1538-4357/ac31b0 +van de Voort, F., Pakmor, R., Bieri, R., & Grand, R. J. J. +2022, MNRAS, 512, 5258, doi: 10.1093/mnras/stac710 +Vasileiadis, A., Nordlund, ˚A., & Bizzarro, M. 2013, ApJL, +769, L8, doi: 10.1088/2041-8205/769/1/L8 +Vassh, N., Vogt, R., Surman, R., et al. 2019, Journal of +Physics G Nuclear Physics, 46, 065202, +doi: 10.1088/1361-6471/ab0bea +Vincenzo, F., & Kobayashi, C. 2020, MNRAS, 496, 80, +doi: 10.1093/mnras/staa1451 +Wallner, A., Faestermann, T., Feige, J., et al. 2015, Nature +Communications, 6, 5956, doi: 10.1038/ncomms6956 +Wallner, A., Feige, J., Kinoshita, N., et al. 2016, Nature, +532, 69, doi: 10.1038/nature17196 +Wallner, A., Froehlich, M. B., Hotchkis, M. A. C., et al. +2021, Science, 372, 742, doi: 10.1126/science.aax3972 +Wanajo, S. 2013, ApJL, 770, L22, +doi: 10.1088/2041-8205/770/2/L22 +Wanajo, S., Sekiguchi, Y., Nishimura, N., et al. 2014, +ApJL, 789, L39, doi: 10.1088/2041-8205/789/2/L39 +Wang, X., Clark, A., Ellis, J., et al. 2021a, in APS Meeting +Abstracts, Vol. 2021, APS Division of Nuclear Physics +Meeting Abstracts, MD.008 +Wang, X., Clark, A. M., Ellis, J., et al. 2021b, arXiv +e-prints, arXiv:2112.09607. +https://arxiv.org/abs/2112.09607 +Wehmeyer, B., Fr¨ohlich, C., Cˆot´e, B., Pignatari, M., & +Thielemann, F. K. 2019, MNRAS, 487, 1745, +doi: 10.1093/mnras/stz1310 +Wehmeyer, B., Pignatari, M., & Thielemann, F. K. 2015, +MNRAS, 452, 1970, doi: 10.1093/mnras/stv1352 +Winteler, C., K¨appeli, R., Perego, A., et al. 2012, ApJL, +750, L22, doi: 10.1088/2041-8205/750/1/L22 +Woosley, S. E., & Weaver, T. A. 1995, ApJS, 101, 181, +doi: 10.1086/192237 +Yag¨ue L´opez, A., Cˆot´e, B., & Lugaro, M. 2021, ApJ, 915, +128, doi: 10.3847/1538-4357/ac02bf + +SLRs: Surfing the wave? +19 +APPENDIX +A. TESTING THE RESOLUTION DEPENDENCE +The spatial resolution of our model is chosen to avoid +a mass bias for newly born stars. In order to sample +the entire IMF in the range 0.1 M⊙ ≤ M < 50 M⊙, a +star-forming sub-cell has to contain at least 50 M⊙ of +gas. +This intrinsically limits the spatial resolution of +the model since a sufficiently large number of sub-cells +fulfilling this mass criterion has to be found at every time +step. For this reason, we have chosen a spatial resolution +of (50pc)3. If the resolution was increased further, it +would become more difficult to find a sufficiently large +number of sub-cells to fulfil the mass criterion every time +step. We can still test the spatial resolution dependence +by decreasing it. +To do this, we set up two models where we decreased +the sub-cell resolution to (80pc)3 and (125pc)3. The re- +sults in Figure 8 show that the abundances have less +variability. This is not surprising, as the lower the reso- +lution, the more the results will converge toward those +of a one-zone model, which would show a line for all +abundances. Since the number of very low abundance +sub-cells is strongly reduced in a model with lower res- +olution, the lower 100% statistics boundary converges +faster than the upper 100% statistics boundary. This +also slightly increases the median in the abundances, +since it is strongly dependent on the number of sub-cells +featuring a given abundance. This effect can be observed +prominently in the 244Pu evolution in the right panel of +Figure 8. +For the time resolution, the time step of 1 Myr was +chosen because the code cannot handle effects emerg- +ing from, e.g., hydrodynamics and cooling. +After 1 +Myr, the gas ejected from a SN explosion has completely +halted and sufficiently cooled down, so it can be used to +form the next generation of stars. Choosing time steps +smaller than 1 Myr would require to model precisely the +trajectory of a SN blast wave in the ISM within a hydro- +dynamic framework, which is not currently implemented +in this model. As we have done above for the spatial res- +olution, we tested also the time-resolution dependence +of the results by lowering it. We set up a model where +we used 4 Myr time step. The results can be found in +Figure 9. In general, the upper 100% statistics for all +isotopes behave in a similar manner as in the standard +case. However, the 95% and 68% statistics are signif- +icantly affected by the lowering of the time resolution. +More nucleosynthesis sites explode per time step in the +lower resolution model. This means that more cells are +affected during each time step, which effectively leads to +a stronger mixing of the entire volume, and less cells re- +taining lower abundance values. In other words, the like- +lihood of each cell to be affected by nucleosynthesis and +mixing events at every time step is higher. This leads +to the shrinking of the spectrum of the statistics, while +the 100% statistics seem to change only marginally. +We caution that the tests presented here would need +to be calibrated to some galactic properties (e.g., age- +metallicity-relation, GCE of α-elements), as done for the +standard model presented in the main text, before they +can be compared to the actual ISM. +B. INCREASING THE AMOUNT OF REFLECTED +MATERIAL +The Sedov-Taylor approach assumes that all material +swept-up by a SN explosion is deposited solely on a bub- +ble shell surrounding the SN explosion. The PINBALL +model also explored in this work is motivated by the +findings of Fry et al. (2020). When a magnetic field is +present in the ISM around the SN blast wave, some ma- +terial inside the outward SN shock front can be deflected +in the backward direction. If that material reaches the +opposite shock front, it might experience another change +of direction caused by the magnetic field, causing the +material to move through the inside of the SN shock +front, like a pinball. Fry et al. (2020) also conclude that +the fraction of reflected material in a SN blast wave is +dependent on the magnetic field strength and the size of +the magnetic grains that condense behind the SN blast +wave, and is difficult to constrain. To investigate the +potential impact of such pinball remnant geometry, we +have introduced a PINBALL model in this work, by as- +suming that 1% of the material swept-up by the SN +blast wave remains inside the remnant bubble, and not +on the shell. This number is motivated by Fry et al. +(2020) and we found that this choice does not strongly +affect the evolution of SLRs. Here, we test whether a +much larger fraction (50 %) of deflected material would +more strongly alter the abundance evolution. The re- +sults in Figure 10 show that even such a high fraction +of deflected material would not strongly affect the over- +all evolution of the SLRs. Therefore, this choice might +affect SLR densities only locally in cells, but not their +overall abundance statistics. This is in contrast to the +HN model, which increases the overall size of the rem- +nant bubble. However, we caution that such high frac- +tions of deflected material would affect the ISM density +distribution, which in turn would alter the star forma- + +20 +Wehmeyer et al. +10 +11 +10 +7 +Mn-53 +10 +8 +10 +6 +Fe-60 +10 +11 +10 +8 +Hf-182 +0 +2 +4 +6 +8 +10 +12 +14 +Time in Gyr +10 +14 +10 +12 +Pu-244 +10 +11 +10 +7 +Mn-53 +10 +8 +10 +6 +Fe-60 +10 +11 +10 +8 +Hf-182 +0 +2 +4 +6 +8 +10 +12 +14 +Time in Gyr +10 +14 +10 +12 +Pu-244 +Figure 8. Same as Figure 1, but with lower sub-cell resolution of an edge length of 80 pc (left panel), and 125 pc (right panel). +10 +11 +10 +7 +Mn-53 +10 +8 +10 +6 +Fe-60 +10 +11 +10 +8 +Hf-182 +0 +2 +4 +6 +8 +10 +12 +14 +Time in Gyr +10 +14 +10 +12 +Pu-244 +Figure 9. Same as Figure 1, but with time resolution of 4 +Myr per time step. +tion history due to the exponent of the Schmidt law (see +Section 2.1) used in this model. This might yield results +which are not congruent with Milky Way Galaxy prop- +erties (e.g., age-metallicity-relation, GCE of α-elements +and so forth). +10 +11 +10 +7 +Mn-53 +10 +8 +10 +6 +Fe-60 +10 +11 +10 +8 +Hf-182 +0 +2 +4 +6 +8 +10 +12 +14 +Time in Gyr +10 +14 +10 +12 +Pu-244 +Figure 10. +Same as Figure 1, but with 50 % deflected +material instead of 1 % as done in the PINBALL model. +C. ROBUSTNESS OF THE DEEP-SEA +DETECTION FIT +To examine the robustness of our one sub-cell fitting +approach applied in Section 3.2, we tested how much the +mean squared distance of the logarithm of the abun- +dances changes if the deep-sea detection data of the +SLRs had a different time (x-axis) or abundance (y-axis) +spacing. For the Standard model (with time and spatial +resolution as in the main text), the value of this mean + +SLRs: Surfing the wave? +21 +squared distance for the actual deep-sea measurements +is S = 4.54, better than 83% of fits3, with S given by +S = 1 +N +� � +ln R(ti) − ln Yi +ln(Yi + Ei) − ln Yi +�2 +. +(C1) +where R(ti) is the abundance of the run at time ti, and +Yi and Ei the deep sea measurement and uncertainty, +respectively, at time ti. +If the time spacing between the individual deep-sea de- +tection data points was five times larger, it is easier to +find a sub-cell that can match the detection data (see top +left panel of Fig. 11 for an example, with S = 3.14, i.e., +better than 85% of fits). This is because the sub-cells +have much more options for the individual evolution be- +tween data points by either radioactive decay or the surf +effect discussed in Section 4.3. In other words, sub-cells +have the opportunity to undergo the surf effect multi- +ple times between two deep-sea detection data points +when the spacing is larger, which increases the possible +spectrum of SLR densities that sub-cells can exhibit. If +the time spacing between detection data was smaller, +the situation would be the opposite. Sub-cells have less +possibilities for individual evolution (via radioactive de- +cay or the surf effect), and thus the spectrum of possi- +ble abundances is lower. This makes it harder to find a +sub-cell that fits the observations even if decreasing the +time spacing between detection data only by a factor of +two (see top right panel of Fig. 11 for an example, with +S = 6.23, better than 82% of fits). +For the abundance spacing, we find a similar result. If +the spacing between detected abundances was larger, we +would have to find sub-cells that undergo stronger den- +sity fluctuations than those fitting the actual detection. +This decreases the number of sub-cells that provide a +fit (see lower left panel of Fig. 11 for an example, with +S = 17.48, better than 81% of fits). Instead, the more +we decrease the spacing between the detected abun- +dances, the closer to a straight line the requirement for +a single sub-cell to fit these detection is. Interestingly, +especially after the right (latest) peak in the modified +60Fe data, these modified data points appear to scatter +around the natural decay of that SLR. Since this area +of the deep-sea detection data features many points, the +weighting of the fit leads to preferring “undisturbed” +sub-cells (i.e., sub-cells that only experience radioactive +decay instead of the surfing effect), which makes it eas- +3 Determined from assuming the abundances follow a normal dis- +tribution with σ equal to the uncertainty. Sampling the abun- +dances to obtain the resulting distribution of S allows us to com- +pare a given S with all possible fits. The goal is to provide a +reference for the goodness of fit. +ier to find sub-cells to fit these hypothetical detection +data (see bottom right panel of Fig. 11 for an example, +with S = 2.89, better than 86% of fits). + +22 +Wehmeyer et al. +(a) Five times larger time spacing +(b) Two times smaller time spacing +(c) Ten times larger abundance spacing +(d) Ten times smaller abundance spacing +Figure 11. Same as Figure 2, with hypothetical different time and abundance spacing for the deep-sea detections. + +10 +3 +10 +10 +10 +LO +10 +11 +10 +10- +10 +10- +10-11 +10-12 +10-13 +10-14 +13.03 +13.04 +13.05 +13.06 +13.07 +13.08 +time「Gvri10 +10 +10 +10 +10 +10 +10-7 +10 +10 +10-9 +10-10 +10-11 +10-12 +10-13 +10-14 +13.020 +13.022 +13.024 +13.026 +13.028 +13.030 +13.032 +time「Gyri10 +3 +10 +10 +10 +10 +10 +10-7 +！ +10- +10 +++? +10-9 +10-11 +10-12 +10-13 +10-14 +13.0150 +13.0175 +13.020013.022513.0250 +13.0275 +13.0300 +13.0325 +time「Gyr]10 +10 +10 +10 +10 +10 +11 +10- +6 +10-7 +10 +10- +10-10 +10-11 +10-12 +10-13 +10-14 +13.0425 +13.0450 +13.047513.050013.0525 +13.0550 +13.0575 +13.0600 +time「Gvr] \ No newline at end of file diff --git a/ttE3T4oBgHgl3EQfkApM/content/tmp_files/load_file.txt b/ttE3T4oBgHgl3EQfkApM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1ec29afa0b00f315e400f941891f845dbeb42306 --- /dev/null +++ b/ttE3T4oBgHgl3EQfkApM/content/tmp_files/load_file.txt @@ -0,0 +1,1729 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf,len=1728 +page_content='Draft version January 12, 2023 Typeset using LATEX twocolumn style in AASTeX63 Inhomogeneous enrichment of radioactive nuclei in the Galaxy: Deposition of live 53Mn, 60Fe, 182Hf, and 244Pu into deep-sea archives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Surfing the wave?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Benjamin Wehmeyer,1, 2 Andr´es Yag¨ue L´opez ,1, 3 Benoit Cˆot´e ,4, 1 Maria K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Pet˝o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1 Chiaki Kobayashi ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2 and Maria Lugaro 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 6 1Konkoly Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Research Centre for Astronomy and Earth Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' E¨otv¨os Lor´and Research Network (ELKH),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Excellence Centre of the Hungarian Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Konkoly-Thege Mikl´os ´ut 15-17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' H-1121 Budapest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Hungary 2Centre for Astrophysics Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' University of Hertfordshire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' College Lane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Hatfield AL10 9AB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' UK 3Computer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Computational and Statistical Sciences (CCS) Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Center for Theoretical Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Los Alamos National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Los Alamos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' NM 87545,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' USA 4Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' University of Victoria,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' BC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' V8W 2Y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Canada 5ELTE E¨otv¨os Lor´and University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Budapest 1117,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Hungary 6School of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Monash University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' VIC 3800,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Australia (Received 2022 April 28;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Revised 2022 December 31;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Accepted 2023 January 02) Submitted to ApJ ABSTRACT While modelling the galactic chemical evolution (GCE) of stable elements provides insights to the formation history of the Galaxy and the relative contributions of nucleosynthesis sites, modelling the evolution of short-lived radioisotopes (SLRs) can provide supplementary timing information on recent nucleosynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' To study the evolution of SLRs, we need to understand their spatial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Using a 3-dimensional GCE model, we investigated the evolution of four SLRs: 53Mn, 60Fe, 182Hf, and 244Pu with the aim of explaining detections of recent (within the last ≈ 1 − 20 Myr) deposition of live 53Mn, 60Fe, and 244Pu of extrasolar origin into deep-sea reservoirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We find that core-collapse supernovae (CCSNe) are the dominant propagation mechanism of SLRs in the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This results in the simultaneously arrival of these four SLRs on Earth, although they could have been produced in different astrophysical sites, which can explain why live extrasolar 53Mn, 60Fe, and 244Pu are found within the same, or similar, layers of deep-sea sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We predict that 182Hf should also be found in such sediments at similar depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Keywords: astrochemistry — Galaxy: abundances — ISM: abundances — local interstellar matter — nuclear reactions, nucleosynthesis, abundances 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' INTRODUCTION Studying the galactic chemical evolution (GCE) of stable elements provides insights on the formation of the Galaxy and the nucleosynthesis processes that pro- duced the chemical elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Audouze & Tinsley 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Matteucci & Greggio 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Gibson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Nomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Prantzos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The abundance of short-lived radioactive isotopes (SLRs), with half-lives of less than a few 100 Corresponding author: Benjamin Wehmeyer benjamin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='wehmeyer@csfk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='org Myrs, instead follows the trend of star formation rate, and is determined by the balance between their pro- duction and decay (the steady-state equilibrium, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Clayton 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' SLRs can be observed live in the inter- stellar medium (ISM, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Diehl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2010), in Earth deep-sea sediments accumulating in the last 10 Myr with a constant rate (Wallner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2015, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Korschinek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Wallner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2021) and extinct in me- teorites (Dauphas & Chaussidon 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Lugaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2018), and offer additional information on the timing of processes and our Solar System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Comparing the SLR abundances in meteorites to the abundances predicted by GCE models can constrain the last stellar events con- tributing SLRs to the molecular cloud in which the Solar arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='04593v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='GA] 11 Jan 2023 ID2 Wehmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' System formed, and determine the time the Solar Sys- tem matter remained isolated from the Galactic medium into such molecular cloud (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Clayton 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Meyer & Clayton 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Huss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Lugaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2014, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Cˆot´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2019a, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Trueman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Further- more, comparing SLR abundances in the ISM at any two points in time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', at the time of Solar System for- mation from meteorites and today from deep-sea sedi- ments) to GCE models can allow us to constrain the ori- gin of a given SLR based on differences in the event rates of stellar processes producing such a given SLR (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Ho- tokezaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In addition, the SLR abundances are reported in deep-sea sediments with high resolution temporal profiles (with 200 kyr sampling), which showed live 60Fe (half-life t1/2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='62 Myr) originating from the ISM together with resolvable 244Pu (t1/2 = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='0 Myr) and 53Mn (t1/2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='74 Myr) anomalies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Wall- ner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2015, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Korschinek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Wallner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' These profiles identified at least 2 distinct signals of SLR deposition to Earth, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='5 and at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='7 Myr ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The temporal overlap of the 53Mn and 60Fe signals supports the involvement of a core- collapse supernova (CCSN) origin of these signals, while the temporal overlap with 244Pu signals may indicate a rare supernova event or a more complicated trans- port history of material in the ISM (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Hotokezaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' To best interpret all these observations, we need to better understand the dependence of SLR abundances on GCE model parameters, the temporal and spatial stochasticity of enrichment events from the different stellar sources, and the transport mechanism of material in the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' To accommodate SLRs in more recent GCE models, Cˆot´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2019a) derived the be- haviour of the ratio of SLRs to their stable reference isotopes over the lifetime of the Galaxy using the two- zone GCE model Omega+ (Cˆot´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This study explained and quantified the effects of galactic inflows and outflows, delay-time distributions of enrich- ment events from different stellar sources, the Galactic star formation history, and the gas-to-star mass ratio, assuming a continuous production rate of SLRs and a homogeneous ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This quantification of uncertainties on the average composition of GCE models was recently extended to consider inhomogeneities in the ISM due to the temporal stochasticity of stellar enrichment events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Cˆot´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2019b) developed a Monte Carlo simula- tion for the temporal evolution of SLRs in a local region of the interstellar gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The study developed a general statistical framework to quantify the uncertainty (prob- ability distributions) of SLR abundances based on the stochastic delay time between star formation and en- richment, and explained the dependence of SLR abun- dances on the average time between enriching events, the delay time distribution, and the mean life of the SLRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Cˆot´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2019b) also quantified the proba- bility of whether an SLR could sample only one stellar event or whether a particular SLR abundance represents a steady-state in the ISM, where frequent production of nuclides is balanced out by their decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Yag¨ue L´opez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2021) studied the abundance ratios of two SLRs using the same statistical framework, which also depend on the relative mean life of the SLRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The ratios of SLRs with similar, and short enough half-lives are largely in- dependent of GCE model uncertainties, and therefore are unique tools to study the nucleosynthesis at their formation sites, provided that their production occurs synchronously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The theoretical considerations in Yag¨ue L´opez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2021) have been successfully applied to the 129I/247Cm ratio to understand the physical condi- tion of the last rapid neutron capture process event that contributed to the Solar System inventory prior to its formation (Cˆot´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The focus of the present study is to use a full inhomogeneous GCE code to con- sider not only temporal discretization (as done in the papers mentioned above) but also spatial discretization of SLRs and the role of inhomogeneities in the ISM to develop a better understanding of transport of matter in our Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In particular, we aim to explain the possibly synchronous delivery of live 53Mn, 60Fe, and 244Pu onto the ocean floor as recorded in the deep-sea sediments deposited in the last 10 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' So far, it has been difficult to draw conclusions about the impact of these detections using GCE models that assume homo- geneous mixing, since spatial discretization effects have a significant impact on the abundances of the detected SLRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Since these problems are difficult or impossible to study with 1-dimensional models, here we use the 3- dimensional GCE model from Wehmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We simulate the spatial and temporal evolution of four SLR abundances over the lifetime of the Galaxy: 53Mn, 60Fe, 182Hf (t1/2 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='90 Myr), and 244Pu to draw con- clusions about their most recent and ongoing (within the last Myrs) deposition to Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The comparison of our model results to SLR abundances in the early Solar Sys- tem derived from meteorites and its implications will be published in a separate study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In Section 2, we summarize the main model- ing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1 we discuss the abundance evolution of the selected SLRs over the entire lifetime of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2, we zoom in on the more re- cent time period closer to present day, and compare our simulation results with deep-sea sediment detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2, we compare the propagation of the different SLRs in the simulation volume and derive the SLRs: Surfing the wave?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 3 dominant transport mechanism for SLRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We present a schematic interpretation of our findings in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3, discuss the impact of yields in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='4 and provide our conclusions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' THE MODEL For this study, we use the 3-dimensional GCE model described in Wehmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2015, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Below, we recall the most important modelling assumptions, and highlight relevant updates to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' General setup A simulation cube of (2kpc)3 is divided into 403 sub- cubes (or cells) with an edge length of 50 pc each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Dur- ing each calculated time step of 1 Myr, the following operations are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Gas with primordial composition falls into the sim- ulation volume according to the prescription of Wehmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2015), which permits for a lin- ear rise of infalling material until 2 Gyr, and then an exponential decrease of the infall rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' When falling into the simulation volume, the gas is ho- mogeneously distributed among all sub-cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The total gas mass of the simulation volume is used to determine the number of stars to be born based on a Schmidt law with power α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='5 (Schmidt 1959;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Kennicutt 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Larson 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The mass of the new born stars is sampled from a Salpeter initial mass function (IMF) (Salpeter 1955) with a slope of −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='35, and mass limits of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1 M⊙ ≤ M < 1 M⊙1 for low-mass stars (LMS), 1 M⊙ ≤ M < 10 M⊙ for intermediate-mass stars (IMS), and 10 M⊙ ≤ M ≤ 50M⊙ for high-mass stars (HMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The newly born stars inherit the chemical composition of the gas out of which they were formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Once the number and masses of the new born stars are known, their birth location is chosen randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Due to supernova explosions (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3), the gas density distribution becomes more and more inhomogeneous with time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', the density distri- bution function steepens), and preference of star formation is given to cells with higher gas den- sities due to the Schmidt law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' To prevent mis- sampling of the IMF, only cells containing at least 50 M⊙ of gas are permitted to form stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This constraint is also a limiting factor to increase the 1 Concerning star masses, we refer to the zero age main sequence (ZAMS) mass of the star throughout this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' resolution of the sub-cells: every time step, a suffi- ciently large number of sub-cells has to be available for star formation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', has to fulfil this minimum mass requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' If the resolution of the sub- cells was increased, not enough sub-cells available for star formation would be found during a time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This would alter the star formation rate and lead to other problems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', with the applicability of the model on the GCE of α-elements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Decreas- ing the spatial resolution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', using larger sub-cell size) is instead feasible and we tested the depen- dence of our model on this in Appendix A: We find that decreasing the sub-cell resolution results in the abundance spectrum getting smaller (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', converging towards a line), which resembles rather a one-zone model behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The lifetime for every newly born star is calcu- lated using the formula by the Geneva group (see Schaller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Schaerer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1993a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Char- bonnel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1993): log(t) = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='79 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='24Z) − (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='35Z)log(M) + (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='74 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='11Z)log2(M), (1) where t is the expected lifetime of a star in Myr, Z is the metallicity with respect to solar, and M the stellar mass in solar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' If a time step contains stars that have reached the end of their lifetime, stellar death is simulated fol- lowing the description detailed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Stellar deaths 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' LMS and IMS During their lifetime, LMS & IMS add significant amounts of C and N to the galactic inventory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' They do not reach burning stages more advanced than He burning, and thus do not contribute to iron-group elements significantly, however they produce slow neutron capture (s-process) elements such as Sr, Y, Zr, Ba, La, Ce, and Pb, as well as Hf of interest here during their thermally-pulsing asymptotic giant branch (AGB) phase (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', K¨appeler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Bisterzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' When LMS & IMS die, they eject some of their initial abun- dances, plus their nucleosynthesis products via stellar winds, except for the fraction that remains locked in the degenerate core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Their death may result in a planetary nebula, and then a white dwarf remnant, and it is far less violent/energetic compared to the death of a HMS be- cause it is wind-driven instead of explosion-driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The 4 Wehmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' ejecta of IMS typically pollute only volumes with di- ameters of the order of light years (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', the Cat’s eye nebula NGC 6543 has a radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1 light year, Reed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1999, and the Helix nebula NGC 7293 a radius of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='43 light years, O’Dell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' LMS & IMS have two main functions in our simulation: to lock up gas during their lifetime, and to produce heavy isotopes via the s-process, including 182Hf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' These isotopes are injected at the location of the source, since the energy injection by the site is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Thermonuclear supernovae Since many stars are born in binary systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Duchˆene & Kraus 2013), a significant fraction of IMS interact with a companion and undergo a supernova of type Ia (SNIa), which are the dominant source of Fe in the galactic disk (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Matteucci & Greggio 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' To include SNeIa, we use the analytical prescription of Greggio (2005), which reduces all stellar and binary evo- lution parameters to the factor PSNIa = 6 · 10−3, repre- senting the probability of an IMS to be born in a system that fulfills all necessary prerequisites to later end up in a SNIa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' When the system has reached the end of its lifetime, we eject stable isotopes in the amounts calcu- lated by Iwamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (1999, model CDD2), together with 10−4 M⊙ of 53Mn, at the same location (which is in agreement with, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Seitenzahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' When a supernova explosion occurs, a shock wave pushes the ejecta into the ISM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' we model this by moving the gas mass of the inner cells into a shell, with an enclosed (pre-explosion) mass of 5 · 104 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This mass corresponds to an explosion energy of 1 Bethe, according to Sedov-Taylor blast wave theory (Ryan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Shigeyama & Tsujimoto 1998), which implies that the radius of a remnant depends strongly on the ISM density surrounding the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The pushed out gas is distributed in a chemically homogeneous shell around the remnant, leaving behind a “bubble” in the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We follow the approach of Wehmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2015, 2019) and eject a constant yield of elements per SNIa, independent of metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' While this approximation is somewhat inaccurate (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Timmes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Thiele- mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Travaglio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Bravo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Seitenzahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020b), it does not strongly affect the outcomes of our simulations, which are focused on SLRs that are mostly influenced by solar metallicity yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' HMS HMSs experience every stellar burning stage and pro- duce significant amounts of α- as well as iron-group el- ements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Woosley & Weaver 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Limongi & Chieffi 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Ritter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' When a HMS has reached the end of its life time, we let it ex- plode as a CCSN, analogous to the explosion of a SNIa: We eject stable elements (yields from Thielemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Nomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1997) and SLRs (according to Sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='5), and move the surrounding gas into a shell around the explosion, depending on the injected kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' As described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='6, we consider a range of different explosion energies and remnant ge- ometries to also account for the potential effect of hy- pernovae (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Nomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2004, 2013) and altered CCSN remnant geometries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Fry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' For our purposes, HMS and their CCSNe are the exclusive contributors of the SLR 60Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Neutron Star Mergers Most HMS are born in binary systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Sana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Duchˆene & Kraus 2013), where both stars eventu- ally undergo CCSN explosions, leaving behind two neu- tron stars (NSs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' There is a possibility that these two NSs are still gravitationally bound after the two CC- SNe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Tauris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' If in a suitable orbit, these two objects reduce their separation distance via the emission of gravitational waves, until they coalesce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Such a merger event provides conditions to synthesize r- process elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Freiburghaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Panov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Korobkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Bauswein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Rosswog 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Rosswog et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2013, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Wanajo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Eichler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Just et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Vassh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' For the purpose of our simulation, we reduce all the mentioned probabilities to a factor PNSM, which represents the fraction of HMSs that fulfil all the needed prerequisites to later undergo a merger event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This ap- proach simplifies the detailed physics of population syn- thesis, the explosion dynamics of CCSNe, and binary survival probabilities, and allows us to reduce all these details to one free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We choose PNSM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='04 as in Wehmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Using a Salpeter initial mass function with an integrated slope of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='35, and a standard cosmic star formation history with constant NSM delay times (see Cˆot´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2017 for details), this probability translates to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='03·10−4 NSM events per unit solar mass of stars formed, which would produce a the- oretical gravitational wave event rate of ∼ 1800 Gpc−3 yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Although this is ∼ 2 times higher than the latest upper limit of 810 Gpc−3 yr−1 derived by LIGO/Virgo (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2021), our main conclusions are not af- fected by the exact choice of NSM rate (see discussion in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' For our purposes, NSMs are the only source of 244Pu in the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Analogous to the explo- sion of a SNIa or a CCSN, when an NSM occurs in our model, 244Pu is ejected to a spherical shell around the source, assuming an explosion energy of 1 Bethe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' SLRs: Surfing the wave?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 5 CCSNe mostly occur asymmetrically, resulting in a natal kick for the newly born NS, which could lead to offsets in the NSM locations from the original CCSNe that formed the two NSs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', van de Voort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' However, various studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Beniamini & Piran 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Tauris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2017) have shown that the second born NS in the majority of NS binaries in our Galaxy were formed by much weaker explosions (possibly ultra-stripped SNe, see Tauris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Mor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2023) that resulted in very weak kicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Indeed, Perets & Beniamini (2021) showed that the offset locations of short GRBs — when divided according to galaxy type — support the idea that kicks play a subdominant role in setting binary neutron star merger offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In addition, if binary NS formation is often preceded by such weak explosions, this could lead to the amount of swept up mass by such explosions to be significantly smaller than obtained for standard CCSNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Further, the coalescence time of two NSs can be approx- imated with a t−x distribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Belczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Cˆot´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2017), which could also potentially lead to a larger spectrum in abundances (see Cˆot´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Here, we use instead a constant coalescence time of 108 years for newly born NSs to merge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In this re- spect, our work will mostly highlight the “surfing” effect on SLRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' SLR sources and yields In this work, we focus on four SLRs: 53Mn, 60Fe, 182Hf, and 244Pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' For a quick overview over these iso- topes’ origins in our model, and their half-lives, see Ta- ble 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The deep-sea detections of three of these isotopes (53Mn, 60Fe, and 244Pu) can be translated into their corresponding ISM density at time of deposition into the deep-sea archive (see Wallner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2021, Supple- mental Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In our simulation, the four isotopes are produced in the four separate individual sites de- scribed above exclusively: SNIa for 53Mn, CCSNe for 60Fe, IMSs for 182Hf, and NSMs for 244Pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' These asso- ciations are reasonable in first order approximation (see details below), furthermore, the advantage of assigning each of these isotopes to a different site is that we are able to use them as unique tracers of each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Hence we are able to study the production parameters of each site (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', occurrence frequency, yields) almost completely independent of the production parameters of the other sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We adopted the following yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' For 53Mn, 10−4 M⊙ is ejected per SNIa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The cal- culations of Seitenzahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2013) resulted in a range of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='06 · 10−5 M⊙ up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='95 · 10−4 M⊙ of 53Mn produced by a SNIa, so our chosen value is well within these limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' It is important to note that the Seitenzahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2013) results were obtained using Chandra-mass models, while our double-degenerate models would probably not pro- duce as much 53Mn, which would lead to a possible over-estimation of 53Mn in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Although 53Mn can in principle also be produced by CC- SNe (Lugaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2016), the expected yields are rather low and the production is not necessarily efficient around solar metallicities in the galaxy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2015, 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Hence, we as- sume that no 53Mn is ejected from CCSNe in our model, and test a simplified approach where 53Mn is ejected by SNIa exclusively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' For 60Fe from CCSNe, we used progenitor mass- dependent yields from the solar metallicity mod- els by Limongi & Chieffi (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 60Fe might also be produced in electron-capture supernovae (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Wanajo 2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' at solar metallicity, however, the predicted rate for this site is expected to be low (∼ 1 − 5%) in comparison to CCSNe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Poe- larends et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Doherty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020a), and the ejecta mass is ∼ 100 times lower than that of CCSNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Therefore, we do not consider it as a source of 60Fe in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' For 182Hf from IMSs, we used progenitor mass- dependent yields from the solar metallicity mod- els by Lugaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 182Hf is also pro- duced in NSMs, but since their occurrence fre- quency is much lower than IMSs, they are far less relevant for the overall production of 182Hf, and we thus simplify the model to produce 182Hf only in IMSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Further, this simplification allows us to test whether 182Hf would still arrive in deep-sea sedi- ments conjointly with the CCSN produced 60Fe and the NSM produced 244Pu, even though it is not produced by either of them in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The yields for 244Pu from NSMs are highly un- certain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' If we assume the order of 10−2 M⊙ to- tal mass ejection by a single NSM (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Korobkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2012), and a mass fraction of 10−6 for 244Pu (which is within the range predicted by Eichler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2015 using different fission fragment and nu- clear mass models), we obtain 10−8 M⊙ 244Pu ejected by a single NSM, the value used for our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Black hole – neutron star mergers might also produce 244Pu if the mass of the black hole is low enough, or its spin is sufficiently strong, however, this site is probably more relevant at early galactic stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' At later galactic stages, the occurrence rate of NSMs likely exceeds the oc- currence rate of black hole – neutron star merg- 6 Wehmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Isotope Half-life t1/2 Source 53Mn 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='74 Myr SNIa 60Fe 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='62 Myr CCSN 182Hf 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='90 Myr IMS 244Pu 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='0 Myr NSM Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Overview of the isotopes, their half-lives, and their sources in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' See Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='5 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' ers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Wehmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This is consis- tent with population synthesis models which typ- ically predict that NSMs occur more frequently than black hole – neutron star mergers at high metallicities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Dominik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Chruslin- ska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Hence, in our simulations, we assume that 244Pu is dominated by NSMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In this work, we omit also other possible r-process sites such as magnetorotationally driven CCSNe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Winteler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Nishimura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' M¨osta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Reichert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2021) and rare, peculiar CCSNe, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', quark-hadron phase transition CC- SNe (Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020), as these sites still lack observational confirmation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We comment on the dependence of the results on CCSN, IMS, and NSM yields in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Supernova ejecta dynamics To estimate the influence of hypernovae, featuring substantially higher explosion energies than regular CC- SNe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Nomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2004, 2013), and the effect of a varied CCSN bubble remnant geometry and SLR distribution within the explosion shell due to magneto- hydrodynamical effects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Fry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020), we set up four different scenarios (Table 2) to study the im- plications of different assumptions for the interaction of CCSN ejecta with the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Standard case (as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' All CCSNe explode with a kinetic energy of 1 Bethe, and therefore pollute 5·104 M⊙ of ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We assume that all SLRs are deposited on a chemically well mixed shell located on the edge of the blast wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Increased explosion energy case (hypernova model, HN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' All CCSNe have an increased explosion en- ergy and pollute 2 · 105 M⊙ of ISM, this estimates the effect on the SLR abundance evolution if all CCSNe exploded as hypernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The remnant ge- ometry is the same as in the standard case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Modified geometry case (PINBALL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' To estimate the impact of a potential “pinball model”-style remnant geometry (Fry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020), where SLRs are reflected backwards towards the center of the explosion after the remnant bubble has halted, all CCSNe explode with a kinetic energy of 1 Bethe, but we assume that 1% of the swept-up ISM (and therefore SLRs) contained therein will remain ho- mogeneously distributed inside the explosion bub- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Combination of increased explosion energy and modified geometry (HN PINBALL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' As a combi- nation of models HN and PINBALL, all CCSNe pollute 2 · 105 M⊙ of ISM and 1% of the swept- up ISM are distributed homogeneously inside the explosion bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Once these values/scenarios are chosen at the beginning of a simulation, they remain constant throughout the en- tire run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This is a simplification because supernova (SN) explosion radii are sound speed dependent, and thus also dependent on the local density (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Chamandy & Shukurov 2020), which in a galaxy strongly fluctu- ates, especially in its earlier evolution stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' How- ever, for our modelling, we use a Sedov-Taylor approach, which simplifies these assumptions to an almost con- stant swept-up mass mostly determined by the CCSN explosion energy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Shigeyama & Tsujimoto 1998, who employed a constant sound speed of 10km/s), this swept-up mass parameter remains constant during a run once it is chosen in the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This approach might underestimate the spectrum of the SLR densities, espe- cially at earlier galactic stages, but we chose to keep this swept-up mass parameter constant to be able to focus on identifying the dependence of SLR density on mixing caused by explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Further, our approach limits the time resolution of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We deliberately chose a time step size of 1 Myr because this allows us to sim- plify all thermo- and hydrodynamic processes into the value of the swept-up mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' One Myr after explosion, the shock front has stopped, and the ejected and swept- up material has cooled down sufficiently to be available for star formation in the following time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' With bet- ter time resolution (with shorter time steps), the ex- plosion ejecta and the swept-up material would have to be closely followed, which would require a detailed thermo- and hydro-dynamic modeling, not available in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We tested, instead, the dependence of our model on a decreased rather than increased time resolu- tion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', longer time steps) in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' RESULTS In this section, we report on the evolution of SLRs over the lifetime of the Galaxy (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1), and compare our results to deep-sea sediment abundances (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' SLRs: Surfing the wave?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 7 Model name ISM polluted Remnant geometry Standard 5 · 104 M⊙ Standard HN 2 · 105 M⊙ Standard PINBALL 5 · 104 M⊙ PINBALL model HN PINBALL 2 · 105 M⊙ PINBALL model Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Overview of the different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The left column states the model name, the middle column how much ISM is polluted by a single CCSN, and the third column how the swept-up material is distributed after the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' See Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='6 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Model name t 53Mn 1/100 t 60Fe 1/100 t 182Hf 1/100 t 244Pu 1/100 Standard 1790 254 585 432 HN 138 192 480 370 PINBALL 192 228 580 426 HN PINBALL 152 194 483 360 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Delay time (tSLR 1/100, in Myr) for each SLR between the onset of the simulation and the time when the median of the abundance of a given SLR reaches 1% of its maximum within the first 3 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Evolution of SLRs At every time step, we store and analyse the gas con- tents and the SLR abundances in every cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Figure 1 shows the statistical distribution of the abundances of the four isotopes over the lifetime of the Galaxy for the four cases described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Overall, the ISM densities of the four SLRs follow the galactic star forma- tion rate with an early rise and slow decrease, although each of them shows a different shift to later times relative to the peak of the star formation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The magnitude of these shifts (where we define tSLR 1/100 as the delay between the onset of the simulation and the time when the me- dian of the abundance of a given SLR reaches 1% of its maximum within the first 3 Gyr) is linked to the delay time between the formation of the progenitor stars and the enrichment events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We have summarized the values of tSLR 1/100 for each model in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In our simulations, CCSNe (ejecting 60Fe) have the shortest delay times, because they are produced from HMSs with the shortest life times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This leads to the 60Fe curve having the smallest shift towards later times (t 60Fe 1/100 = 254 Myr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 244Pu has the second lowest de- lay time(t 244Pu 1/100 = 432 Myr), since this SLR is ejected as soon as two short-living HMSs have died and the two resulting NSs have spiraled inwards towards their common center of mass for the coalescence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 182Hf has a longer delay time than 244Pu (t 182Hf 1/100 = 585 Myr), because IMSs stars live longer than HMSs plus our as- sumed coalescence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Since we require two IMSs in a binary system to have reached the end of their lives for a SNIa to occur, the time scale to produce the SNIa ejecta will always be longer than the lifetime of the secondary star in the binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Thus, 53Mn has the largest shift of all SLRs (t 53Mn 1/100 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='79 Gy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Because of the large variations of gas contents and SLR abundances among the cells, we present the SLR densi- ties in the cells for every time step as statistical distri- butions, where the shaded areas represent 100% (light grey), 95% (dark grey) and 68% (black) of the SLR den- sities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' For 60Fe, the 68% cell-to-cell fluctuation span two orders of magnitude at almost all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The spectrum of possible 60Fe densities is the most narrow right after the time of highest star formation (at the point of high- est number of HMS deaths) at t≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='5 Gy, and broadens with later times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The reason why many stellar deaths lead to a narrower spectrum in abundances stems from the fact that the SLR has less time to decay between enrichment events, therefore the minimum abundance value immediately before a subsequent nucleosynthesis event is closer to the maximum value of that abundance just after the nucleosynthesis event, as compared to a case when stellar deaths are more apart in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In this latter case, the SLR has much more time to decay before the subsequent injection of that SLR into the ISM, and hence decays to lower values than in the former case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' A secondary effect is that if more stars die in a given time span, there is also more production of 60Fe, which leads to a higher overall abundance of 60Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' A corresponding behavior can also be observed for the other SLRs, at the respective points in time when the highest number of stars contribute to each SLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We show the implications of the four shock remnant models introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='6 in Figure 1: Stronger explosions (HN model) lead to a larger spectrum of SLR density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This effect can be well observed when compar- ing the difference in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', the 68% band of 60Fe between the standard and the HN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In the latter, more cells are affected by a single SN explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Because of the larger radius of the remnant in the HN model, there are more cells inside the remnant, and these are cleared of their SLR content, which leads to a larger number of lower SLR density cells in every time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' For 244Pu, this also leads to a significantly lower density median, since NSMs are very rare, the additional num- ber of cells with low 244Pu content strongly affects the median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In the HN model, instead, more cells are pol- luted per time step and thus the mixing is more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Therefore, also the delay for the SLRs to approach their steady-state value is lower than in the Standard model (t 53Mn 1/100 = 138 Myr, t 60Fe 1/100 = 192 Myr, t 182Hf 1/100 = 480 Myr, 8 Wehmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='Mn-53 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='Fe-60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='Hf-182 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='Time in Gyr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='Time in Gyr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='Pu-244 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='(b) HN model ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='Fe-60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='Hf-182 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='Time in Gyr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='Pu-244 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='(c) PINBALL model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='Mn-53 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='Fe-60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='Hf-182 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='Time in Gyr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='Pu-244 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='(d) HN PINBALL model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' SLR densities in g/cm3 in the simulation volume, where the distribution at each time step represents the spatial distribution of SLR densities in the (40)3 sub-cells, for the four cases of Table 1, as indicated below each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The median is shown as a solid yellow line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The black (dark grey, light grey) shaded areas represent the 68% (95%, 100%) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' SLRs: Surfing the wave?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 9 t 244Pu 1/100 = 370 Myr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The PINBALL model case leads to ejecta distributed more homogeneously inside the supernova bubble, how- ever, the effect appears to influence the SLR density statistics only marginally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The delays of the SLRs to approach their steady-state value (t 53Mn 1/100 = 192 Myr, t 60Fe 1/100 = 228 Myr, t 182Hf 1/100 = 580 Myr, t 244Pu 1/100 = 426 Myr) are all longer than in the HN model case, but show only a slight correction when compared to the Standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Even when the fraction of reflected material is increased to a much higher value (50 %), it only affects SLR densi- ties locally in sub-cells, but not their overall abundance statistics, as described in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In the combined case with both PINBALL geometry and high explosion energy (HN PINBALL model), instead the flattening effect of the more homogeneous distribu- tion of SLRs (due to the PINBALL treatment) combined with the larger explosion bubbles (due to the HN treat- ment) becomes more prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In this model, ejecta from every nucleosynthesis site are distributed through- out the entire, larger explosion bubble, which results in more cells being polluted by an SLR per nucleosynthe- sis event than in any other model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' For the more rare sites (SNeIa and NSMs producing 53Mn and 244Pu, re- spectively), this also leads to larger fluctuations in the median between time steps (compared to the standard model), since the SLRs produced in these rare events are distributed throughout the volume much quicker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This behavior can also be observed in the delay times for the SLRs to approach the steady-state value (t 53Mn 1/100 = 152 Myr, t 60Fe 1/100 = 194 Myr, t 182Hf 1/100 = 483 Myr, t 244Pu 1/100 = 360 Myr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This means that the ISM homogenizes faster, eliminating quickly the spikes in SLR densities gener- ated by a single rare nucleosynthesis event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In other words, since this last model homogenizes the ISM so quickly, the median behaves more like a one-zone-model instead of a single cell in the simulation volume, the latter will be discussed in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Recent evolution Figure 2 shows a zoom-in on the evolution of the SLRs, closer to the current day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' To the figure, we added the ISM densities inferred from deep-sea sediment detec- tions from Wallner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2016), Korschinek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2020) and Wallner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2021), as well as the SLR density evolution in one of the sub-cells of the simulation which best fits the ISM densities derived from the deep-sea de- tections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We introduced a factor λ for a vertical shift for all derived ISM densities of every isotope of the deep-sea detections in every model for fitting, since we were more interested in fitting the shape of the detection curves, rather than the actual values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Further, we introduced a Model name λ53Mn λ60Fe λ244Pu ∆t (Myr) Standard 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='211 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='5 174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='63 HN 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='125 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='01 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='63 PINBALL 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='0439 431.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='63 HN PINBALL 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='476 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='669 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='63 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Vertical (λ) and horizontal (time) shift (∆t) factors for the vertical and the horizontal shift of the ISM densities of the four isotopes as inferred by their deep-sea detections used for our fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' factor ∆t to account for a horizontal (time) shift in the ISM densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The values for the two factors for each model can be found in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We test the possibility for a sub-cell to fit the detection data in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' First, we consider the top left panel in Figure 2 (the standard case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' If we follow the density evolution of the best-fitting cell, the effects of radioactive decay of the SLRs in that cell are visible (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', around 13330 Myr in the 60Fe evolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' However, another effect often dominates: since we assume that SN ejecta behave in a Sedov-Taylor-like expansion pattern, any stellar ex- plosion clears its neighboring cells completely of their contents as the blast wave travels through those cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Together with the gas contents, also the SLRs are car- ried away from those cells to pile up in the SN remnant shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This results in a very low or even zero gas and isotope content of these cells, which is the main cause of the major discontinuities seen in the green lines in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Further, the sudden increases in the green lines can also be explained by blast waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Again, all cells within the radius of a shock wave are emptied (or almost emptied in the PINBALL models), and their gas and isotope con- tent from before the explosion is deposited on a shell around the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' If an observed cell is located on the shell around an explosion, its gas and isotope con- tent thus strongly increases, which leads to an upward jump in isotope abundances in the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This effect of blast waves is observed in all SLRs in our models and might explain the data of Wallner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2021), who found that 60Fe and 244Pu arrive on Earth synchronously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' If in our model 60Fe suddenly arrives in an ISM cell due to a nearby stellar explosion, it is very likely that also some 244Pu arrives synchronously in that cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In the top right panel of Figure 2 (increased explosion energy), the sweep-up effect discussed above is more fre- quent as demonstrated by the fact that the lower 68% confidence band has a lower boundary), which happens because more space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', more simulation cells) is af- fected by a each CCSN, therefore, the clearing out ef- 10 Wehmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' fect occurs more frequently than in the standard case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' For the PINBALL model case (bottom left panel of Fig- ure 2), we observe the same as in the previous Section, that this model does not have a strong effect on the statistics of the SLR densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' A difference to the stan- dard model, however, can be observed here when con- sidering the evolution in a single cell (green line), which is more variable than in the standard model case (when the radioactive decay effect is subtracted from the evolu- tion of the line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This is because more cells are affected per CCSN explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This leads to all cells being af- fected more often in a given time interval, which leads to the density in each cell varying at a higher frequency as compared to the standard case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In the lower right panel of Figure 2 (HN PINBALL model), the evolution of the single cell (green line) oscil- lates even stronger in comparison to all the other mod- els, because the most cells are affected by each single CCSN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This results in any given cell being affected by SLR density changes much more often in a given time span, which results in the density curve oscillating at the highest frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In this model, the SLR density in a given cell is completely determined by external events, rather than by radioactive decay, whereas decay is the dominant effect in the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In all four models, it is possible to find cells that repro- duce the shape of the measurement data reasonably well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The underlying reason for the observation that all four isotopes show a synchronous increase in a given cell is that CCSNe are the dominant propagation mechanism for all SLRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Even if e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', an NSM (ejecting 244Pu) hap- pened far away from a given location in the ISM, CCSNe will be responsible for the ejecta propagation, as we will see in the following Sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' DISCUSSION In this section, we analyze the pollution intervals to a given parcel of the ISM expected for different SLRs (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1), and the propagation of SLRs in the ISM (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We suggest a schematic interpretation of the findings of Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Further, we test the impact of varying yields and event occurrence frequencies on our results in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Pollution intervals We have noted in the previous section that the in- creases in different SLR densities often coincide (green lines in Figure 2), although they are produced by differ- ent nucleosynthesis sites with very distinct occurrence frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' To investigate the cause of this behavior, we examine how often a given gas cell is polluted with a given SLR (or equivalently, how much time elapses between two consecutive pollution events in that cell), and compare this to the time that elapses between two consecutive positive changes in the gas mass of the cell2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' To exclude infalling gas (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1) from triggering a positive gas mass change event in a cell, we set a fidu- cial threshold for such a positive change in gas to 100 solar masses of gas or more (which is higher than the highest amount of infalling gas at all times).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We mon- itored the time between two consecutive SLR and gas mass enrichment events, δSLR (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', δ53Mn, δ60Fe, δ182Hf, or δ244Pu), and δmass, respectively, in every cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Then, we calculate the average of each of these pollution times ( < δSLR > for every SLR) in each cell over the entire lifetime of the Galaxy, and generate a histogram over the averages of all cells, which is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' All the histogram curves show a Gaussian behavior (due to the central limit theorem) and overlap with each other at large portions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This means that all the SLRs arrive at a given location at a very similar frequency, and in conjunction with every arrival of gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This is further evidence that the probability of all SLRs to arrive syn- chronously at a given location is very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The dominant propagation mechanism To further examine the cause of the synchronous ar- rival of SLRs, we explore the contributions of the dif- ferent nucleosynthesis sites towards the propagation of matter throughout our simulation volume and the age of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' For every nucleosynthesis event, we monitor the coordinates of all the cells that are affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' From this, we calculate the time that elapses between two con- secutive events that affect each given cell for each type of explosion, δsite (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', δCCSN, δNSM, and δSNIa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In other words, we examine the time that elapses between the cell being affected by the shock fronts of two dif- ferent explosions of the same type of event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We then calculate the mean of these elapsed times for each ex- ploding nucleosynthesis site, < δCCSN >, < δNSM >, and < δSNIa >, and plot these medians in Figure 4, together with < δmass > from Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' None of the < δsite > distributions is congruent with < δmass >, but < δCCSN > is the closest to < δmass >, which means that CCSNe affect the most cells, relative to the other types of explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We therefore conclude that CC- SNe are the dominant propagation mechanism of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' When considering that < δmass >≈< δSLR > from Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1, we further conclude that CCSNe are the dom- inant propagation mechanism of all SLRs in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This means that the ejecta from all the nucleosynthesis 2 We examine this for the standard model only, although the con- clusions drawn could be extended to the other models as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' SLRs: Surfing the wave?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 11 (a) Standard model (b) HN model (c) PINBALL model (d) HN PINBALL model Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Same as Figure 1, but zoomed-in to ±10 Myr around the respective ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In addition to what is shown in Figure 1, ISM densities for 53Mn, 60Fe, and 244Pu, inferred from deep-sea sediment detections are shown as red symbols with error bars (shifted by factors λ and ∆t), and the abundance evolution of the one simulation cell in each model that best fits these deep-sea detection ISM abundances is shown in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' sites included in our model travel as dynamically as the ejecta of CCSNe, even if their occurrence rate is much lower than that of CCSNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Schematic interpretation As seen in the previous sections, although each of the SLRs is produced in a different nucleosynthesis site, the density of all SLRs most likely increase synchronously in each given cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This is consistent with the observa- tion of the three SLRs detected in deep-sea sediments, 53Mn (Korschinek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020), 60Fe (Wallner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2016) and 244Pu (Wallner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Especially for the latter two, it has been shown in Wallner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2021) that they are both deposited in deep-sea sediments of the same depth, which leads to the conclusion that they arrived synchronously, even if they could have been produced in separate, independent nucleosynthesis sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This could be interpreted in the following way, for the example of 60Fe from HMS and 244Pu from NSMs (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' An NSM explodes in an area of the Galaxy that is relatively close to the Solar System, however, it is separated from it by a region of high density, which could have been cre- ated by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', a previous CCSN or a superbubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Once the NSM explodes, the shock wave of the explosion, and hence the ejecta, will stop at the high density gas region, so none of the ejecta of this explosion event reaches the Solar System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' If there is a HMS inside the region be- hind the shock wave (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', within the explosion bubble) of that NSM, that star will end its life in a CCSN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Once it explodes, it will create a second shock wave, carrying 10-6 3 5 10-7 G-u 10-8 10-9 > 10-10 10-11 10-6 510-7 L 10-9 10-9 182 Hf-1 10-11 Pu-244 10-12 10-13 10-14 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='305 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='310 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='315 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='320 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='325 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='330 time [Gyr]10-6 10-7 5 10-8 u 10-9 > 10-10 10-11 10-6 0 10-7 e 10-8 L 10-9 2 10-9 8 H 10-11 Pu-244 10-12 10-13 10-14 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='425 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='430 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='435 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='440 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='445 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='450 time [Gyr]10-7 3 5 10-9 Mn-5 10-11 10-13 10-7 e 10-8 10-9 2 Hf-182 10-10 10-12 10-11 Pu-244 10-13 10-15 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='050 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='055 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='060 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='065 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='070 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='075 time [Gyr]10-6 3 10-7 5 10-8 10-9 10-10 > 10-11 10-6 10-7 10-8 e F 10-9 2 8 10-10 10-12 Pu-244 10-12 10-13 10-14 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='215 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='220 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='225 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='230 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='235 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='240 time [Gyr]12 Wehmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='0 Myr 100 101 102 103 104 N Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Distribution of the means of the elapsed time of arrival of the different focus SLRs in all cells: < δ53Mn > (red), < δ60Fe > (yellow), < δ182Hf > (green), < δ244Pu > (magenta), and < δmass > (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Cell distribution of the means of the elapsed time between two consecutive events of the same kind: < δCCSN > (blue), < δNSM > (yellow), < δSNIa > (green), and < δmass > (magenta, same as blue color curve in Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' with it its CCSN ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This second shock wave might just have enough energy to push the high density region (where the NSM ejecta were stopped earlier) towards the Solar System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Since the NSM ejecta are still conserved in this stopped high density area which is now pushed further by the subsequent CCSN, both ejecta from NSM and CCSN are conjointly pushed towards the Solar Sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The result is a synchronous arrival of CCSN and NSM ejecta on Earth, hence 60Fe and 244Pu in Earth’s inventory increase simultaneously, as observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In other words, NSM and other ejecta could “surf the wave” of the CCSN explosion shock fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Since we have seen in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2 that all SLRs show a very sim- ilar < δSLR >, this propagation argument can also be made for the other rare nucleosynthesis site ejecta, 53Mn from SNIa, and also for the more locally deposited 182Hf from IMSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In order to determine how often this “surf- ing” effect occurs, we calculate the fraction of CCSNe that sweep up significant amounts of SLRs in their blast wave: 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='35% / 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='94% / 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='89% / 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='81% of all CCSNe carry 53Mn, 60Fe, 182Hf, and 244Pu, respectively, which means that almost all CCSNe contribute significantly to the propagation of all SLRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' To examine how this number is correlated with each nucleosynthesis site, we also consider a model where we lowered the frequency of NSMs by a factor of 100 (with PNSM = 4 · 10−4, which would correspond to a theoretical gravitational wave rate of ∼ 18 Gpc−3 yr−1, roughly four times be- low the current uncertainties of the LIGO/Virgo de- tections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In this model, the fraction of CCSNe that sweep up significant amounts of the NSM ejecta 244Pu decreases to 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='15%, while the other values remain al- most unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Effects of varying yields and frequency of the events To test the impact of different yields and event fre- quencies on the results, first, we tested a case where the mass dependency of the yields was kept, but all their values were uniformly decreased by a factor of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The result confirms that a linear change of a factor of ≈ 10 is reflected in the whole distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Second, we quanti- fied the effect of the progenitor mass-dependency of the 60Fe and 182Hf yields on the ISM evolution of the SLRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' To this aim we changed the 60Fe and 182Hf yields from our fiducial yields from the literature (as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2), to a case where we took a constant yield for each progenitor equal to the IMF-weighted average over the entire ZAMS mass range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The results are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' At the end of the simulation at time 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='475 Gyr, stellar births and deaths are almost in equilibrium, and a comparison made at this point allows us to minimize the impact of stochas- tic star formation or deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Using the constant yields, the 60Fe and 182Hf median increase by factor of ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2 and 3, respectively, at this late time of the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The overall trend however differs: in the case of 60Fe the median calculated with the constant yields is always above the fiducial case, while for 182Hf the two lines cross at a time of roughly 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='5 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The reason is that for the 182Hf ejected by IMSs, in the fiducial case the more massive IMSs eject more of this SLR, they also die earlier and therefore more 182Hf per IMS is ejected at earlier galactic stages, as compared to the model where all IMSs eject the averaged yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' At later stages, in- 104 mass> < CCSN> 103 < NSM> < SNla> 102 101 100 101 102 MyrSLRs: Surfing the wave?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 13 Solar System Bubble of higher gas density NSM progenitor Solar System NSM remnant bubble CCSN progenitor Solar System CCSN remnant bubble Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Schematic interpretation of the the result that all SLRs arrive at the Solar System conjointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Top: The Solar System and NSM progenitor separated from each other by an area of high density ISM, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', the edges of a previous SN bubble or a superbubble, which is created when multiple CCSNe share the same shock front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Middle: The NSM pro- genitor explodes and pollutes the higher density area with its ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Bottom: A CCSN explodes inside the bubble, push- ing the NSM ejecta into the Solar System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' CCSN and NSM ejecta arrive simultaneously at the Solar System location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' stead, the lower mass IMSs become more predominant, and in the case of constant yields, they eject more 182Hf than in the fiducial case resulting in a higher final 182Hf median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The upper limit of the full abundance statistics also in- creases for 182Hf, however, the increase is lower, ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2, than that of the median value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This is due to a clump- ing effect more significantly present in the case of the fiducial yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The more massive IMSs that contribute relatively more 182Hf are actually fewer than their lower- mass counterparts, therefore their ejections are less ho- mogeneously distributed than in the case where all IMSs all eject the same amount of 182Hf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' For 60Fe from HMSs, the effect is weaker because the life time of the most massive HMSs compared to the least massive HMSs is not so different than the differ- ence in life times of the most and least massive IMSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In the fiducial yields case, the most massive HMSs eject far more 60Fe than the lower mass ones, which leads to a larger fluctuation of abundances (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', larger upper limit of the full abundance distribution) in the former case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' However, the median of abundances is lower in the former case, because the majority of HMSs are (due to the IMF) at the lower mass end of the HMS mass spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This leads to a lower abundance median in this case compared to the case where all HMSs eject the same, IMF-averaged yield of 60Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' As discussed in Cˆot´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2019b) and Yag¨ue L´opez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2021), the occurrence frequency of stellar events influences how much time an SLR has to decay before a subsequent nucleosynthesis event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Since this frequency is governed by the rate of an event, this implies that also the event rates play a major role for the evolution of the SLR abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Although the rates of HMSs, IMSs, and SNIa have been determined by observations, the ob- servational error bar on the occurrence rate of NSMs as suggested by gravitational wave detections is uncertain within a factor of ≈10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We illustrate the effect of chang- ing the NSM rate on 244Pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In Figure 7, we present the evolution of 244Pu in our fidu- cial model versus a model with NSM occurrence rates reduced by a factor of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The median of 244Pu ISM densities is reduced by a factor of ≈ 20, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', it is two times lower than the case when the yields where reduced by a factor of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The reason is that a decrease in the rate results in less cells polluted by NSM ejecta per time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' A decreased yield, instead, results in the same amount of cells polluted per time step, but with a lower 244Pu density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Since the median is more sensitive to the amount of cells being polluted than the amount of 244Pu in the cells, the median value of the abundance is then lower in the former case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 14 Wehmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 0 2 4 6 8 10 12 14 Time in Gyr 10 8 10 7 10 6 Fe-60 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Effect of varied 182Hf (left) and 60Fe (right) yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The solid lines represent the abundance medians and the shaded areas the full abundance distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The fiducial case is in blue and the case with the constant IMF-averaged yields is in magenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Same as Figure 6, but showing the effect of de- creasing the occurrence frequency of NSMs by a factor of 10 (magenta), relative to the fiducial case in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' CONCLUSIONS We simulated the evolution of four SLRs from four separate nucleosynthesis sites: 53Mn from SNIa, 60Fe from CCSNe, 182Hf from IMSs, and 244Pu from NSMs, over the lifetime of our Galaxy using a 3-dimensional GCE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We conclude that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The integrated (over the entire simulation volume) median ISM density of the four SLRs follows the Galactic star formation rate over time (with an individual delay for each SLR), but their density at a given location fluctuates strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 60Fe has the lowest delay since HMSs have the shortest lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Second is 244Pu since the life- times of two HMSs plus the coalescence time scale is still lower than the lifetime of IMSs (ejecting 182Hf, second longest delay time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 53Mn has the longest delay time since it always requires the longer-lived IMS in a IMS binary to die to pro- duce that SLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Even though the SLRs are produced at different sites, their arrival at a given location in the ISM often coincides, because their propagation is dom- inantly governed by the frequency of CCSNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This explains why three different SLRs, 53Mn, 60Fe, and 244Pu, produced in three different nucle- osynthesis sites (SNIa, HMSs, and NSMs) could be detected in layers of similar depth in deep-sea sed- iments, which suggest that they arrived conjointly on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Our model can robustly reproduce these deep-sea detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Neither assuming a PINBALL model, nor that all CCSNe explode as HNe strongly influences the overall evolution of SLRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Reducing the occurrence frequency of NSMs re- duces the median abundances of 244Pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Reduced yields reduce the median abundances proportion- ally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Decreasing CCSN and IMS yields decreases the median abundances of 60Fe and 182Hf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Effects that still need to be addressed in future work are: Those deriving from a different IMF, which would alter the lifetimes of stars and thus change the distribution of the pollution intervals, and from the existence of islands of explodability of massive stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 10-!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 10-10 10-12 10-13 0 2 4 6 8 10 12 14 Time in Gyr10-9 10-10 244 Pu- 10-11 10-12 10-13 0 2 4 6 8 10 12 14 Time in GyrSLRs: Surfing the wave?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 15 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Ertl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Ebinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The presence of failed CCSNe may reduce their dominance in being the major propagation mechanism for SLRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Further, additional sources for the four SLRs should be addressed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', rare sub-types of CCSNe as a source of 244Pu (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Also, further propagation mechanisms not currently included in our model could have an effect on the propagation of SLRs: Diffusion/turbulence: Hotokezaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2015) found that the discrepancies between early Solar System abundance and more recent deep-sea de- tection of 244Pu can be explained by a model that assumes a purely diffusion/turbulence-dominated propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This model was later expanded by Beniamini & Hotokezaka (2020), who concluded that this approach could also applied to r-process elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We did not include diffusion/turbulence in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' If included, diffusion/turbulence might reduce (increase) the SLR densities in those cells with the highest (lowest) SLR abundances, which would lead to a reduced spectrum of SLR densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' However, these considerations would go beyond the scope of this work, where we wanted to highlight the “surfing” effect of SLRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We plan to work on a comparison between CCSN-dominated, and diffusion-dominated propagation for SLRs in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Superbubbles: A large bubble in the ISM can be created by multiple CCSN explosions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The effect of superbubbles is very difficult to estimate in our GCE model with lim- ited spatial and time resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The implemen- tation of superbubbles requires the implementa- tion of thermodynamics/fluid dynamics (as done in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Vasileiadis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Fujimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2018), but this would go beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Nonetheless, more detailed effects that could be expected in hydrodynamical simulations can be estimated with our set of models with vary- ing parameters, and all of our models show the surf effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Galactic outflows: Galactic outflows could po- tentially built up a reservoir of gas outside the Galaxy that may be unaffected by GCE for a pe- riod of time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', as done with Omega+ in Cˆot´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2018), or may be enriched differently in hy- drodynamical simulations (Vincenzo & Kobayashi 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The implications of this effect on SLR abundances are difficult to estimate in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Although SLR abundances (due to the absence of nucleosynthesis sites increasing the SLR abun- dances in the extragalactic reservoir) would poten- tially decrease exponentially in that extragalactic reservoir due to radioactive decay, gas being incor- porated back from the reservoir into the Galaxy might slightly enhance the abundance of SLRs in the simulation volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' However, since our main aim for this work was to showcase the “surfing” effect for SLRs, we decided to use primordial in- flow into the simulation volume as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Future further detection of live radioisotopes in the deep-sea floor might provide further constraints on the propagation mechanism of SLRs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2021a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This will also be of further interest for the GCE of r-process elements, since it is yet unclear whether the behaviour of different classes of elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', iron group and r-process elements) is coupled or not (Beniamini & Hotokezaka 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' A direct compar- ison between different or combined propagation mecha- nisms would therefore help to further confirm or rule out whether the propagation of SLRs and r-process elements is CCSN- or diffusion-dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Our study will also be extended towards a full cosmological zoom-in simulation of a galaxy (Kobayashi & Nakasato 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Vincenzo & Kobayashi 2020), which will allow for additional sub- galactic-scale mixing effects and mechanisms to be ad- dressed (Wehmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Additionally, our models can be applied to study the abundances of SLRs at the time of the formation of the Solar System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors thank Anton Wallner for providing the deep-sea measurement data relevant for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We further thank Adrienne Ertel and Jesse Miller for con- structive discussion with regard to the PINBALL model at the 2019 JINA-CEE Frontiers meeting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This work is supported by the ERC Consolidator Grant (Hun- gary) funding scheme (Project RADIOSTAR, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 724560).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We also thank the COST actions “ChETEC” (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 16117) and “ChETEC-INFRA” (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 101008324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' MKP received funding from the Euro- pean Union’s Horizon 2020 research and innovation pro- gramme under the Marie Sklodowska-Curie grant agree- ment No 753276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' BC and BW acknowledge support from the National Science Foundation (NSF, USA) un- der Grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' PHY-1430152 (JINA Center for the Evo- lution of the Elements) and Grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' OISE-1927130 (IReNA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' CK acknowledges funding from the UK Sci- ence and Technology Facility Council (STFC) through grant ST/R000905/1, & ST/V000632/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The work of AYL was supported by the US Department of En- 16 Wehmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' ergy through the Los Alamos National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Los Alamos National Laboratory is operated by Triad Na- tional Security, LLC, for the National Nuclear Security Administration of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Department of Energy (Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 89233218CNA000001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We thank an anonymous referee for valuable comments that helped improving this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' REFERENCES Abbott, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Abbott, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Abraham, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2021, ApJL, 913, L7, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3847/2041-8213/abe949 Audouze, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Tinsley, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1976, ARA&A, 14, 43, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1146/annurev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='aa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='090176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='000355 Bauswein, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Goriely, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Janka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2013, ApJ, 773, 78, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1088/0004-637X/773/1/78 Belczynski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Repetto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Holz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2016, ApJ, 819, 108, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3847/0004-637X/819/2/108 Beniamini, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Hotokezaka, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020, MNRAS, 496, 1891, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1093/mnras/staa1690 Beniamini, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Piran, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2016, MNRAS, 456, 4089, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1093/mnras/stv2903 Bisterzo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Travaglio, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Gallino, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Wiescher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & K¨appeler, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2014, ApJ, 787, 10, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1088/0004-637X/787/1/10 Bravo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Dom´ınguez, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Badenes, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Piersanti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Straniero, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2010, ApJL, 711, L66, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1088/2041-8205/711/2/L66 Chamandy, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Shukurov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020, Galaxies, 8, 56, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3390/galaxies8030056 Charbonnel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Meynet, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Maeder, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Schaller, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Schaerer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1993, A&AS, 101, 415 Chruslinska, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Nelemans, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Belczynski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2019, MNRAS, 482, 5012, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1093/mnras/sty3087 Clayton, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1984, ApJ, 285, 411, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1086/162518 Cˆot´e, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Belczynski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Fryer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2017, ApJ, 836, 230, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3847/1538-4357/aa5c8d Cˆot´e, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Lugaro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Reifarth, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2019a, ApJ, 878, 156, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3847/1538-4357/ab21d1 Cˆot´e, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Silvia, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', O’Shea, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Smith, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Wise, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2018, ApJ, 859, 67, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3847/1538-4357/aabe8f Cˆot´e, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Yag¨ue, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Vil´agos, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Lugaro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2019b, ApJ, 887, 213, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3847/1538-4357/ab5a88 Cˆot´e, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Eichler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Yag¨ue L´opez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2021, Science, 371, 945, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='aba1111 Curtis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Ebinger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Fr¨ohlich, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2019, ApJ, 870, 2, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3847/1538-4357/aae7d2 Dauphas, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Chaussidon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2011, Annual Review of Earth and Planetary Sciences, 39, 351, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1146/annurev-earth-040610-133428 Diehl, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Lang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Martin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2010, A&A, 522, A51, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1051/0004-6361/201014302 Doherty, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Gil-Pons, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Siess, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Lattanzio, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Lau, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2015, MNRAS, 446, 2599, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1093/mnras/stu2180 Dominik, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Belczynski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Fryer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2012, ApJ, 759, 52, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1088/0004-637X/759/1/52 Duchˆene, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Kraus, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2013, ARA&A, 51, 269, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1146/annurev-astro-081710-102602 Ebinger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Curtis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Fr¨ohlich, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2019, ApJ, 870, 1, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3847/1538-4357/aae7c9 Eichler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Arcones, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Kelic, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2015, ApJ, 808, 30, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1088/0004-637X/808/1/30 Ertl, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Janka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Woosley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Sukhbold, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Ugliano, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2016, ApJ, 818, 124, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3847/0004-637X/818/2/124 Fischer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Wu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Wehmeyer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020, ApJ, 894, 9, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3847/1538-4357/ab86b0 Freiburghaus, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Rosswog, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Thielemann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1999, ApJL, 525, L121, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1086/312343 Fry, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Fields, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Ellis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020, ApJ, 894, 109, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3847/1538-4357/ab86bf Fujimoto, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Krumholz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Tachibana, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2018, MNRAS, 480, 4025, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1093/mnras/sty2132 Gibson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Fenner, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Renda, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Kawata, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='-c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2003, PASA, 20, 401, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1071/AS03052 Greggio, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2005, A&A, 441, 1055, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1051/0004-6361:20052926 Hotokezaka, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Piran, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Paul, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2015, Nature Physics, 11, 1042, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1038/nphys3574 Huss, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Meyer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Srinivasan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Goswami, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Sahijpal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2009, GeoCoA, 73, 4922, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='gca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='039 Iwamoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Brachwitz, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Nomoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1999, ApJS, 125, 439, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1086/313278 Just, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Bauswein, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Ardevol Pulpillo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Goriely, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Janka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2015, MNRAS, 448, 541, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1093/mnras/stv009 K¨appeler, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Gallino, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Bisterzo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Aoki, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2011, Reviews of Modern Physics, 83, 157, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1103/RevModPhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='157 Kennicutt, Robert C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1998, ApJ, 498, 541, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1086/305588 Kobayashi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Karakas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Lugaro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020a, ApJ, 900, 179, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3847/1538-4357/abae65 SLRs: Surfing the wave?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 17 Kobayashi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Karakas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Umeda, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2011, MNRAS, 414, 3231, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1365-2966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='18621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='x Kobayashi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Leung, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Nomoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020b, ApJ, 895, 138, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3847/1538-4357/ab8e44 Kobayashi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Nakasato, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2011, ApJ, 729, 16, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1088/0004-637X/729/1/16 Kobayashi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Nomoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Hachisu, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2015, ApJL, 804, L24, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1088/2041-8205/804/1/L24 Kobayashi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Umeda, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Nomoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Tominaga, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Ohkubo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2006, ApJ, 653, 1145, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1086/508914 Korobkin, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Rosswog, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Arcones, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Winteler, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2012, MNRAS, 426, 1940, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1365-2966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='21859.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='x Korschinek, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Faestermann, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Poutivtsev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020, PhRvL, 125, 031101, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='031101 Krause, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Burkert, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Diehl, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2018, A&A, 619, A120, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1051/0004-6361/201732416 Larson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1991, in Astronomical Society of the Pacific Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 20, Frontiers of Stellar Evolution, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Lambert, 571 Limongi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Chieffi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2006, ApJ, 647, 483, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1086/505164 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2018, ApJS, 237, 13, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3847/1538-4365/aacb24 Lugaro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Ott, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Kereszturi, ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2018, Progress in Particle and Nuclear Physics, 102, 1, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='ppnp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='002 Lugaro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Pignatari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Ott, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2016, Proceedings of the National Academy of Science, 113, 907, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1519344113 Lugaro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Heger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Osrin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2014, Science, 345, 650, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1253338 Matteucci, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Greggio, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1986, A&A, 154, 279 Meyer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Clayton, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2000, SSRv, 92, 133, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1023/A:1005282825778 Mor, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Livne, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Piran, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2023, MNRAS, 518, 623, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1093/mnras/stac2775 M¨osta, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Roberts, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Halevi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2018, ApJ, 864, 171, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3847/1538-4357/aad6ec Nishimura, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Sawai, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Takiwaki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Yamada, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Thielemann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2017, ApJL, 836, L21, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3847/2041-8213/aa5dee Nomoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Hashimoto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Tsujimoto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1997, NuPhA, 616, 79, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1016/S0375-9474(97)00076-6 Nomoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Kobayashi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Tominaga, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2013, ARA&A, 51, 457, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1146/annurev-astro-082812-140956 Nomoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Maeda, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Mazzali, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2004, in Astrophysics and Space Science Library, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 302, Astrophysics and Space Science Library, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Fryer, 277–325, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1007/978-0-306-48599-2 10 O’Dell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', McCullough, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Meixner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2004, AJ, 128, 2339, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1086/424621 Panov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Korneev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Thielemann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2008, Astronomy Letters, 34, 189, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1007/s11443-008-3006-1 Perets, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Beniamini, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2021, MNRAS, 503, 5997, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1093/mnras/stab794 Poelarends, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Herwig, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Langer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Heger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2008, ApJ, 675, 614, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1086/520872 Prantzos, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Abia, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Cristallo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Limongi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Chieffi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020, MNRAS, 491, 1832, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1093/mnras/stz3154 Reed, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Balick, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Hajian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1999, AJ, 118, 2430, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1086/301091 Reichert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Obergaulinger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Eichler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Aloy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Arcones, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2021, MNRAS, 501, 5733, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1093/mnras/stab029 Ritter, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Herwig, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Jones, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2018, MNRAS, 480, 538, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1093/mnras/sty1729 Rosswog, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2013, Nature, 500, 535, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1038/500535a Rosswog, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Korobkin, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Arcones, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Thielemann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Piran, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2014, MNRAS, 439, 744, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1093/mnras/stt2502 Rosswog, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Piran, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Nakar, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2013, MNRAS, 430, 2585, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1093/mnras/sts708 Ryan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Norris, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Beers, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1996, ApJ, 471, 254, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1086/177967 Salpeter, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1955, ApJ, 121, 161, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1086/145971 Sana, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', de Mink, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', de Koter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2012, Science, 337, 444, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1223344 Schaerer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Charbonnel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Meynet, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Maeder, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Schaller, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1993a, A&AS, 102, 339 Schaerer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Meynet, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Maeder, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Schaller, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1993b, A&AS, 98, 523 Schaller, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Schaerer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Meynet, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Maeder, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1992, A&AS, 96, 269 Schmidt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1959, ApJ, 129, 243, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1086/146614 Seitenzahl, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Ciaraldi-Schoolmann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', R¨opke, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2013, MNRAS, 429, 1156, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1093/mnras/sts402 Shigeyama, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Tsujimoto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1998, ApJL, 507, L135, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1086/311699 Sukhbold, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Ertl, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Woosley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Brown, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Janka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2016, ApJ, 821, 38, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3847/0004-637X/821/1/38 Tauris, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Langer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Podsiadlowski, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2015, MNRAS, 451, 2123, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1093/mnras/stv990 18 Wehmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Tauris, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Kramer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Freire, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2017, ApJ, 846, 170, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3847/1538-4357/aa7e89 Thielemann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Brachwitz, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', H¨oflich, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Martinez-Pinedo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Nomoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2004, NewAR, 48, 605, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='newar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='038 Thielemann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Nomoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Hashimoto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1996, ApJ, 460, 408, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1086/176980 Timmes, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Brown, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Truran, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2003, ApJL, 590, L83, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1086/376721 Travaglio, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Hillebrandt, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Reinecke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2005, A&A, 443, 1007, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1051/0004-6361:20052883 Trueman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Cˆot´e, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Yag¨ue L´opez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2022, ApJ, 924, 10, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3847/1538-4357/ac31b0 van de Voort, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Pakmor, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Bieri, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Grand, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2022, MNRAS, 512, 5258, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1093/mnras/stac710 Vasileiadis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Nordlund, ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Bizzarro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2013, ApJL, 769, L8, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1088/2041-8205/769/1/L8 Vassh, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Vogt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Surman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2019, Journal of Physics G Nuclear Physics, 46, 065202, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1088/1361-6471/ab0bea Vincenzo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Kobayashi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2020, MNRAS, 496, 80, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1093/mnras/staa1451 Wallner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Faestermann, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Feige, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2015, Nature Communications, 6, 5956, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1038/ncomms6956 Wallner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Feige, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Kinoshita, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2016, Nature, 532, 69, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1038/nature17196 Wallner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Froehlich, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Hotchkis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2021, Science, 372, 742, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='aax3972 Wanajo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2013, ApJL, 770, L22, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1088/2041-8205/770/2/L22 Wanajo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Sekiguchi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Nishimura, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2014, ApJL, 789, L39, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1088/2041-8205/789/2/L39 Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Clark, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Ellis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2021a, in APS Meeting Abstracts, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2021, APS Division of Nuclear Physics Meeting Abstracts, MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='008 Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Clark, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Ellis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2021b, arXiv e-prints, arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='09607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='org/abs/2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='09607 Wehmeyer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Fr¨ohlich, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Cˆot´e, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Pignatari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Thielemann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2019, MNRAS, 487, 1745, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1093/mnras/stz1310 Wehmeyer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Pignatari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Thielemann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2015, MNRAS, 452, 1970, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1093/mnras/stv1352 Winteler, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', K¨appeli, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Perego, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2012, ApJL, 750, L22, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1088/2041-8205/750/1/L22 Woosley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Weaver, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 1995, ApJS, 101, 181, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1086/192237 Yag¨ue L´opez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', Cˆot´e, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', & Lugaro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 2021, ApJ, 915, 128, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3847/1538-4357/ac02bf SLRs: Surfing the wave?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 19 APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' TESTING THE RESOLUTION DEPENDENCE The spatial resolution of our model is chosen to avoid a mass bias for newly born stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In order to sample the entire IMF in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1 M⊙ ≤ M < 50 M⊙, a star-forming sub-cell has to contain at least 50 M⊙ of gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This intrinsically limits the spatial resolution of the model since a sufficiently large number of sub-cells fulfilling this mass criterion has to be found at every time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' For this reason, we have chosen a spatial resolution of (50pc)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' If the resolution was increased further, it would become more difficult to find a sufficiently large number of sub-cells to fulfil the mass criterion every time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We can still test the spatial resolution dependence by decreasing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' To do this, we set up two models where we decreased the sub-cell resolution to (80pc)3 and (125pc)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The re- sults in Figure 8 show that the abundances have less variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This is not surprising, as the lower the reso- lution, the more the results will converge toward those of a one-zone model, which would show a line for all abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Since the number of very low abundance sub-cells is strongly reduced in a model with lower res- olution, the lower 100% statistics boundary converges faster than the upper 100% statistics boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This also slightly increases the median in the abundances, since it is strongly dependent on the number of sub-cells featuring a given abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This effect can be observed prominently in the 244Pu evolution in the right panel of Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' For the time resolution, the time step of 1 Myr was chosen because the code cannot handle effects emerg- ing from, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', hydrodynamics and cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' After 1 Myr, the gas ejected from a SN explosion has completely halted and sufficiently cooled down, so it can be used to form the next generation of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Choosing time steps smaller than 1 Myr would require to model precisely the trajectory of a SN blast wave in the ISM within a hydro- dynamic framework, which is not currently implemented in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' As we have done above for the spatial res- olution, we tested also the time-resolution dependence of the results by lowering it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We set up a model where we used 4 Myr time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The results can be found in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In general, the upper 100% statistics for all isotopes behave in a similar manner as in the standard case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' However, the 95% and 68% statistics are signif- icantly affected by the lowering of the time resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' More nucleosynthesis sites explode per time step in the lower resolution model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This means that more cells are affected during each time step, which effectively leads to a stronger mixing of the entire volume, and less cells re- taining lower abundance values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In other words, the like- lihood of each cell to be affected by nucleosynthesis and mixing events at every time step is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This leads to the shrinking of the spectrum of the statistics, while the 100% statistics seem to change only marginally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' We caution that the tests presented here would need to be calibrated to some galactic properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', age- metallicity-relation, GCE of α-elements), as done for the standard model presented in the main text, before they can be compared to the actual ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' INCREASING THE AMOUNT OF REFLECTED MATERIAL The Sedov-Taylor approach assumes that all material swept-up by a SN explosion is deposited solely on a bub- ble shell surrounding the SN explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The PINBALL model also explored in this work is motivated by the findings of Fry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' When a magnetic field is present in the ISM around the SN blast wave, some ma- terial inside the outward SN shock front can be deflected in the backward direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' If that material reaches the opposite shock front, it might experience another change of direction caused by the magnetic field, causing the material to move through the inside of the SN shock front, like a pinball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Fry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2020) also conclude that the fraction of reflected material in a SN blast wave is dependent on the magnetic field strength and the size of the magnetic grains that condense behind the SN blast wave, and is difficult to constrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' To investigate the potential impact of such pinball remnant geometry, we have introduced a PINBALL model in this work, by as- suming that 1% of the material swept-up by the SN blast wave remains inside the remnant bubble, and not on the shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This number is motivated by Fry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (2020) and we found that this choice does not strongly affect the evolution of SLRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Here, we test whether a much larger fraction (50 %) of deflected material would more strongly alter the abundance evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The re- sults in Figure 10 show that even such a high fraction of deflected material would not strongly affect the over- all evolution of the SLRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Therefore, this choice might affect SLR densities only locally in cells, but not their overall abundance statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This is in contrast to the HN model, which increases the overall size of the rem- nant bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' However, we caution that such high frac- tions of deflected material would affect the ISM density distribution, which in turn would alter the star forma- 20 Wehmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 10 11 10 7 Mn-53 10 8 10 6 Fe-60 10 11 10 8 Hf-182 0 2 4 6 8 10 12 14 Time in Gyr 10 14 10 12 Pu-244 10 11 10 7 Mn-53 10 8 10 6 Fe-60 10 11 10 8 Hf-182 0 2 4 6 8 10 12 14 Time in Gyr 10 14 10 12 Pu-244 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Same as Figure 1, but with lower sub-cell resolution of an edge length of 80 pc (left panel), and 125 pc (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 10 11 10 7 Mn-53 10 8 10 6 Fe-60 10 11 10 8 Hf-182 0 2 4 6 8 10 12 14 Time in Gyr 10 14 10 12 Pu-244 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Same as Figure 1, but with time resolution of 4 Myr per time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' tion history due to the exponent of the Schmidt law (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='1) used in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This might yield results which are not congruent with Milky Way Galaxy prop- erties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', age-metallicity-relation, GCE of α-elements and so forth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 10 11 10 7 Mn-53 10 8 10 6 Fe-60 10 11 10 8 Hf-182 0 2 4 6 8 10 12 14 Time in Gyr 10 14 10 12 Pu-244 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Same as Figure 1, but with 50 % deflected material instead of 1 % as done in the PINBALL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' ROBUSTNESS OF THE DEEP-SEA DETECTION FIT To examine the robustness of our one sub-cell fitting approach applied in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='2, we tested how much the mean squared distance of the logarithm of the abun- dances changes if the deep-sea detection data of the SLRs had a different time (x-axis) or abundance (y-axis) spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' For the Standard model (with time and spatial resolution as in the main text), the value of this mean SLRs: Surfing the wave?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 21 squared distance for the actual deep-sea measurements is S = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='54, better than 83% of fits3, with S given by S = 1 N � � ln R(ti) − ln Yi ln(Yi + Ei) − ln Yi �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (C1) where R(ti) is the abundance of the run at time ti, and Yi and Ei the deep sea measurement and uncertainty, respectively, at time ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' If the time spacing between the individual deep-sea de- tection data points was five times larger, it is easier to find a sub-cell that can match the detection data (see top left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 11 for an example, with S = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='14, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', better than 85% of fits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This is because the sub-cells have much more options for the individual evolution be- tween data points by either radioactive decay or the surf effect discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' In other words, sub-cells have the opportunity to undergo the surf effect multi- ple times between two deep-sea detection data points when the spacing is larger, which increases the possible spectrum of SLR densities that sub-cells can exhibit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' If the time spacing between detection data was smaller, the situation would be the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Sub-cells have less possibilities for individual evolution (via radioactive de- cay or the surf effect), and thus the spectrum of possi- ble abundances is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This makes it harder to find a sub-cell that fits the observations even if decreasing the time spacing between detection data only by a factor of two (see top right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 11 for an example, with S = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='23, better than 82% of fits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' For the abundance spacing, we find a similar result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' If the spacing between detected abundances was larger, we would have to find sub-cells that undergo stronger den- sity fluctuations than those fitting the actual detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' This decreases the number of sub-cells that provide a fit (see lower left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 11 for an example, with S = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='48, better than 81% of fits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Instead, the more we decrease the spacing between the detected abun- dances, the closer to a straight line the requirement for a single sub-cell to fit these detection is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Interestingly, especially after the right (latest) peak in the modified 60Fe data, these modified data points appear to scatter around the natural decay of that SLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Since this area of the deep-sea detection data features many points, the weighting of the fit leads to preferring “undisturbed” sub-cells (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=', sub-cells that only experience radioactive decay instead of the surfing effect), which makes it eas- 3 Determined from assuming the abundances follow a normal dis- tribution with σ equal to the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Sampling the abun- dances to obtain the resulting distribution of S allows us to com- pare a given S with all possible fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' The goal is to provide a reference for the goodness of fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' ier to find sub-cells to fit these hypothetical detection data (see bottom right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 11 for an example, with S = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='89, better than 86% of fits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 22 Wehmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' (a) Five times larger time spacing (b) Two times smaller time spacing (c) Ten times larger abundance spacing (d) Ten times smaller abundance spacing Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' Same as Figure 2, with hypothetical different time and abundance spacing for the deep-sea detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 10 3 10 10 10 LO 10 11 10 10- 10 10- 10-11 10-12 10-13 10-14 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='03 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='04 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='05 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='06 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='07 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='08 time「Gvri10 10 10 10 10 10 10-7 10 10 10-9 10-10 10-11 10-12 10-13 10-14 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='020 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='022 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='024 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='026 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='028 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='030 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='032 time「Gyri10 3 10 10 10 10 10 10-7 ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 10- 10 ++?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content=' 10-9 10-11 10-12 10-13 10-14 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='0150 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='0175 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='020013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='022513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='0250 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='0275 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='0300 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='0325 time「Gyr]10 10 10 10 10 10 11 10- 6 10-7 10 10- 10-10 10-11 10-12 10-13 10-14 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='0425 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='0450 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='047513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='050013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='0525 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE3T4oBgHgl3EQfkApM/content/2301.04593v1.pdf'} +page_content='0550 13.' metadata={'source': 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